ARITMÉTICA MODULAR, CÓDIGOS ELEMENTARES E CRIPTOGRAFIA

Livre

0
3
111
2 years ago
Preview
Full text
(1)❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆ ❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r✱ ❈ó❞✐❣♦s ❊❧❡♠❡♥t❛r❡s ❡ ❈r✐♣t♦❣r❛✜❛ ❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❇❛rr❡t♦ ❆❣♦st♦ ❞❡ ✷✵✶✹ ❙ã♦ ❈r✐stó✈ã♦✲❙❊

(2)

(3) ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆ ❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❇❛rr❡t♦ ❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r✱ ❈ó❞✐❣♦s ❊❧❡♠❡♥t❛r❡s ❡ ❈r✐♣t♦❣r❛✜❛ ❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ✭P❘❖❋▼❆❚✮✳ ❖❘■❊◆❚❆❉❖❘✿ ❈❛r❞♦s♦ Pr♦❢✳ ❉r✳ ❏✳ ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à ✈❡rsã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡❢❡♥❞✐❞❛ ♣❡❧❛ ❛❧✉♥❛ ❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❇❛rr❡t♦✱ ♦r✐❡♥t❛❞❛ ♣❡❧♦ Pr♦❢✳ ❉r✳ ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦✳ ❆❣♦st♦ ❞❡ ✷✵✶✹ ❙ã♦ ❈r✐stó✈ã♦✲❙❊

(4) ❋■❈❍❆ ❈❆❚❆▲❖●❘➪❋■❈❆ ❊▲❆❇❖❘❆❉❆ P❊▲❆ ❇■❇▲■❖❚❊❈❆ ❈❊◆❚❘❆▲ ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊ ❇✷✼✸❛ ❇❛rr❡t♦✱ ❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❆r✐t♠ét✐❝❛ ♠♦❞✉❧❛r✱ ❝ó❞✐❣♦s ❡❧❡♠❡♥t❛r❡s ❡ ❝r✐♣t♦❣r❛✜❛ ✴ ❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❇❛rr❡t♦❀ ♦r✐❡♥t❛❞♦r ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦ ✕ ❙ã♦ ❈r✐stó✈ã♦✱ ✷✵✶✹✳ ✶✶✶ ❢✳ ✿ ✐❧✳ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❡r❣✐♣❡✱ ✷✵✶✹✳ ❖ ✶✳ ▼❛t❡♠át✐❝❛ ✲ ❊st✉❞♦ ❡ ❡♥s✐♥♦✳ ✷✳ ❆r✐t♠ét✐❝❛✳ ✸✳ ❈r✐♣t♦✲ ❣r❛✜❛✳ ✹✳ ❈ó❞✐❣♦ ❞❡ ❜❛rr❛s✳ ■✳ ❈❛r❞♦s♦✱ ❏♦sé ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛✱ ♦r✐❡♥t✳ ■■✳ ❚ít✉❧♦✳ ❈❉❯✿ ✺✶✶✳✶

(5)

(6) ❉❡❞✐❝♦ ❡ ❛ ❡st❡ ♠✐♥❤❛ ✐♥❝❡♥t✐✈♦✱ tr❛❜❛❧❤♦ ❢❛♠í❧✐❛ ❢♦rç❛✱ à ❉❡✉s ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦✱ ❛♠✐③❛❞❡✱ ♣❛❝✐ê♥❝✐❛ ❡ ❛♠♦r✳ ❊ss❡ s♦♥❤♦ só ❢♦✐ ♣♦ssí✈❡❧ ♣♦rq✉❡ t✐♥❤❛ ✈♦❝ês ❛♦ ♠❡✉ ❧❛❞♦✳ ✈

(7) ❆❣r❛❞❡❝✐♠❡♥t♦s ■♥✐❝✐♦ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣❡❧♦ ❞♦♠ ❞❛ ✈✐❞❛ ❡ ♣♦r ♠❡ ♣r❡s❡♥t❡❛r ❝♦♠ ♣❡ss♦❛s ❡s♣❡❝✐❛✐s✱ s❡♠ ♦s ✐♥❝❡♥t✐✈♦s ❡ ♦ ❛♣♦✐♦ ❞❡st❛s ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦✳ ❆ ♠✐♥❤❛ ♠ã❡✱ ❊❞♥❛✱ ❛ ♠❛✐s ❣❡♥❡r♦s❛ ❞❡ t♦❞❛s ❛s ♠ã❡s✳ ❆ s❡♥❤♦r❛ é ♠❡✉ ❡①❡♠♣❧♦ ❞❡ ✈✐❞❛✳ ❖❜r✐❣❛❞❛ ♣♦r s❡♠♣r❡ ❛❝r❡❞✐t❛r ❡♠ ♠✐♠✳ ❚❡ ❛♠♦ ♠✉✐t♦✦ ❆♦ ♠❡✉ ♣❛✐✱ ❋❡r❞✐♥❛♥❞♦✱ ♦ ♠❛✐s ❜❛♥❞♦s♦ ❡ sá❜✐♦ ❞❡ t♦❞♦s ♦s ♣❛✐s✳ ❖❜r✐❣❛❞❛ ♣♦r s❡ ❢❛③❡r s❡♠♣r❡ ♣r❡s❡♥t❡ ❡♠ ♠✐♥❤❛ ✈✐❞❛✱ ♠❡ ✐♥❝❡♥t✐✈❛♥❞♦ ❡ ❛❝r❡❞✐t❛♥❞♦ ♥♦ ♠❡✉ ♣♦t❡♥❝✐❛❧✳ ❖ s❡♥❤♦r é ♦ ♠❡✉ ❤❡ró✐✳ ❚❡ ❛♠♦ ♠✉✐t♦✦ ❆♦ ♠❡✉ q✉❡r✐❞♦ ❡s♣♦s♦✱ ❆é③✐♦✱ ♣♦r ❢❛③❡r ❞♦ ♠❡✉ s♦♥❤♦✱ ♥♦ss♦ s♦♥❤♦✳ P♦r s❡♠♣r❡ ❡stá ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❛♣♦✐❛♥❞♦ ❡ ♠❡ ❢❛③❡♥❞♦ ❛❝r❡❞✐t❛r q✉❡ s♦✉ ❝❛♣❛③ ❞❡ ✐r ❛❧é♠ ❞♦ q✉❡ ✐♠❛❣✐♥♦✳ ❖❜r✐❣❛❞❛ ♣♦r s❡r ♣❛❝✐❡♥t❡✱ ❝♦♠♣r❡❡♥s✐✈♦✱ ❛♠♦r♦s♦ ❡ ❛♠✐❣♦✳ ❱♦❝ê ♠❡ t♦r♥❛ ❝♦♠♣❧❡t❛✳ ❚❡ ❛♠♦ ❞❡♠❛✐s✦✦✦ ❆ ❆❧❡①❛♥❞r❡✱ ♠❡✉ ♠❛✐♦r ♣r❡s❡♥t❡✳ ❆ r❛③ã♦ ♣♦r q✉❡♠ ✈✐✈♦✳ ▼❛♠ã❡ t❡ ❛♠❛ ♠✉✐t♦✳ ❆♦s ♠❡✉s ✐r♠ã♦s✱ ❘❡❣✐♥❛❧❞♦ ❡ ❘❛❢❛❡❧❛✱ ♣❡❧♦s ✐♥❝❡♥t✐✈♦s ❡ ❝♦♥✜❛♥ç❛✳ ❱♦❝ês sã♦ ❢✉♥❞❛♠❡♥t❛✐s ♥❛ ♠✐♥❤❛ ✈✐❞❛✳ ❆♠♦ ✈♦❝ês✦ ❆♦s ♠❡✉s ❛✈ós✱ t✐♦s✱ s♦❜r✐♥❤♦s✱ ❝✉♥❤❛❞♦s✱ s♦❣r♦s ❡ ♣r✐♠♦s ♣♦r t♦❞♦ ❛♣♦✐♦ ❡ ♣♦r ✈✐❜r❛r❡♠ s❡♠♣r❡ ❝♦♠✐❣♦✳ ❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❆♥❞❡rs♦♥✱ ♣♦r ❛❝r❡❞✐t❛r ♥❛ ♠✐♥❤❛ ❝❛♣❛❝✐❞❛❞❡ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❞✐s♣♦♥í✈❡❧ ❛ ❛❥✉❞❛r✳ ❱♦❝ê ❢♦✐ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ▼❡✉ ♠✉✐t♦ ♦❜r✐❣❛❞❛✦ ❊♥✜♠✱ ❛ t♦❞♦s q✉❡ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛ ❝♦♥tr✐❜✉✐r❛♠ ❝♦♠ ♠❛✐s ❡ss❛ ❡t❛♣❛ ❝♦♥❝❧✉✐❞❛ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳ ✈✐

(8) ❘❡s✉♠♦ ❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ tr❛t❛r ❞❡ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❞♦s ✐♥t❡✐r♦s ❡ ❡✈✐❞❡♥❝✐❛r ❛❧❣✉♥s t✐♣♦s ❞❡ ❝ó❞✐❣♦s ❡❧❡♠❡♥t❛r❡s✱ ❛ ❡①❡♠♣❧♦ ❞♦s ❈ó❞✐❣♦s ❞❡ ❈és❛r✱ ❆✜♠✱ ❞❡ ❱✐❣❡♥èr❡✱ ❞❡ ❍✐❧❧✱ ❘❙❆✱ ❞❡ ❘❛❜✐♥✱ ▼❍ ❡ ❊❧●❛♠❛❧✱ ❡①✐st❡♥t❡s ♥❛ ❝r✐♣t♦❣r❛✜❛✱ r❡ss❛❧t❛♥❞♦ ❛ ♠❛t❡♠át✐❝❛ q✉❡ ❡①✐st❡ ♣♦r trás ❞♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ ❝❛❞❛ ✉♠ ❞❡❧❡s✳ ❊st✉❞❛♠♦s ❝♦♥❝❡✐t♦s ❞❡ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❡ ♦s ❛♣❧✐❝❛♠♦s ❛♦ ❡st✉❞♦ ❞❡ ♠❛tr✐③❡s ❡ ❞❡t❡r♠✐♥❛♥t❡s q✉❡ s❡ ❢❛③❡♠ ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ss❡s ❝ó❞✐❣♦s ❡ ♣❛r❛ ❛ ❡✈♦❧✉çã♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✳ ❆♣r❡s❡♥t❛♠♦s ❛✐♥❞❛ ❛❧❣✉♥s ❝ó❞✐❣♦s ❡♥❝♦♥tr❛❞♦s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✱ ❜✉s❝❛♥❞♦ ❡st✐♠✉❧❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦ ❧❡✐t♦r ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦s ❝ó❞✐❣♦s✳ P♦r ✜♠✱ ❛ tít✉❧♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❝♦♠♣❧❡♠❡♥t❛r✱ ❡①♣♦♠♦s ✉♠ ❜r❡✈❡ ❛♣❛♥❤❛❞♦ ❤✐stór✐❝♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✳ P❛❧❛✈r❛s ❈❤❛✈❡s✿ ❈r✐♣t♦❣r❛✜❛✱ ❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r✱ ❈ó❞✐❣♦ ❞❡ ❈és❛r✱ ❈ó❞✐❣♦ ❆✜♠✱ ❈ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡✱ ❈ó❞✐❣♦ ❍✐❧❧✱ ❈ó❞✐❣♦ ❘❙❆✱ ❈ó❞✐❣♦ ❞❡ ❘❛❜✐♥✱ ❈ó❞✐❣♦ ▼❍✱ ❈ó❞✐❣♦ ❊❧●❛♠❛❧✳ ✈✐✐

(9) ❆❜str❛❝t ❚❤❡ ♠❛✐♥ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ tr❡❛t t❤❡ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ♦❢ ✇❤♦❧❡ ♥✉♠❜❡rs✱ ❛♥❞ s❤♦✇ ❡✈✐❞❡♥❝❡ ♦❢ s♦♠❡ t②♣❡s ♦❢ ❡❧❡♠❡♥t❛r② ❝♦❞❡ s✉❝❤ ❛s ❈❡s❛r✬s✱ ❆✜♠✱ ♦❢ ❱✐❣❡♥❡r❡✬s✱ ❍✐❧❧✬s✱ ❘❙❆✱ ❘❛❜✐♥✬s✱ ▼❍ ❛♥❞ ❊❧●❛♠❛❧✱ t❤♦s❡ ❢♦✉♥❞ ✐♥ ❝r②♣t♦❣r❛♣❤②✱ ❤✐❣❤❧✐❣❤t✐♥❣ t❤❡ ♠❛t❤❡♠❛t✐❝s ✇❤✐❝❤ ❡①✐sts ❜❡❤✐♥❞ t❤❡ ❢✉♥❝t✐♦♥ ♦❢ ❡❛❝❤ ♦❢ t❤❡♠✳ ❲❡ ❤❛✈❡ st✉❞✐❡❞ t❤❡ ❝♦♥❝❡♣ts ♦❢ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝ ❛♥❞ ❛♣♣❧✐❡❞ t❤❡♠ t♦ t❤❡ st✉❞② ♦❢ ♠❛tr✐❝❡s ❛♥❞ ❞❡t❡r♠✐♥❛♥ts t❤❛t ❛r❡ ♥❡❝❡ss❛r② ❢♦r t❤❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡s❡ ❝♦❞❡s ❛♥❞ ❢♦r t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ ❝r②♣t♦❣r❛♣❤②✳ ❲❡ ❛❧s♦ ♣r❡s❡♥t s♦♠❡ ❝♦❞❡s ❢♦✉♥❞ ✐♥ ♦✉r ❞❛②✲t♦✲❞❛② ❧✐❢❡✱ ❛✐♠✐♥❣ t♦ st✐♠✉❧❛t❡ t❤❡ ❝✉r✐♦s✐t② ♦❢ t❤❡ r❡❛❞❡r ✐♥t♦ ❞✐s❝♦✈❡r✐♥❣ t❤❡s❡ ❝♦❞❡s✳ ❋✐♥❛❧❧②✱ ❢♦r ❝♦♠♣❧❡♠❡♥t❛r② ✐♥❢♦r♠❛t✐♦♥ ♣✉r♣♦s❡s✱ ✇❡ r❡✈❡❛❧ ❛ ❜r✐❡❢ ❝♦❧❧❡❝t❡❞ ❤✐st♦r② ♦❢ ❝r②♣t♦❣r❛♣❤②✳ ❑❡② ✇♦r❞s✿ ❝r②♣t♦❣r❛♣❤②✱ ♠♦❞✉❧❛r ❛r✐t❤♠❡t✐❝✱ ❈❛❡s❛r✬s ❝♦❞❡✱ ❆✜♠ ❝♦❞❡✱ ❱✐❣❡♥ér❡✬s ❝♦❞❡✱ ❍✐❧❧✬s ❝♦❞❡✱ ❘❙❆ ❝♦❞❡✱ ❘❛❞✐♥✬s ❝♦❞❡✱ ▼❍ ❝♦❞❡✱ ❊❧●❛♠❛❧ ❝♦❞❡✳ ✈✐✐✐

(10) ❙✉♠ár✐♦ ❘❡s✉♠♦ ✈✐✐ ❆❜str❛❝t ✈✐✐✐ ■♥tr♦❞✉çã♦ ✶ ✶ ❆ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ❞❡ ❆❧❣✉♥s ❈ó❞✐❣♦s ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ✶✳✺ ✶✳✻ ✶✳✼ ✶✳✽ ✶✳✾ ✶✳✶✵ ✶✳✶✶ ✶✳✶✷ ✶✳✶✸ ✶✳✶✹ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ❖ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❙✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❉❡t❡r♠✐♥❛♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼❛tr✐③ ❆❞❥✉♥t❛ ✲ ▼❛tr✐③ ■♥✈❡rs❛ ✳ ▼❛tr✐③❡s ❊❧❡♠❡♥t❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼❛tr✐③❡s ❡ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ❈ó❞✐❣♦s ❊❧❡♠❡♥t❛r❡s ❡ ❈r✐♣t♦❣r❛✜❛ ✷✳✶ ✷✳✷ ✷✳✸ ✷✳✹ ✷✳✺ ✷✳✻ ✷✳✼ ✷✳✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦ ❞❡ ❈és❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦s ❆✜♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦ ❞❡ ❍✐❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❉❡❝♦❞✐✜❝❛♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷ ◗✉❡❜r❛♥❞♦ ✉♠ ❈ó❞✐❣♦ ❞❡ ❍✐❧❧ ❙✐st❡♠❛ ❘❙❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦ ❞❡ ❘❛❜✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❖ ▼ét♦❞♦ ▼❍ ✭▼❡r❦❧❡ ❡ ❍❡❧❧♠❛♥✮ ✳ ✷✳✼✳✶ ❖ Pr♦❜❧❡♠❛ ❞❛ ▼♦❝❤✐❧❛ ✳ ✳ ✳ ✷✳✼✳✷ ❈♦❞✐✜❝❛♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✼✳✸ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❛ ❘❡s♦❧✉çã♦ ❉❡❝♦❞✐✜❝❛♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈ó❞✐❣♦ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❞♦ Pr♦❜❧❡♠❛ ❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼♦❝❤✐❧❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✸ ✺ ✼ ✶✷ ✶✺ ✶✼ ✷✸ ✷✺ ✷✾ ✸✵ ✸✼ ✹✵ ✹✹ ✹✻ ✺✵ ✺✷ ✺✸ ✺✼ ✺✾ ✻✷ ✻✹ ✻✽ ✼✷ ✼✹ ✼✹ ✼✺ ✳ ✼✺ ✳ ✼✽

(11) ✷✳✽✳✶ ✷✳✽✳✷ ❊t❛♣❛ ❞❡ ❈♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ❊t❛♣❛ ❞❡ ❉❡❝♦❞✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✸ ❖ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s ❝ó❞✐❣♦s ❝♦♠ ê♥❢❛s❡ ♥❛ ♠❛t❡♠át✐❝❛ ♠♦❞✉❧❛r ✸✳✶ ❈ó❞✐❣♦s ❞❡ ❜❛rr❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✶ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✷ ❖ s✐❣♥✐✜❝❛❞♦ ❞♦s 13 ❞í❣✐t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✶✳✸ ❈♦♠♦ sã♦ ❣❡r❛❞♦s ♦s ❝ó❞✐❣♦s ❞❡ ❜❛rr❛s❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ❈P❋ ✲ ❈❛❞❛str♦ ❞❡ P❡ss♦❛s ❋ís✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✶ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✷ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ❈P❋❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❈◆P❏ ✲ ❈❛❞❛str♦ ◆❛❝✐♦♥❛❧ ❞❛ P❡ss♦❛ ❏✉rí❞✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸✳✶ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸✳✷ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ❈◆P❏❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ ■❙❇◆ ✲ ■♥t❡r♥❛t✐♦♥❛❧ ❙t❛♥❞❛r❞ ❇♦♦❦ ◆✉♠❜❡r ❡♠ ♣♦rt✉❣✉ês ◆ú♠❡r♦ P❛❞rã♦ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ▲✐✈r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✶ ❍✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✷ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ISBN − 10❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✸ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ISBN − 13❄ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✽✶ ✽✶ ✽✷ ✽✹ ✽✺ ✽✻ ✽✻ ✽✾ ✽✾ ✾✵ ✾✷ ✾✷ ✾✷ ✾✹ ❆ ❇r❡✈❡ ❍✐stór✐❝♦ ❞❛ ❈r✐♣t♦❣r❛✜❛ ✾✼ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✾✾ ①

(12) ■♥tr♦❞✉çã♦ ❈♦♠♦ s✉r❣✐r❛♠ ♦s ❝ó❞✐❣♦s❄ P❛r❛ q✉❡ s❡r✈❡♠❄ ◗✉❛❧ ❛ ♠❛t❡♠át✐❝❛ q✉❡ ❡①✐st❡ ♣♦r trás ❞❡ss❡s ❝ó❞✐❣♦s❄ ❊ss❛s ♣❡r❣✉♥t❛s ♣❛r❡❝❡♠ ♥ã♦ s❡r tã♦ ❝♦♠✉♥s ❛ss✐♠ ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛✳ ◆♦ ❡♥t❛♥t♦✱ q✉❛♥❞♦ ♦❧❤❛♠♦s ❛♦ ♥♦ss♦ r❡❞♦r ♣❡r❝❡❜❡♠♦s q✉❡ ♦s ❝ó❞✐❣♦s ❡stã♦ ❡♠ t♦❞♦s ♦s ❧✉❣❛r❡s✳ ❍♦❥❡ ❡♠ ❞✐❛ ♠✉✐t❛s ❝♦✐s❛s sã♦ ✐❞❡♥t✐✜❝❛❞❛s ❛ ♣❛rt✐r ❞❡ ❝ó❞✐❣♦s✳ ❆♦ ❝♦♠❡ç❛r ❡st✉❞❛r s♦❜r❡ ❡❧❡s✱ ♣❡r❝❡❜❡♠♦s ♦ q✉ã♦ ❛♥t✐❣♦s sã♦✳ ❈♦♠ ♦ ❛✈❛♥ç♦ t❡❝♥♦❧ó❣✐❝♦ ❡ ♦ s✉r❣✐♠❡♥t♦ ❞❡ ❢❡✐①❡s ❞❡ ❧✉③ ❡ s❝❛♥♥❡rs t♦r♥♦✉✲s❡ ♣♦ssí✈❡❧ tr❛♥s♠✐t✐r ❞❛❞♦s ❞✐r❡t♦ ❡ r❛♣✐❞❛♠❡♥t❡ ❛♦s ❝♦♠♣✉t❛❞♦r❡s✱ ❝r✐❛♥❞♦ ❛ss✐♠ ❝♦♥❞✐çõ❡s ♣❛r❛ ❛ ✉t✐❧✐③❛çã♦ ❞❛ ❝♦❞✐✜❝❛çã♦✳ ❊♠ 1952✱ s✉r❣✐✉ ❛ ♣r✐♠❡✐r❛ ♣❛t❡♥t❡ ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛ ❡ ❡♥tã♦ ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦ ❡ss❡s ❝ó❞✐❣♦s ❢♦r❛♠ s❡ ♠♦❞❡r♥✐③❛♥❞♦ ❛té s✉r❣✐r ♦ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s q✉❡ t❡♠♦s ❤♦❥❡✳ ❆❣✉ç❛♥❞♦ ❛✐♥❞❛ ♠❛✐s ♥♦ss❛ ❝✉r✐♦s✐❞❛❞❡ ♥♦s ❞❡♣❡r❛♠♦s ❝♦♠ ♦ s❡❣✉✐♥t❡ q✉❡st✐♦♥❛♠❡♥t♦✿ ◆❛ s♦❝✐❡❞❛❞❡ ❜r❛s✐❧❡✐r❛✱ ♦ q✉❡ ❞✐❢❡r❡ ✉♠❛ ♣❡ss♦❛ ❞❛ ♦✉tr❛❄ ❯♠❛ ♣r♦✈á✈❡❧ r❡s♣♦st❛ s❡r✐❛ ♦ ♥♦♠❡ ❝♦♠ ♦ q✉❛❧ ❛ ♣❡ss♦❛ ❢♦✐ r❡❣✐str❛❞❛✱ ♦ ♣r♦❜❧❡♠❛ é q✉❡ ❡①✐st❡♠ ✈ár✐❛s ♣❡ss♦❛s ❝✉❥♦ ♦s ♥♦♠❡s sã♦ ✐❣✉❛✐s✳ ❊♥tã♦ ❝♦♠♦ ❞✐❢❡r❡♥❝✐á✲❧♦s ❞✐❛♥t❡ ❞❛ s♦❝✐❡❞❛❞❡❄ ❊♠ ✶✾✻✽✱ s✉r❣❡ ♦ ❈P❋ ✭❈❛❞❛str♦ ❞❡ P❡ss♦❛s ❋ís✐❝❛s✮ ✉♠ ♦✉tr♦ t✐♣♦ ❞❡ ❝ó❞✐❣♦✳ ■♥✐❝✐❛❧♠❡♥t❡ ❝r✐❛❞♦ ♣❛r❛ s❡r ✉♠ ❞♦❝✉♠❡♥t♦ ❞❡ ❛rr❡❝❛❞❛çã♦ ❞❡ ✐♠♣♦st♦ ❞❡ r❡♥❞❛✱ ♣♦ré♠ ❤♦❥❡✱ é ♠✉✐t♦ ♠❛✐s ❞♦ q✉❡ ✐ss♦✱ ❡❧❡ é ✉♠ ❞♦❝✉♠❡♥t♦ ❢♦r♠❛❞♦ ♣♦r ✶✶ ❞í❣✐t♦s✱ ú♥✐❝♦ ❡ ✐♥tr❛♥s❢❡rí✈❡❧✱ q✉❡ ✐❞❡♥t✐✜❝❛ ❝❛❞❛ ♣❡ss♦❛ ❡ ❛s ❞✐❢❡r❡♠ ♠❡s♠♦ q✉❛♥❞♦ ❡❧❛s ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ♥♦♠❡ ❞❡ r❡❣✐str♦✳ ◆♦ ❡♥t❛♥t♦✱ ♥ã♦ ❡①✐st❡ ❛♣❡♥❛s ❡ss❡s ❞♦✐s ❝ó❞✐❣♦s ❝✐t❛❞♦s✱ ❝♦♠ ❡ss❡ ❛✈❛♥ç♦ t❡❝♥♦❧ó❣✐❝♦✱ ♦✉tr♦s ❝ó❞✐❣♦s ❢♦r❛♠ ❛♣❛r❡❝❡♥❞♦✱ ❝♦♠♦✿ ❈◆P❏✱ ■❙❇◆ ❡ ♦✉tr♦s✳ ❋❛③❡♥❞♦ ✉♠❛ ❛♥á❧✐s❡ ♠❛✐s ♣r♦❢✉♥❞❛ ❞❡ss❡s ❝ó❞✐❣♦s✱ ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ♣r❡♦❝✉♣❛çã♦ q✉❡ ❡①✐st❡ ❡♠ t♦❞♦s ❡❧❡s é ❛ s❡❣✉r❛♥ç❛ ♥❛s ✐♥❢♦r♠❛çõ❡s✳ ❊♥tr❡t❛♥t♦✱ ❛✈❛♥ç❛♥❞♦ ♠❛✐s ♥♦s ❡st✉❞♦s✱ ✜❝❛♠♦s ❞✐❛♥t❡ ❞❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦ ✭♦ ❝♦♥t♦ ❛ s❡❣✉✐r ❢♦✐ r❡t✐r❛❞♦ ❞❡ ❬✹❪✮✿ ❯♠ ❝❛s❛❧✱ ❆❧✐❝❡ ❡ ❇♦❜✱ q✉❡ ✈✐✈❡♠ ✐s♦❧❛❞♦s ❡ ❛♣❡♥❛s ♣♦❞❡♠ s❡ ❝♦♠✉♥✐❝❛r ❛tr❛✈és ❞♦ ❝♦rr❡✐♦✳ ❊❧❡s s❛❜❡♠ q✉❡ ♦ ❝❛rt❡✐r♦ é ✉♠ tr❡♠❡♥❞♦ ✏❢♦❢♦q✉❡✐r♦✑ ❡ q✉❡ ❧ê t♦❞❛s ❛s s✉❛s ❝❛rt❛s✳ ❆❧✐❝❡ t❡♠ ✉♠❛ ♠❡♥s❛❣❡♠ ♣❛r❛ ❇♦❜ ❡ ♥ã♦ q✉❡r q✉❡ ❡❧❛ s❡❥❛ ❧✐❞❛✳ ◗✉❡ é q✉❡ ♣♦❞❡ ❢❛③❡r❄ ❊❧❛ ♣❡♥s♦✉ ❡♠ ❧❤❡ ❡♥✈✐❛r ✉♠ ❝♦❢r❡ ❝♦♠ ❛ ♠❡♥s❛❣❡♠✱ ❢❡❝❤❛❞♦ ❛ ❝❛❞❡❛❞♦✳ ▼❛s ❝♦♠♦ ❧❤❡ ❢❛rá ❝❤❡❣❛r ❛ ❝❤❛✈❡❄ ◆ã♦ ♣♦❞❡ ❡♥✈✐❛r ❞❡♥tr♦ ❞♦ ❝♦❢r❡✱ ♣♦✐s ❛ss✐♠ ❇♦❜ ♥ã♦ ♦ ♣♦❞❡rá ❛❜r✐r✳ ❙❡ ❡♥✈✐❛r ❛ ❝❤❛✈❡ ❡♠ s❡♣❛r❛❞♦✱ ♦ ❝❛rt❡✐r♦ ♣♦❞❡ ❢❛③❡r ✉♠❛ ❝ó♣✐❛✳ ❉❡♣♦✐s ❞❡ ♠✉✐t♦ ♣❡♥s❛r✱ ❡❧❛ t❡♠ ✉♠❛ ✐❞é✐❛✳ ❊♥✈✐❛r✲ ❧❤❡ ♦ ❝♦❢r❡ ❢❡❝❤❛❞♦ ❝♦♠ ✉♠ ❝❛❞❡❛❞♦✳ ❙❛❜❡ q✉❡ ❇♦❜ é ❡s♣❡rt♦ ❡ ❛❝❛❜❛rá ♣♦r ♣❡r❝❡❜❡r ❛ s✉❛ ✐❞❡✐❛✳ ❈♦♠ ♠❛✐s ✉♠❛ ✐❞❛ ❡ ✉♠❛ ✈♦❧t❛ ❞♦ ❝♦rr❡✐♦✱ ❡ s❡♠ ♥✉♥❝❛ t❡r❡♠ tr♦❝❛❞♦ ❝❤❛✈❡s✱ ❛ ♠❡♥s❛❣❡♠ ❝❤❡❣❛ ❛té ❇♦❜✱ q✉❡ ❛❜r❡ ♦ ❝♦❢r❡ ❡ ❛ ❧ê✳ ❈♦♠♦ é q✉❡ ✈♦❝ê ❛❝❤❛ q✉❡ r❡s♦❧✈❡r❛♠ ♦ ♣r♦❜❧❡♠❛❄ P❡♥s❡ ✶

(13) ❜❡♠ ♥♦ ❛ss✉♥t♦✱ t❡♥t❡ r❡s♣♦♥❞❡r ❛ q✉❡stã♦✳ ➱ s✐♠♣❧❡s. . . ❞❡♣♦✐s q✉❡ ✈♦❝ê ❞❡s❝♦❜r✐r✱ é ❝❧❛r♦✳ ❖ ✏tr✉q✉❡✑ ✉s❛❞♦ ❢♦✐ ♦ s❡❣✉✐♥t❡✿ ❇♦❜ ❝♦❧♦❝♦✉ ✉♠ ♦✉tr♦ ❝❛❞❡❛❞♦ ♥♦ ❝♦❢r❡ ❡ ❡❧❡ t✐♥❤❛ ❛ ❝❤❛✈❡ ❞❡ss❡ s❡❣✉♥❞♦ ❝❛❞❡❛❞♦✳ ❉❡✈♦❧✈❡ ♦ ❝♦❢r❡ ❛ ❆❧✐❝❡ ♣♦r ❝♦rr❡✐♦✱ ❞❡st❛ ✈❡③ ❢❡❝❤❛❞♦ ❝♦♠ ♦s ❞♦✐s ❝❛❞❡❛❞♦s✳ ❆❧✐❝❡ r❡♠♦✈❡ ♦ s❡✉ ❝❛❞❡❛❞♦✱ ❝♦♠ ❛ ❝❤❛✈❡ q✉❡ ♣♦ss✉✐ ❡ r❡❡♥✈✐❛ ♦ ❝♦❢r❡ ♣❡❧♦ ❝♦rr❡✐♦ só ❝♦♠ ♦ ❝❛❞❡❛❞♦ ❝♦❧♦❝❛❞♦ ♣♦r ❇♦❜✳ ➱ ❝❧❛r♦ q✉❡ ❇♦❜ t❡♠ ❛♣❡♥❛s q✉❡ ❛❜r✐r ♦ ❝♦❢r❡✱ ❝♦♠ ❛ s✉❛ ♣ró♣r✐❛ ❝❤❛✈❡ ❡ ❧❡r ❛ ♠❡♥s❛❣❡♠ ❡♥✈✐❛❞❛ ♣❡❧❛ s✉❛ ❛♠❛❞❛✳ ❖ ❝❛rt❡✐r♦ ♥ã♦ t❡♠ ❝♦♠♦ s❛❜❡r ♦ ❝♦♥t❡ú❞♦ ❞♦ ❝♦❢r❡✳ ❆ q✉❡stã♦ ♣r✐♥❝✐♣❛❧ r❡❧❛t❛❞❛ ❛❝✐♠❛ é ❝♦♠♦ tr❛♥s♠✐t✐r ✉♠❛ ♠❡♥s❛❣❡♠ ❞❛ ❢♦♥t❡ ❆ ♣❛r❛ ❛ ❢♦♥t❡ ❇✱ ❞❡ ♠♦❞♦ q✉❡ ❛s ❢♦♥t❡s ♥ã♦ ❛✉t♦r✐③❛❞❛s ♥ã♦ t❡♥❤❛♠ ❛❝❡ss♦ ❛♦s ❝♦♥t❡ú❞♦s ❞❛ ♠❡♥s❛❣❡♠✳ P❛r❛ ❡①✐st✐r ✉♠❛ ❝♦♠✉♥✐❝❛çã♦ s❡❣✉r❛ é ✐♠♣♦rt❛♥t❡ ♦ ❡st✉❞♦ ❞❡ té❝♥✐❝❛s ♠❛t❡♠át✐❝❛s r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ❛ ❝♦♥✜❞❡♥❝✐❛❧✐❞❛❞❡✱ ✐♥t❡❣r✐❞❛❞❡ ❡ ❛✉t❡♥t✐❝❛çã♦✱ q✉❡ ♣❡r♠✐t❛ ✉♠❛ tr❛♥s❢♦r♠❛çã♦ ❞❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ ❡♠ ✉♠ ❝ó❞✐❣♦ s❡❝r❡t♦✳ ❆ ♣❛rt✐r ❞❡ss❛ s✐t✉❛çã♦✱ ♣❡r❝❡❜❡♠♦s q✉❡ t♦❞♦s ♦s ♦✉tr♦s ❝ó❞✐❣♦s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ t✐♥❤❛♠ ❝♦♠♦ ú♥✐❝❛ ❡ ❡①❝❧✉s✐✈❛ ♣r❡♦❝✉♣❛çã♦ ❢❛③❡r ❝♦♠ q✉❡ ❛ ✐♥❢♦r♠❛çã♦ ❝❤❡❣✉❡ ❝♦♠ s❡❣✉r❛♥ç❛✳ ◆♦t❛♠♦s q✉❡ ♦s ♣r✐♠❡✐r♦s ❝ó❞✐❣♦s ❝✐t❛❞♦s ❞✐❢❡r❡♠ ❞❡ss❛ ú❧t✐♠❛ s✐t✉❛çã♦✳ ❊❧❡s ❢❛③❡♠ ♣❛rt❡s ❞❡ r❛♠♦s ❞✐❢❡r❡♥t❡s ❞❛ ♠❛t❡♠át✐❝❛✱ ♦s ♣r✐♠❡✐r♦s ❝ó❞✐❣♦s ✭❝ó❞✐❣♦ ❞❡ ❜❛rr❛✱ ❈P❋✱ ❈◆P❏✱ ❡♥tr❡s ♦✉tr♦s✮ ♣❡rt❡♥❝❡♠ ❛ t❡♦r✐❛ ❞♦s ❝ó❞✐❣♦s ❡♥q✉❛♥t♦ ❡ss❛ ú❧t✐♠❛ s✐t✉❛çã♦✱ ♦ ❝♦♥t♦✱ tr❛t❛✲s❡ ❞❡ ❝ó❞✐❣♦ s❡❝r❡t♦ ❡ ♣❡rt❡♥❝❡ ❛ ❝r✐♣t♦❣r❛✜❛✳ ❚❡♦r✐❛ ❞♦s ❝ó❞✐❣♦s ❡ ❈r✐♣t♦❣r❛✜❛ sã♦ r❛♠♦s ❞✐st✐♥t♦s ❡ s❡r✈❡♠ ♣❛r❛ ♣r♦♣ós✐t♦s ❞✐❢❡r❡♥t❡s✳ ◆❛ t❡♦r✐❛ ❞♦s ❝ó❞✐❣♦s✱ ❝♦♠♦ ❥á ❝✐t❛♠♦s✱ ❛ ú♥✐❝❛ ♣r❡♦❝✉♣❛çã♦ é q✉❡ ❛ ♠❡♥s❛❣❡♠ ❝❤❡❣✉❡ ❝♦♠ s❡❣✉r❛♥ç❛ ❛♦ s❡✉ ❞❡st✐♥♦✳ ❊♥q✉❛♥t♦ ♥❛ ❝r✐♣t♦❣r❛✜❛✱ ❛ q✉❡stã♦ ♣r✐♥❝✐♣❛❧ é ❝♦♠♦ tr❛♥s♠✐t✐r ✉♠❛ ♠❡♥s❛❣❡♠ ❞❡ ✉♠❛ ❢♦♥t❡ ♣❛r❛ ♦✉tr❛✱ ❞❡ ♠♦❞♦ q✉❡ ❛s ❢♦♥t❡s ♥ã♦ ❛✉t♦r✐③❛❞❛s ♥ã♦ t❡♥❤❛♠ ❛❝❡ss♦ ❛ ❝♦♥t❡ú❞♦s ❞❛ ♠❡♥s❛❣❡♠✱ ✉t✐❧✐③❛♥❞♦ ❞✐✈❡rs❛s ❡str❛té❣✐❛s✱ r❡❣r❛s ❡ ❢ór♠✉❧❛s q✉❡ ♣❡r♠✐t❡♠ ❛ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦ ❞❛ ♠❡♥s❛❣❡♠ ♦❢❡r❡❝❡♥❞♦ ✉♠❛ ❝♦♠✉♥✐❝❛çã♦ s❡❣✉r❛✳ ❇❛s❡❛♥❞♦✲s❡ ♥❛s ♣❡sq✉✐s❛s r❡❛❧✐③❛❞❛s ♥♦ ❝❛♠♣♦ ❞♦s ❝ó❞✐❣♦s é q✉❡ ❡♥❢❛t✐③❛♠♦s ❡st❡ tr❛❜❛❧❤♦ ♥♦ ❡st✉❞♦ ❞❛ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❡ss❡♥❝✐❛❧ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦s ❞✐✈❡rs♦s ❝ó❞✐❣♦s ❞❡ ❝r✐♣t♦❣r❛✜❛✳ ❉✐✈✐❞✐♠♦s ❡st❡ tr❛❜❛❧❤♦ ❡♠ 3 ❝❛♣ít✉❧♦s✱ s❡♥❞♦ ♦ ♣r✐♠❡✐r♦ ❞❡❧❡s tr❛t❛♥❞♦ ❞❛ ♠❛t❡♠át✐❝❛ ❡❧❡♠❡♥t❛r ❞❡ ❛❧❣✉♥s ❝ó❞✐❣♦s✱ ♦♥❞❡ tr❛❜❛❧❤❛♠♦s ❛ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡✱ ✈❡t♦r❡s✱ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r✱ t✉❞♦ ❞❡ ❢♦r♠❛ s✉❝✐♥t❛ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ❛s ❞✐✈❡rs❛s ♠❛♥❡✐r❛s ❞❡ ❝♦❞✐✜❝❛r ❡ ❞❡❝♦❞✐✜❝❛r ❛s ♠❡♥s❛❣❡♥s✳ ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛♠♦s ✈ár✐♦s ❝ó❞✐❣♦s ❝r✐♣t♦❣rá✜❝♦s ✉s❛❞♦s ♥❛s ❞✐✈❡rs❛s ❝♦♠✉♥✐❝❛çõ❡s ❞❡ ♠❡♥s❛❣❡♥s s❡❝r❡t❛s✳ ◆♦ t❡r❝❡✐r♦ ❡ ú❧t✐♠♦ ❝❛♣ít✉❧♦✱ t❡♠♦s ❛❧❣✉♥s ❝ó❞✐❣♦s✱ r❡❢❡r❡♥t❡s à ❚❡♦r✐❛ ❞♦s ❈ó❞✐❣♦s✱ ♣❛r❛ ❛❣✉ç❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❞♦ ❧❡✐t♦r✱ ♠♦str❛♥❞♦ t♦❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ❡①✐st❡ ♣♦r trás ❞❡ss❡s✳ ❋✐♥❛❧♠❡♥t❡✱ ❛ tít✉❧♦ ❞❡ ✐♥❢♦r♠❛çã♦ ❝♦♠♣❧❡♠❡♥t❛r✱ ❡①♣♦♠♦s ✉♠ ❜r❡✈❡ ❛♣❛♥❤❛❞♦ ❤✐stór✐❝♦ ❞❛ ❝r✐♣t♦❣r❛✜❛ ✭❆♣ê♥❞✐❝❡ ❆✮✳ ✷

(14) ❈❛♣ít✉❧♦ ✶ ❆ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ❞❡ ❆❧❣✉♥s ❈ó❞✐❣♦s ❊st❡ ❝❛♣ít✉❧♦ ❢♦✐ ❜❛s❡❛❞♦ ♥♦s t❡①t♦s ❬✸✱ ✺✱ ✽✱ ✾✱ ✶✵✱ ✶✶✱ ✶✹❪ ❡ t❡♠ ♣♦r ♦❜❥❡t✐✈♦ s✉❜s✐❞✐❛r ♦ ❡st✉❞♦ ❞♦s ❝ó❞✐❣♦s q✉❡ s❡rã♦ tr❛t❛❞♦s ♥❡st❡ tr❛❜❛❧❤♦✳ ❋❛r❡♠♦s✱ ♣♦rt❛♥t♦✱ ✉♠❛ ❜r❡✈❡ ❡①♣♦s✐çã♦ ❞♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ♠❛t❡♥át✐❝♦s ♥❡❝❡ssár✐♦s✳ ❚❡r❡♠♦s ✶ ❝♦♠♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❛ ❛r✐t♠ét✐❝❛ ❡❧❡♠❡♥t❛r✱ ❥á ❡st✉❞❛❞❛ ❞❡s❞❡ ❊✉❝❧✐❞❡s q✉❡ ❢♦✐ ✐♠♣♦rt❛♥t❡ ❡ ♥♦rt❡♦✉ ♦s ❝r✐❛❞♦r❡s ❞♦s ❞✐✈❡rs♦s ❝ó❞✐❣♦s ❡①✐st❡♥t❡s ❤♦❥❡✳ ✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡ a ❡ b sã♦ ✐♥t❡✐r♦s✱ ❞✐③❡♠♦s q✉❡ a ❞✐✈✐❞❡ b✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r a|b✱ q✉❛♥❞♦ ❡①✐st✐r ✉♠ ✐♥t❡✐r♦ c t❛❧ q✉❡ b = ac✳ ❙❡ a ♥ã♦ ❞✐✈✐❞❡ b ❡s❝r❡✈❡♠♦s a ∤ b✳ ❊①❡♠♣❧♦ ✶✳✷✳ P❡❧❛ ❞❡✜♥✐çã♦✱ 2|6 ♣♦✐s 6 = 2×3❀ 5|10 ♣♦✐s 10 = 5×2 ❡ 1|a (∀a ∈ Z) ♣♦✐s a = 1 × a✳ ◆♦ ❡♥t❛♥t♦✱ 0 ∤ b✱ ❝♦♠ b 6= 0✱ ♣♦✐s 0 × c = 0 6= b✳ Pr♦♣♦s✐çã♦ ✶✳✸✳ ❈♦♥s✐❞❡r❡ a✱ b ❡ c ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❙❡ a|b ❡ b|c✱ ❡♥tã♦ a|c✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ a|b ❡ b|c✱ ❡①✐st❡♠ ✐♥t❡✐r♦s b = k1 a ❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❡ b ♥❛ ❡q✉❛çã♦ k1 ❡ k2 ❡ c = k2 b✳ c = k2 b t❡r❡♠♦s ❝♦♠ c = k2 k1 a✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡①✐st✐r k = k1 k2 ✐♥t❡✐r♦ t❛❧ q✉❡ c = ka✳ ▲♦❣♦✱ a|c✳ ❊①❡♠♣❧♦ ✶✳✹✳ ❈♦♠♦ 3|12 ❡ 12|48✱ ❡♥tã♦ 3|48✳ ✶ P♦✉❝♦ s❡ ❝♦♥❤❡❝❡ s♦❜r❡ ❛ ✈✐❞❛ ❡ ❛ ♣❡rs♦♥❛❧✐❞❛❞❡ ❞❡ ❊✉❝❧✐❞❡s✳ Pr♦✈❛✈❡❧♠❡♥t❡ s✉❛ ❢♦r♠❛çã♦ ♠❛t❡♠át✐❝❛ t❡♥❤❛ s❡ ❞❛❞♦ ♥❛ ❡s❝♦❧❛ ♣❧❛tô♥✐❝❛ ❞❡ ❆t❡♥❛s✳ ❊❧❡ ❢♦✐ ♣r♦❢❡ss♦r ❞♦ ▼✉s❡✉ ❡♠ ❆❧❡①❛♥❞r✐❛✳ ❊✉❝❧✐❞❡s ❡s❝r❡✈❡✉ ❝❡r❝❛ ❞❡ ✉♠❛ ❞ú③✐❛ ❞❡ tr❛t❛❞♦s ❡ ✉♠ ❧✐✈r♦ s♦❜r❡ s❡çõ❡s ❝ô♥✐❝❛s❀ ♣♦ré♠✱ ♠❛✐s ❞❛ ♠❡t❛❞❡ ❞♦ q✉❡ ❡❧❡ ❡s❝r❡✈❡✉ s❡ ♣❡r❞❡✉✳ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s ♥ã♦ tr❛t❛♠ ❛♣❡♥❛s ❞❡ ❣❡♦♠❡tr✐❛✱ ♠❛s t❛♠❜é♠ ❞❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s ❡ á❧❣❡❜r❛ ❡❧❡♠❡♥t❛r✳ ❖ ❧✐✈r♦ é ❝♦♠♣♦st♦ ❞❡ q✉❛tr♦❝❡♥t♦s ❡ s❡ss❡♥t❛ ❡ ❝✐♥❝♦ ♣r♦♣♦s✐çõ❡s ❞✐str✐❜✉í❞❛s ❡♠ tr❡③❡ ❧✐✈r♦s ♦✉ ❝❛♣ít✉❧♦s✱ ❞♦s q✉❛✐s ♦s s❡✐s ♣r✐♠❡✐r♦s sã♦ s♦❜r❡ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡❧❡♠❡♥t❛r✱ ♦s três s❡❣✉✐♥t❡s s♦❜r❡ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ♦ ❧✐✈r♦ ✐♥❝♦♠❡♥s✉rá✈❡✐s ❡ ♦s três ú❧t✐♠♦s tr❛t❛♠ s♦❜r❡ ❣❡♦♠❡tr✐❛ ♥♦ ❡s♣❛ç♦✳ t❛♠❜é♠ ✉♠❛ ❡①♣♦s✐çã♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ♥✉♠ér✐❝❛ ♦✉ ♣✐t❛❣ór✐❝❛✳ ✸ X s♦❜r❡ ❆❧é♠ ❞✐ss♦✱ ❡♥❝♦♥tr❛♠♦s

(15) Pr♦♣♦s✐çã♦ ✶✳✺✳ ❈♦♥s✐❞❡r❡ a✱ b✱ c✱ m ❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❙❡ c|a ❡ c|b ❡♥tã♦ c|(ma + nb)✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ c|a ❡ c|b ❡♥tã♦ a = k1 c ❡ b = k2 c ✳ ▼✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❡st❛s ❞✉❛s ❡q✉❛çõ❡s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦r m ❡ n t❡r❡♠♦s ma = mk1 c ❡ nb = nk2 c✳ ❙♦♠❛♥❞♦✲s❡ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ♦❜t❡♠♦s ma + nb = (mk1 + nk2 )c, ♦ q✉❡ ♥♦s ❞✐③ q✉❡ c|(ma + nb)✳ ❊①❡♠♣❧♦ ✶✳✻✳ ❈♦♠♦ 3|15 ❡ 3|42✱ ❡♥tã♦ 3|(8 × 15 − 7 × 42). ❚❡♦r❡♠❛ ✶✳✼✳ ❈♦♥s✐❞❡r❡ a, d ❡ n ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ ❆ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ t❡♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ (i) n|n❀ (ii) d|n ⇒ ad|an❀ (iii) a 6= 0 ad|an ⇒ d|n❀ ❡ (iv) 1|n❀ (v) n|0❀ (vi) d|n (vii) d|n (viii) d|n n 6= 0 ⇒ |d| ≤ |n|❀ ❡ ❡ ❡ n|d ⇒ |d| = |n|❀ d 6= 0 ⇒ (n/d)|n✳ (i)✿ ❈♦♠♦ n = 1n s❡❣✉❡ ❞❛ ❞❡✜♥✐ç❛õ q✉❡ n|n✳ (ii)✿ ❙❡ d|n ❡♥tã♦ n = cd ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ c✳ ▲♦❣♦ an = cad✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❉❡♠♦♥str❛çã♦✳ ❞❡♠♦♥str❛çã♦✳ (viii)✿ ❙❡ d|n ❡♥tã♦ n = k1 d ❡ ♣♦rt❛♥t♦ n/d é ✉♠ ✐♥t❡✐r♦✳ ❈♦♠♦ (n/d)d = n s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡ (n/d)|n✳ ❖s ❞❡♠❛✐s ✐t❡♥s t❛♠❜é♠ sã♦ ❝♦♥s❡q✉ê♥❝✐❛s ✐♠❡❞✐❛t❛s ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳ ✹

(16) ✶✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ❖ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✱ q✉❡ ✈❡r❡♠♦s ❛ s❡❣✉✐r✱ ❢♦✐ ✉s❛❞♦ ♣♦r ❊✉❝❧✐❞❡s ♥♦ s❡✉ ❧✐✈r♦ ❊❧❡♠❡♥t♦s ❡ ❡st❛❜❡❧❡❝❡ ✉♠❛ ❞✐✈✐sã♦ ❝♦♠ r❡st♦✳ ❖ ❡st✉❞♦ ❞♦ ❛❧❣♦r✐t♠♦ ♥❡st❛ s❡çã♦ s❡ ❜❛s❡✐❛ ♥♦ q✉❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✳ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦ ✏❚♦❞♦ s✉❜❝♦♥❥✉♥t♦ ♥ã♦ ✈❛③✐♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣♦ss✉✐ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✑ ✳ P❛r❛ ✐❧✉str❛r ♦ ♣r✐♥❝í♣✐♦✱ ♦❜s❡r✈❡ ♣♦r ❡①❡♠♣❧♦ q✉❡ ♦s s✉❜❝♦♥❥✉♥t♦s A = {4, 5, 8, 9} ♣♦ss✉❡♠ ❝♦♠♦ ♠❡♥♦r❡s ❡❧❡♠❡♥t♦s 4 ❚❡♦r❡♠❛ ✶✳✽✳ ❉❛❞♦s ❞♦✐s ✐♥t❡✐r♦s ❡ r B = {2, 4, 6, 8, 10, · · · } ❡ ❡ 2✱ a ❡ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ d ✱ d > 0✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣❛r ❞❡ ✐♥t❡✐r♦s q t❛✐s q✉❡ a = qd + r, ✭q é ❝❤❛♠❛❞♦ ❞❡ q✉♦❝✐❡♥t❡ ❡ r ❝♦♠ 0 ≤ r < d (r = 0 ⇔ d|a) ❞❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ a ✭✶✳✶✮ d✮✳ ♣♦r ❉❡♠♦♥str❛çã♦✳ ❊①✐stê♥❝✐❛✿ ❢♦r♠❛ a − dx✱ S ♦ ❝♦♥❥✉♥t♦ x ∈ Z✱ ✐st♦ é✿ ❙❡❥❛ ❝♦♠ ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ♥ã♦✲♥❡❣❛t✐✈♦s q✉❡ sã♦ ❞❛ S = {a − dx : x ∈ Z ❖ ❝♦♥❥✉♥t♦ S é ♥ã♦ ✈❛③✐♦✳ ❉❡ x = −|a| ♦❜t❡♠♦s ❡ ❢❛t♦✱ s❡♥❞♦ a − dx ≥ 0}. d > 0✱ t❡♠♦s ❝♦♥s✐❞❡r❛♥❞♦ d ≥ 1 ❡✱ ♣♦rt❛♥t♦✱ a − dx = a + d|a| ≥ a + |a| ≥ 0. ▲♦❣♦✱ a − d(−|a|) ∈ S ✳ ❆❣♦r❛✱ s❡♥❞♦ ❡①✐st❡ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ r ❞❡ S S ♥ã♦ ✈❛③✐♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱ ❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ q t❛❧ q✉❡ r ≥ 0 e r = a − dq, ♦✉ s❡❥❛✱ r≥d a = dq + r ❝♦♠ ❛❧❣✉♠ t❡rí❛♠♦s ❆❧é♠ ❞✐ss♦✱ t❡♠♦s r < d✳ ❉❡ ❢❛t♦✱ s❡ ❢♦ss❡ 0 ≤ r − d = a − dq − d = a − d(q + 1) < r. ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡rí❛♠♦s S✳ q ∈ Z✳ r−d∈S ❞❡ ♠♦❞♦ q✉❡ r ♥ã♦ s❡r✐❛ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ ❞❡ P♦rt❛♥t♦✱ t❡♠♦s ❣❛r❛♥t✐❞❛ ❛ ♣❛rt❡ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✭✶✳✶✮✳ ❯♥✐❝✐❞❛❞❡✿ P❛r❛ ❞❡♠♦♥str❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❞♦✐s ♦✉tr♦s ✐♥t❡✐r♦s q1 ❡ r1 q ❡ r✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡♠ t❛✐s q✉❡ a = dq1 + r1 e 0 ≤ r1 < d. ❊♥tã♦✱ t❡r❡♠♦s✿ dq1 + r1 = dq + r ⇒ r1 − r = d(q − q1 ) ⇒ d|(r1 − r). ✺

(17) P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s −d < −r ≤ 0 e 0 ≤ r1 < d, q✉❡ ✐♠♣❧✐❝❛ −d < r1 − r < d, ♦✉ s❡❥❛✱ |r1 − r| < d. d|(r1 − r) ❡ |r1 − r| < d✳ ▲♦❣♦✱ r1 − r = 0✳ q1 d = qd✱ s❡❣✉❡ q✉❡ q = q1 ✳ ▲♦❣♦✱ r1 = r ❡ q1 = q ✳ ❆ss✐♠✱ ❈♦r♦❧ár✐♦ ✶✳✾ ❝♦♠ d 6= 0✱ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ✭❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❡♥tã♦ ❡①✐st❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s q ❡ r✱ ❙❡ a ❡ d d 6= 0 ❡ ❛❣♦r❛ sã♦ ❞♦✐s ✐♥t❡✐r♦s ❡ sã♦ ú♥✐❝♦s✱ t❛✐s q✉❡ a = dq + r, 0 ≤ r < |d|. ❉❡♠♦♥str❛çã♦✳ ❡♥tã♦ |d| > 0✱ ❙❡ d > 0✱ ❛ ❝♦♥❝❧✉sã♦ é ♦❜t✐❞❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✽✳ ❆❣♦r❛✱ s❡ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✽✱ ❡①✐st❡♠ ú♥✐❝♦s ✐♥t❡✐r♦s q1 ❡ a = |d|q1 + r, ◆❡st❡ ❝❛s♦✱ ♥♦t❡ q✉❡ |d| = −d✱ r d < 0 t❛✐s q✉❡ 0 ≤ r < |d|. ❞❡ ♠♦❞♦ q✉❡ a = d(−q1 ) + r, 0 ≤ r < |d|. P♦rt❛♥t♦✱ ❡①✐st❡♠ ú♥✐❝♦s ✐♥t❡✐r♦s q = −q1 ❡ r t❛✐s q✉❡ a = dq + r, 0 ≤ r < |d|. ❖s ✐♥t❡✐r♦s a✱ d✱ q ❡ r sã♦ ❝❤❛♠❛❞♦s r❡s♣❡❝t✐✈❛♠❡♥t❡ ❞❡ ❞✐✈✐❞❡♥❞♦✱ ❞✐✈✐s♦r✱ q✉♦❝✐❡♥t❡ ❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r ❊①❡♠♣❧♦ ✶✳✶✵✳ ❆❝❤❛r ♦ q✉♦❝✐❡♥t❡ q d✳ ❡ ♦ r❡st♦ r ♥❛ ❞✐✈✐sã♦ ❞❡ q✉❡ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✳ ❊❢❡t✉❛♥❞♦ ❛ ❞✐✈✐sã♦ ✉s✉❛❧ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞❡ a ❡ b✱ a = −83 ♣♦r b = 12 ♦❜t❡♠♦s 83 = 12 × 6 + 11, ♦✉ ❛✐♥❞❛✱ −83 = 12 × (−6) − 11. ❈♦♠♦ ♦ t❡r♠♦ r = −11 < 0 b = 12 ❛♦ s✉❜tr❛✐♥❞♦ ♦ ✈❛❧♦r ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ 0 ≤ r < 12✱ ❡♥tã♦ s♦♠❛♥❞♦ ❡ s❡❣✉♥❞♦ ♠❡♥❜r♦ ❞❛ ✐❣✉❛❧❞❛❞❡ ❛♥t❡r✐♦r✱ ♦❜t❡♠♦s −83 = 12 × (−6) − 12 + 12 − 11 = 12 × (−7) + 1. ▲♦❣♦✱ ❝♦♠♦ 0 ≤ r = 1 < 12✱ ♦ q✉♦❝✐❡♥t❡ é ✻ q = −7 ❡ ♦ r❡st♦ é r = 1✳

(18) ✶✳✸ ❖ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❖ ❧✐✈r♦ V II ❞❛ ♦❜r❛ ✏❖s ❊❧❡♠❡♥t♦s✑ ❞❡ ❊✉❝❧✐❞❡s ❝♦♠❡ç❛ ❝♦♠ ♦ ♣r♦❝❡ss♦ ♣❛r❛ ❛❝❤❛r ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♦✉ ♠❛✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❤♦❥❡ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❛❧❣♦r✐t♠♦ ❡✉❝❧✐❞✐❛♥♦✱ ❡ ♦ ✉s❛ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ❞♦✐s ✐♥t❡✐r♦s sã♦ s✐✳ ♣r✐♠♦s ❡♥tr❡ ❉❡✜♥✐çã♦ ✶✳✶✶✳ ❙❡❥❛♠ a ❡ b ❞♦✐s ✐♥t❡✐r♦s ♥ã♦ ❝♦♥❥✉♥t❛♠❡♥t❡ ♥✉❧♦s ✭a 6= 0 ♦✉ b = 6 0✮✳ ❈❤❛♠❛✲s❡ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❞❡ a ❡ b ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ d (d > 0) q✉❡ s❛t✐s✜③❡r às s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✭✶✮ d|a ❡ d|b❀ ✭✷✮ s❡ c|a ❡ c|b✱ ❡♥tã♦ c|d✳ ❖❜s❡r✈❡✲s❡ q✉❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦ (1)✱ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b ❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦ (2)✱ d é ♦ ♠❛✐♦r ❞❡♥tr❡ t♦❞♦s ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ a ❡ b✳ ❖ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✷ ❞❡ a ❡ b é ❞❡♥♦t❛❞♦ ♣❡❧❛ ♥♦t❛çã♦ mdc(a, b) ✳ P♦r ❡①❡♠♣❧♦✱ s❡❥❛♠ a = 6 ❡ b = 8✳ ■♥❞✐❝❛♥❞♦ ♣♦r Dx ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞✐✈✐s♦r❡s ❞❡ x ∈ Z✱ t❡♠♦s D6 = {−6, −3, −2, −1, 1, 2, 3, 6} ❡ D8 = {−8, −4, −2, −1, 1, 2, 4, 8}, ❞❡ ♠♦❞♦ q✉❡ D6 ∩ D8 = {−1, −2, 1, 2}. ❆❣♦r❛ ♦❜s❡r✈❛♠♦s q✉❡✿ ✶✮ 2|6, 2|8❀ ✷✮ s❡ c|6 ❡ ❝♦♠✉♠ ▲♦❣♦✱ ♦ c|8✱ ❞❡ 6 ❡♥tã♦ ❡ c ♣♦❞❡ s❡r 8✳ −1, −2, 1, 2✳ ◆♦ ❡♥t❛♥t♦✱ 2 é ♠á①✐♠♦ ❞✐✈✐s♦r mdc(6, 8) = 2✳ ➱ ✐♠❡❞✐❛t♦ ♦❜s❡r✈❛r q✉❡✿ • mdc(a, b) = mdc(b, a)❀ • mdc(|a|, |b|) = mdc(a, b)✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝♦♥✈❡♥❝✐♦♥❛♠♦s✿ • mdc(0, 0) = 0✳ ◆♦t❡ q✉❡ ♥❡ss❡ ú❧t✐♠♦ ❝❛s♦ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ♥ã♦ é ♦ ♠❛✐♦r ❞♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s✿ ❝♦♠♦ 1|0, 2|0, 3|0, ... ♥ã♦ ❤á ✉♠ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ♣❛r❛ 0 ❛♣❡♥❛s ✉♠❛ ❝♦♥✈❡♥çã♦ ❛❞❡q✉❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s✿ • mdc(a, 1) = 1❀ • s❡ a 6= 0✱ ❡♥tã♦ ♦ mdc(a, 0) = |a|❀ ✷ ◆❛ ❧✐t❡r❛t✉r❛ é ❝♦♠✉♠ ✉s❛✲s❡ ❛ ♥♦t❛çã♦ ♣❛r❛ ✼ mdc(a, b) s✐♠♣❧❡s♠❡♥t❡ ❝♦♠♦ (a, b) ❡ 0❀ ✐ss♦ é

(19) • mdc(a, b). ❊①❡♠♣❧♦ ✶✳✶✷✳ ❛✮ mdc(8, 1) = 1 ❜✮ mdc(−3, 0) = | − 3| = 3 ❝✮ mdc(−6, 12) = | − 6| = 6 Pr♦♣♦s✐çã♦ ✶✳✶✸✳ ❙❡ a|b✱ ❡♥tã♦ mdc(a, b) = |a|✳ ❉❡ ❢❛t♦✱ |a||a ❡ |a||b ✭s❡❣✉❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧❛❞❛❞❡ ❡ ❞❛ ❤✐♣ót❡s❡✮✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ c > 0✱ s❡ c|a ❡ c|b✱ é ó❜✈✐♦ q✉❡ c||a|✳ ❉❡♠♦♥str❛çã♦✳ a = bq + r ❡ d = mdc(a, b)✱ d = mdc(b, r)✱ ❡♥tã♦ d = mdc(a, b)✳ Pr♦♣♦s✐çã♦ ✶✳✶✹✳ s❡ ❉❡♠♦♥str❛çã♦✳ q✉❡ d|bq ✳ ▲♦❣♦ ❙❡ ❡♥tã♦ d = mdc(b, r)✳ ❆❧é♠ ❞✐ss♦✱ ❈♦♠♦ d = mdc(a, b)✱ ❡♥tã♦ d|a ❡ d|b✳ ❉❡ss❛ ú❧t✐♠❛ r❡❧❛çã♦ r❡s✉❧t❛ d|(a − bq), ♦✉ s❡❥❛✱ d|r✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ c|b ❡ c|r✱ ❡♥tã♦ c|(bq + r). ❈♦♠♦ bq + r = a✱ ❡♥tã♦ c|a ❡ c|b✱ ♦ q✉❡ ✐♠♣❧✐❝❛ c|d✱ ♣♦✐s d = mdc(a, b)✳ ❆ s❡❣✉♥❞❛ ❛✜r♠❛çã♦ s❡ ♣r♦✈❛ ❞❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✳ ❘❡t♦r♥❛r❡♠♦s ❛❣♦r❛ ❛ q✉❡stã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ P❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❛♣❧✐❝❛r❡♠♦s✱ s✉❝❡ss✐✈❛♠❡♥t❡✱ ❛ ♣❛rt✐r ❞❡ a ❡ b✱ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ a = bq1 + r1 (0 ≤ r1 < |b|) b = r 1 q2 + r 2 (0 ≤ r2 < r1 ) r 1 = r 2 q3 + r 3 (0 ≤ r3 < r2 ) ✳✳ ✳ ✭✶✳✷✮ ➱ ❝❧❛r♦ q✉❡✱ s❡ ❛❝♦♥t❡❝❡r ❞❡ r1 s❡r ♥✉❧♦✱ ❡♥tã♦ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✸ ♥♦s ❣❛r❛♥t❡ q✉❡ |b| = mdc(a, b) ❡ ♦ ♣r♦❝❡ss♦ t❡r♠✐♥❛ ♥❛ ♣r✐♠❡✐r❛ ❡t❛♣❛✳ ▼❛s✱ ❞❡ q✉❛❧q✉❡r ♠❛♥❡✐r❛✱ ♥❛ s❡q✉ê♥❝✐❛ |b| > r1 > r2 > r3 > ... ❞❡✈❡rá ♦❝♦rr❡r rn+1 = 0✱ ♣❛r❛ ❛❧❣✉♠ í♥❞✐❝❡ n✳ ❉❡ ❢❛t♦✱ s❡ t♦❞♦s ♦s ri ❢♦ss❡♠ ♥ã♦ ♥✉❧♦s✱ ❡♥tã♦ {|b|, r1 , r2 , ...} ♥ã♦ ♣♦ss✉✐r✐❛ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦✱ ♦ q✉❡ ❝♦♥tr❛r✐❛ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✳ ❆ss✐♠✱ ♣❛r❛ ❛❧❣✉♠ n❀ rn−2 = rn−1 q + rn ✽

(20) rn−1 = rn qn+1 . ❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛s Pr♦♣♦s✐çõ❡s ✶✳✶✸ ❡ ✶✳✶✹✱ ♦❜té♠✲s❡ ❡♥tã♦ ♦ s❡❣✉✐♥t❡✿ rn = mdc(rn−1 , rn ) = mdc(rn−2 , rn−1 ) = ... = mdc(b, r1 ) = mdc(a, b), ♦✉ s❡❥❛✱ rn = mdc(a, b). ❖❜s❡r✈❡ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ mdc é ❝♦♥str✉t✐✈❛✳ ❖ ❞✐s♣♦s✐t✐✈♦ ♣rát✐❝♦ q✉❡ ❝♦st✉♠❛ s❡r ❡♠♣r❡❣❛❞♦ ♣❛r❛ ❛♣❧✐❝á✲❧♦ ♥❛ ♣rát✐❝❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣r♦❝❡ss♦ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s ♦✉ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ ➱ ✉s✉❛❧ ❛ s❡❣✉✐♥t❡ ♦r❣❛♥✐③❛çã♦ ❡♠ ❢♦r♠❛ ❞❡ t❛❜❡❧❛ ❞❡ss❡ ❞✐s♣♦s✐t✐✈♦ ❞❡ ❝á❧❝✉❧♦ ❞❡ mdc(a, b)✿ q 1 q 2 q3 qn qn+1 a b r1 r2 . . . rn−1 rn r1 r2 r 3 r4 0 mdc(a, b)✱ ❞✐✈✐❞✐✲s❡ a ♣♦r b r2 é ♦❜t✐❞♦ ♣❡❧❛ ❞✐✈✐sã♦ ❞❡ b ❖ t❡r❝❡✐r♦ r❡st♦ r3 é ♦❜t✐❞♦ ♣❡❧❛ ❞✐✈✐sã♦ ❞❡ r1 ♣♦r r2 ✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡ ❆ t❛❜❡❧❛ s❡ tr❛❞✉③ ♥❛ s❡❣✉✐♥t❡ r❡❣r❛✿ ♣❛r❛ s❡ ✏❛❝❤❛r✑ ♦ ❡ ❡♥❝♦♥tr❛✲s❡ ♦ ✏♣r✐♠❡✐r♦✑ r❡st♦ ♣♦r r1 ✳ r1 ✳ ❖ ✏s❡❣✉♥❞♦✑ r❡st♦ ❛té ❡♥❝♦♥tr❛r ✉♠ r❡st♦ ♥✉❧♦✳ ❖ ú❧t✐♠♦ r❡st♦ ♥ã♦ ♥✉❧♦ é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ♣r♦❝✉r❛❞♦✳ ❊①❡♠♣❧♦ ✶✳✶✺✳ ❆❝❤❛r mdc(630, 22) ♦ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✳ ❚❡♠♦s✱ s✉❝❡ss✐✈❛♠❡♥t❡✿ 630 = 22 × 28 + 14 22 = 14 × 1 + 8 14 = 8 × 1 + 6 8=6×1+ ▲♦❣♦✱ 2 = mdc(630, 22)✳ 6=2×3+0 ❯s✉❛❧♠❡♥t❡ ♣r♦❝❡❞❡✲s❡ ❛ss✐♠✿ 28 1 630 22 14 14 8 6 P♦rt❛♥t♦✱ ♦ ✷ 1 8 1 3 6 2 0 ✷ mdc(630, 22) = 2✳ ❖ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s t❛♠❜é♠ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❛❝❤❛r ❡①♣r❡ssã♦ ❞♦ mdc(a, b) = rn ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s x ❡ y a ❡ b✱ ♦✉ s❡❥❛✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r t❛✐s q✉❡ mdc(a, b) = ax + by. P❛r❛ ❡♥❝♦♥tr❛r ♦s ♥ú♠❡r♦s x ❡ y ❜❛st❛ ❡❧✐♠✐♥❛r s✉❝❡ss✐✈❛♠❡♥t❡ ♦s r❡st♦s rn−1 , rn−2 , . . . , r3 , r2 , r1 ❡♥tr❡ ❛s n ♣r✐♠❡✐r❛s ✐❣✉❛❧❞❛❞❡s ❞❡ ✭✶✳✷✮✳ ✾ ✭✶✳✸✮

(21) ❊①❡♠♣❧♦ ✶✳✶✻✳ ❡ 22✳ ❆❝❤❛r ❡①♣r❡ssã♦ ❞♦ mdc(630, 22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ 630 ❈♦♠♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✶✺✿ 630 = 22 × 28 + 14 22 = 14 × 1 + 8 14 = 8 × 1 + 6 ✭✶✳✹✮ 8=6×1+2 6 = 2 × 3. ❞❡ 2 = mdc(630, 22)✳ ❆❣♦r❛✱ ♣❛r❛ ♦❜t❡r 2 = mdc(630, 22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r 630 ❡ 22 ❜❛st❛ ❡❧✐♠✐♥❛r ♦s r❡st♦s 6✱ 8 ❡ 14 ❡♥tr❡ ❛s q✉❛tr♦ ♣r✐♠❡✐r❛s ✐❣✉❛❧❞❛❞❡s ❞❡ ✭✶✳✹✮✱ ▲♦❣♦✱ ❞♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ 2 = = = = = = = 8−6×1 8 − (14 − 8 × 1) −14 + 8 × 2 −14 + 2(22 − 14 × 1) 2 × 22 − 3 × 14 2 × 22 − 3 × (630 − 28 × 22) 630(−3) + 22(86), ✐st♦ é✱ 2 = mdc(630, 22) = 630x + 22y ♦♥❞❡ x = −3 ❡ y = 86✳ ❆ r❡♣r❡s❡♥t❛çã♦ ❞♦ ✐♥t❡✐r♦ 22 2 = mdc(630, 22) 630 ❡ ♣r♦❞✉t♦ 630 × 22 ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ♥ã♦ é ú♥✐❝❛✳ ❖❜s❡r✈❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ s♦♠❛♥❞♦ ❡ s✉❜tr❛✐♥❞♦ ♦ ❛♦ s❡❣✉♥❞♦ ♠❡♠❜r♦ ❞❛ ✐❣✉❛❧❞❛❞❡✿ 2 = 630(−3) + 22(86); ♦❜t❡♠♦s✿ 2 = 630(−3 + 22) + 22(86 − 630) = 630 × 19 + 22 × (−544), q✉❡ é ✉♠❛ ♦✉tr❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ✐♥t❡✐r♦ ❞❡ 630 ❡ 2 = mdc(630, 22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r 22✳ ❉❡✜♥✐çã♦ ✶✳✶✼✳ ❉♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b s❡ ❞✐③❡♠ ♣r✐♠♦s ❡♥tr❡ s✐ s❡ mdc(a, b) = 1✳ ◆❡st❡ ❝❛s♦ ❞✐③✲s❡ t❛♠❜é♠ q✉❡ ❊①❡♠♣❧♦ ✶✳✶✽✳ a é ♣r✐♠♦ ❝♦♠ ❉♦✐s ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s ❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡ 1|a ❡ 1|(a + 1)✳ a ❆❣♦r❛✱ s❡ b a + 1 sã♦ s❡♠♣r❡ ♣r✐♠♦s c|a ❡ c|(a + 1)✱ ❡♥tã♦ ❡ c|[(a + 1) − a], ♦✉ s❡❥❛✱ c|1✳ ✶✵ ♦✉ ✈✐❝❡✲✈❡rs❛✳ ❡♥tr❡ s✐✳

(22) Pr♦♣♦s✐çã♦ ✶✳✶✾✳ ❙❡ d = mdc(a, b)✱ ❉❡♠♦♥str❛çã♦✳ ▼✉❧t✐♣❧✐q✉❡♠♦s mdc(sa, sb) = |s|d✱ ❡♥tã♦ |s| ♣♦r ❝❛❞❛ ✉♠❛ ❞❛s ♣❛r❛ t♦❞♦ ✐❣✉❛❧❞❛❞❡s ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♥♦ ♣r♦❝❡ss♦ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s q✉❡ ❧❡✈❛ ❛ ❡ |b|✿ d✱ s ∈ Z✳ ♦❜t✐❞❛s ♣❡❧♦ ❛ ♣❛rt✐r ❞❡ |a| |s||a| = (|s||b|)q1 + |s|r1 |s||b| = (|s|r1 )q2 + |s|r2 ✳ ✳ ✳ |s|rn−2 = (|s|rn−1 )qn + |s|rn |s|rn−1 = (|s|rn )qn+1 . ❆s Pr♦♣♦s✐çõ❡s ✶✳✶✸ ❡ ✶✳✶✹ ♥♦s ❣❛r❛♥t❡♠ ❡♥tã♦ q✉❡ |s|d = |s|rn = mdc(|s|rn−1 , |s|rn ) = ... = mdc(|s||b|, |s|r1 ) = mdc(|s||a|, |s||b|). P♦rt❛♥t♦✱ |s|d = mdc(|s||a|, |s||b|) = mdc(|sa|, |sb|) = mdc(sa, sb). ❈♦r♦❧ár✐♦ ✶✳✷✵✳ ❙❡ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ a, b ∈ Z \ {0} ❡ d = mdc(a, b)✱ a/d ❡ b/d sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ t❡♠♦s q✉❡ mdc (a/d, b/d) = 1✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦  a b d = mdc(a, b) = mdc d , d d d ❡ d 6= 0✱  = d · mdc  a b , d d  ❡♥tã♦✿ mdc ❈♦r♦❧ár✐♦ ✶✳✷✶✳ ❙❡❥❛♠ a, b, c ∈ Z✳ ❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡  ❙❡ a b , d d  a|bc ❡ mdc(a, b) = 1. = 1. mdc(a, b) = 1✱ ❡♥tã♦ a|c✳ ❯s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✾✱ q✉❡ mdc(ac, bc) = |c|. P♦r ❤✐♣ót❡s❡ a|bc✱ ❡ ♦❜✈✐❛♠❡♥t❡ ❈♦r♦❧ár✐♦ ✶✳✷✷✳ ❙❡❥❛♠ ❡♥tã♦ ab|c✳ ❉❡♠♦♥str❛çã♦✳ ❉❡ mdc(ac, bc) = |c|✳ a|ac✳ ❊♥tã♦✱ a, b, c ∈ Z✳ mdc(a, b) = 1 a|c P♦r ❤✐♣ót❡s❡✱ ab ❞✐✈✐❞❡ mdc(ac, bc)✳ ❡ ❞❡❝♦rr❡✱ ❡ b|c✳ ab|cb ▲♦❣♦ a ❙❡ P♦rt❛♥t♦✱ a|mdc(ac, bc)✳ b sã♦ ❞✐✈✐s♦r❡s ❞❡ ab|c✳ ✶✶ c ❡ a|c✳ mdc(a, b) = 1✱ ❡♠ ✈✐rt✉❞❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✾✱ ❆ss✐♠✱ ❡ P♦rt❛♥t♦✱ ab|ac. q✉❡

(23) ❊①❡♠♣❧♦ ✶✳✷✸✳ P❛r❛ q✉❡ ✉♠ ♥ú♠❡r♦ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r q✉❡ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r ❡ ♣♦r 2 3 ♣♦✐s ♦ é ♥❡❝❡ssár✐♦ ❡ s✉❢✉❝✐❡♥t❡ 6 mdc(2, 3) = 1✳ ❆ ❞❡✜♥✐çã♦ ❞❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ❞❡ ♠❛♥❡✐r❛ ó❜✈✐❛ ♣❛r❛ três ♦✉ ♠❛✐s ♥ú♠❡r♦s✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ três ♥ú♠❡r♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♣♦❞❡✲s❡ ❧❛♥ç❛r ♠ã♦ ❞♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿ mdc(a, b, c) = mdc(mdc(a, b), c) = mdc(a, mdc(b, c)) d = mdc(a, b, c)✳ ❊♥tã♦ d|a✱ d|b q✉❡ d|mdc(a, b)✳ ❆ss✐♠✱ Pr♦✈❡♠♦s ❛ ♣r✐♠❡✐r❛ ❞❡ss❛s ✐❣✉❛❧❞❛❞❡s✳ ❙❡❥❛ ❉❛s ❞✉❛s ♣r✐♠❡✐r❛s ❞❡ss❛s r❡❧❛çõ❡s s❡❣✉❡ d|mdc(a, b) ❙❡❥❛✱ ❛❣♦r❛✱ k ❡ d|c✳ d|c. ❡ d1 = mdc(a, b) ❡ ❞❡ c✳ ❈♦♠♦ d1 |a ❡ d1 |b✱ ♣❡❧❛ k|a✱ k|b ❡ k|c✳ ▲♦❣♦ k|d ♣♦✐s d = mdc(a, b, c)✳ ❆ ✉♠ ❞✐✈✐s♦r ❞❡ tr❛♥s✐t✐✈✐❞❛❞❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ ❞❡♠♦♥str❛çã♦ ✜❝❛ ❝♦♠♣❧❡t❛ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳ ❊①❡♠♣❧♦ ✶✳✷✹✳ ❆❝❤❡♠♦s ♦ mdc(6, 8, 20)✳ 1 3 6 2 0 8 ✷ ▲♦❣♦✱ mdc(2, 20) = 2✱ ♣♦✐s 2|20✳ ❯s❛♥❞♦ ♦ ❆❧❣♦rít♠♦ ❞❡ ❊✉❝❧✐❞❡s t❡♠♦s ❊♥tã♦✱ mdc(6, 8, 20) = 2. ✶✳✹ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ◆♦ ❝❛s♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ú♠❡r♦ b é ▼ú❧t✐♣❧♦ ❞❡ ❉❡✜♥✐çã♦ ✶✳✷✺✳ ♥ú♠❡r♦s ✐♥t❡✐r♦s ✭✶✮ a|m ❡ a ❞✐✈✐❞✐r ✉♠ ✐♥t❡✐r♦ ❡ b m é ❞✐t♦ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ❞❡ ❞♦✐s q✉❛♥❞♦✿ b|m❀ ✭✷✮ ♣❛r❛ q✉❛❧q✉❡r k > 0✱ s❡ a|k ❡ b|k, ❡♥tã♦ m|k ✳ ❖❜s❡r✈❡✲s❡ q✉❡✱ ❡♠ ❧✐♥❣✉❛❣❡♠ ❧✐t❡r❛❧✱ ❛ ❝♦♥❞✐çã♦ ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡ ♣♦s✐t✐✈♦ ❞❡ ✏♠í♥✐♠♦✑✳ ❞✐③❡♠♦s t❛♠❜é♠ q✉❡ ♦ a✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ a b a a ❡ ❞❡ q✉❛♥t♦ ❞❡ b b❀ ❡♥q✉❛♥t♦ q✉❡ ❛ ❝♦♥❞✐çã♦ é t❛♠❜é♠ ♠ú❧t✐♣❧♦ ❞❡ m ❡ n m|n ❡ n|m✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ s❡ ✭✷✮ ❞❛ ❞❡✜♥✐çã♦ t❡♠♦s (1) m✱ ❞❛ ❞❡✜♥✐çã♦ ❞✐③ q✉❡ (2) m é ❞✐③ q✉❡ t♦❞♦ ♠ú❧t✐♣❧♦ q✉❡ ❝❛r❛❝t❡r✐③❛ ❛ ♥♦♠❡♥❝❧❛t✉r❛ s❛t✐s❢❛③❡♠ ❛ ❞❡✜♥✐çã♦✱ ❡♥tã♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ▲♦❣♦✱ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ s❡ ❡①✐st✐r✱ ❞❡✈❡ s❡r ú♥✐❝♦✳ m = n ❡ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♠í♥✐♠♦ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ m = mmc(a, b) ♣❛r❛ r❡♣r❡s❡♥t❛r ♦ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡❝♦rr❡ ❞✐r❡t❛♠❡♥t❡ q✉❡ • mmc(a, b) = mmc(b, a); ✶✷

(24) • mmc(|a|, |b|) = mmc(a, b); P♦r ❡①❡♠♣❧♦✱ s❡❥❛♠ ❞❡ x ∈ Z✱ t❡♠♦s a = −6 ❡ b = 8✳ ■♥❞✐❝❛♥❞♦ ♣♦r M−6 = {· · · , −12, −6, 0, 6, 12, 18, 24, · · · } Mx ♦ ❝♦♥❥✉♥t♦ ❞♦s ♠✉❧t✐♣❧♦s M8 = {· · · , −16, −8, 0, 8, 16, 24, · · · }, ❡ ❞❡ ♠♦❞♦ q✉❡ M−6 ∩ M8 = {· · · − 72, −48, −24, 0, 24, 48, · · · }. ❆❣♦r❛ ♦❜s❡r✈❛♠♦s q✉❡✿ −6|24, 8|24❀ ✶✮ −6|n ✷✮ s❡ ❡ 8|c✱ c ❞❡✈❡ s❡r 24, 48, 72, ...✳ −6 ❡ 8✳ ❡♥tã♦ ❝♦♠✉♠ ♣♦s✐t✐✈♦ ❞❡ ▲♦❣♦✱ ♦ ◆♦ ❡♥t❛♥t♦✱ 24 é ♠❡♥♦r ♠ú❧t✐♣❧♦ mdc(−6, 8) = 24✳ ◗✉❛♥t♦ à ❡①✐stê♥❝✐❛ ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ ❝♦♥s✐❞❡r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦ ❝❛s♦ a=0 • 0|0 ❡ b|0 ❡ • 0|m′ b q✉❛❧q✉❡r✳ ❉❡✈❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡ mmc(0, b) = 0✳ ❉❡ ❢❛t♦✿ 0 = b0✳✮ ✭♥♦t❡ q✉❡ b|m′ → 0|m′ ❡ P❛r❛ ♦s ❞❡♠❛✐s ❝❛s♦s ❛ ❣❛r❛♥t✐❛ ❞❡ ❡①✐stê♥❝✐❛ é ❞❛❞❛ ♣❡❧❛ ♣r♦♣♦s✐çã♦ s❡❣✉✐♥t❡✳ Pr♦♣♦s✐çã♦ ✶✳✷✻✳ ❙❡❥❛♠ a, b ∈ Z✳ ❊♥tã♦✱ mmc(a, b) · mdc(a, b) = |ab|. ❉❡♠♦♥str❛çã♦✳ ❡ d|b✱ • t❡♠✲s❡ ❊♥tã♦✱ m = ab/d ∈ N✱ ❝♦♠♦ ❡♥tã♦ • ❱❛♠♦s ♣r✐♠❡✐r♦ ❝♦♥s✐❞❡r❛r d|ab✳ a|m✳ ❆♥❛❧♦❣❛♠❡♥t❡ s❡ a, b ∈ N ❡ d = mdc(a, b)✳ ❉❡s❞❡ a/d, b/d ∈ N✳ ❆ss✐♠✿ q✉❡ d|a ❛❧é♠ ❞❡ ab b = m, a = d d ♠♦str❛ q✉❡ b|m✳ m′ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ a ❡ ❞❡ b✳ ▲♦❣♦✱ m = bs✳ ❊♥tã♦ ar = bs ❡✱ ♣♦rt❛♥t♦✱ ❙❡❥❛ ❡①✐st❡♠ ′ r, s ∈ Z t❛✐s q✉❡ m′ = ar ❡ a b r = s. d d ❉❛í s❡❣✉❡ q✉❡ a/d ❞✐✈✐❞❡ (b/d)s✳ ❈♦♠♦ mdc (a/d, b/d) = 1✱ ✭❈♦r♦❧ár✐♦ ✶✳✷✶ ✲ Pr♦♣♦s✐çã♦ ✶✳✶✾✮✳ ❆ss✐♠✱ a s= t d ♣❛r❛ ❛❧❣✉♠ t ∈ Z✳ ❉❡s❞❡ q✉❡ m′ = bs✱ ♦❜t❡♠♦s a ab m′ = b t = t = mt, d d ♦✉ s❡❥❛✱ m|m′ . ✶✸ t❡♠♦s q✉❡ (a/d)|s

(25) P♦rt❛♥t♦✱ ♣♦r ❞❡✜♥✐çã♦✱ m = mmc(a, b)✳ P❛r❛ ♦ ❝❛s♦ a, b ∈ Z✱ ❜❛st❛ ♥♦t❛r q✉❡ mdc(a, b) = mdc(|a|, |b|) ❡ mmc(a, b) = mmc(|a|, |b|), ♣♦✐s t❡r❡♠♦s mdc(a, b) · mmc(a, b) = mdc(|a|, |b|) · mmc(|a|, |b|) = |a||b| = |ab|. ❈♦r♦❧ár✐♦ ✶✳✷✼✳ ❙❡ ❉❡♠♦♥str❛çã♦✳ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦ mmc(a, b) = |ab|✳ ❉❡ ❢❛t♦✱ ❝♦♠♦ d = mdc(a, b) = 1✱ ❡♥tã♦ mmc(a, b) = |ab|✳ ab = m ∈ Ma ∩ Mb ✳ a, b ∈ N \ {0}✳ P❡❧♦ q✉❡ ❢♦✐ ✈✐st♦✱ d ▼❛s 0 ∈ Ma ∩ Mb ❡ ❝♦♠♦ m > 0✱ ❡♥tã♦ m ♥ã♦ é ♦ ♠❡♥♦r ❞♦s ♠ú❧t✐♣❧♦s ❝♦♠✉♥s ❞❡ a ❡ b✳ ◆❛ ✈❡r❞❛❞❡✱ ♥❡st❡ ❝❛s♦ m = mmc(a, b) é ♦ ♠❡♥♦r ❞♦s ♠ú❧t✐♣❧♦s ❝♦♠✉♥s ♥ã♦ ♥✉❧♦s ❞❡ a ❡ b✳ ❖❜s❡r✈❛çã♦ ✶✳✷✽✳ ❙❡❥❛♠ ❊①❡♠♣❧♦ ✶✳✷✾✳ ❱❛♠♦s ✉s❛r ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣❛r❛ mmc(20, 8)✳ ❛❝❤❛r ❈❛❧❝✉❧❛♥❞♦ ♦ ♠❞❝✱ t❡♠♦s 2 2 20 8 ✹ . 4 0 ❊♥tã♦ mmc(20, 8) = Pr♦♣♦s✐çã♦ ✶✳✸✵✳ ❙❡ m = mmc(a, b)✱ 20 × 8 = 40. 4 ❡♥tã♦ s ∈ Z✳ mmc(sa, sb) = |s|m ♣❛r❛ q✉❛❧q✉❡r ◗✉❛♥❞♦ a = 0 ♦✉ b = 0✱ ❡♥tã♦ m = 0 ❡ sa = 0 ❡ sb = 0❀ ❞❛í mmc(sa, sb) = 0 = sm✳ ❙❡ s = 0✱ ✜❝❛♠♦s ❝♦♠ mmc(0, 0) = 0 ❡ ♦ r❡s✉❧t❛❞♦ t❛♠❜é♠ ❉❡♠♦♥str❛çã♦✳ é ✈❡r❞❛❞❡✐r♦✳ ❙✉♣♦♥❤❛♠♦s a, b ❡ s ♥ã♦ ♥✉❧♦s✳ ❊♥tã♦✱ ♣❡❧❛ ❞✉❛s ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✿ mmc(sa, sb) = |sasb| s2 |ab| |sab| = = = |s| · mmc(a, b). mdc(sa, sb) |s| · mdc(a, b) mdc(a, b) ●❡♥❡r❛❧✐③❛çã♦✿ ❆ ❡①t❡♥sã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♠ N ♣❛r❛ 3 ♦✉ ♠❛✐s ♥ú♠❡r♦s s❡ ❢❛③ ♥❛t✉r❛❧♠❡♥t❡✳ ◆♦ ❝❛s♦ ❞❡ 3 ♥ú♠❡r♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❝á❧❝✉❧♦ ♣♦❞❡ s❡r ❢❡✐t♦ ❝♦♠ ❜❛s❡ ♥❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ❝✉❥❛ ❞❡♠♦♥str❛çã♦ é ✐♠❡❞✐❛t❛✿ mmc(a, b, c) = mmc(a, mmc(b, c)) = mmc(mmc(a, b), c). P♦r ❡①❡♠♣❧♦✿ mmc(3, 5, 20) = mmc(mmc(3, 5), 20) = mmc(15, 20) = 60. ✶✹

(26) ✶✳✺ ◆ú♠❡r♦s Pr✐♠♦s ❆ ♥♦çã♦ ❞❡ ♥✉♠❡r♦ ♣r✐♠♦ ❢♦✐✱ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ✐♥tr♦❞✉③✐❞❛ ♣♦r P✐tá❣♦r❛s✱ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✺✸✵ ❆❈✳ ❆ ❡s❝♦❧❛ ♣✐t❛❣ór✐❝❛ ❞❛✈❛ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ❛♦ ♥ú♠❡r♦ ✉♠✱ q✉❡ ❡r❛ ❝❤❛♠❛❞❛ ❞❡ ✉♥✐❞❛❞❡ ✭❡♠ ❣r❡❣♦✿ ▼♦♥❛❞✮✳ ❖s ❞❡♠❛✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥❛t✉r❛✐s ✲ ♦ ✷✱ ✸✱ ✹✱ ❡t❝ ✲ t✐♥❤❛♠ ❝❛rát❡r s✉❜❛❧t❡r♥♦✱ s❡♥❞♦ ✈✐st♦s ❝♦♠♦ ♠❡r❛s ♠✉❧t✐♣❧✐❝✐❞❛❞❡s ❣❡r❛❞❛s ♣❡❧❛ ✉♥✐❞❛❞❡ ❡ ♣♦r ✐ss♦ r❡❝❡❜✐❛♠ ❛ ❞❡♥♦♠✐♥❛çã♦ ❞❡ ♥ú♠❡r♦ ✭❡♠ ❣r❡❣♦✿ ❆r✐t❤♠♦s✮✳ ❊♥tr❡t❛♥t♦✱ ❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ❛ ❣❡r❛çã♦ ❞♦s ♥ú♠❡r♦s ♥ã♦ ♣❛r❛✈❛ ♣♦r ❛í✳ P✐tá❣♦r❛s t❡r✐❛ ❛t✐♥❛❞♦ q✉❡ ❡①✐st❡♠ ❞♦✐s t✐♣♦s ❞❡ ❛r✐t❤♠ós✿ • ❖s ♣r♦t♦✐ ❛r✐t❤♠ós ✭♥ú♠❡r♦s ♣r✐♠ár✐♦s ♦✉ ♣r✐♠♦s✮✱ q✉❡ sã♦ ❛q✉❡❧❡s q✉❡ ♥ã♦ ♣♦❞❡♠ s❡r ❣❡r❛❞♦s✱ ❛tr❛✈és ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ♣♦r ♦✉tr♦s ❛r✐t❤♠ós✱ ❝♦♠♦ é ♦ ❝❛s♦ ❞❡ • 2, 3, 5, 7...✳ ❖s ❞❡✉t❡ró✐ ❛r✐t❤♠ós ✭♥ú♠❡r♦s s❡❝✉♥❞ár✐♦s✮✱ ♣♦❞❡♠ s❡r ❣❡r❛❞♦s ♣♦r ♦✉tr♦s ❛r✐t❤♠ós✱ ♣♦r ❡①❡♠♣❧♦✱ 4 = 2.2, 6 = 3.2✱ ❡t❝✳ ❆ ♥♦çã♦ ❞❡ ♣r✐♠♦ ❢♦r❛✱ ♠✉✐t♦ ♣r♦✈❛✈❡❧♠❡♥t❡✱ ✐♥tr♦❞✉③✐❞❛ ♣♦r P✐tá❣♦r❛s✳ ➱ ✐♠♣♦ssí✈❡❧ t❡r ❝♦♠♣❧❡t❛ s❡❣✉r❛♥ç❛ ♥❡ss❛ ❛tr✐❜✉✐çã♦✱ ♣♦✐s P✐tá❣♦r❛s ♥ã♦ ❞❡✐①♦✉ ♥❡♥❤✉♠ r❡❣✐str♦ ❡s❝r✐t♦ ❞❡ s❡✉s tr❛❜❛❧❤♦s ❡ ♦s ❞♦❝✉♠❡♥t♦s ♠❛✐s ❛♥t✐❣♦s q✉❡ t❡♠♦s ❢❛❧❛♥❞♦ ❞❡ s✉❛s ✐❞❡✐❛s r❡s✉♠❡♠✲s❡ ❛ ♣❡q✉❡♥♦s ❢r❛❣♠❡♥t♦s ❞❡ t❡①t♦s ❡s❝r✐t♦s ✈ár✐❛s ❣❡r❛çõ❡s ❛♣ós ❡❧❡✳ ❊♥tr❡t❛♥t♦✱ ❡ss❡s ❢r❛❣♠❡♥t♦s✱ ❛♣❡s❛r ❞❡ ❝♦♥t❡r❡♠ ✐♥❢♦r♠❛çõ❡s ♠✉✐t♦ ❡s❝❛ss❛s✱ sã♦ ✉♥â♥✐♠❡s ❡♠ ❛✜r♠❛r q✉❡ P✐tá❣♦r❛s ✐♥✐❝✐♦✉ ♦ ❡st✉❞♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳ ❖ ♠❛✐s ❛♥t✐❣♦ ❧✐✈r♦ ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ ❝❤❡❣♦✉ ❝♦♠♣❧❡t♦ ❛♦s ♥♦ss♦s ❞✐❛s ❡ q✉❡ ❞❡s❡♥✈♦❧✈❡ s✐st❡♠❛t✐❝❛♠❡♥t❡ ♦ ❡st✉❞♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s ❡ ❖s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✳ ❈♦♠♦ s❡ s❛❜❡✱ ❊✉❝❧✐❞❡s s❡❣✉✐✉ ♠✉✐t♦ ❞❡ ♣❡rt♦ ❛s ♦r✐❡♥t❛çõ❡s ♠❛t❡♠át✐❝❛s ❞♦s ♣✐t❛❣ór✐❝♦s✳ ❆ss✐♠✱ ♥ã♦ é s✉r♣r❡❡♥❞❡♥t❡ q✉❡✱ ♥♦ ❝❛♣ít✉❧♦ ❞❡ s✉❛ ♦❜r❛ ❡♠ q✉❡ tr❛t❛ ❞❛ t❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❡❧❡ ❞❡✜♥❛ ♥ú♠❡r♦ ♣r✐♠♦ ❞❡ ✉♠ ♠♦❞♦ ❛❜s♦❧✉t❛♠❡♥t❡ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛s ✐❞❡✐❛s ♣✐tá❣♦r✐❝❛s ❡①♣♦st❛s ❛❝✐♠❛✳ ❊♠ ❖s ❊❧❡♠❡♥t♦s✱ ❱♦❧✳ ❱■■✱ ❉❡✜♥✐çã♦ ✶✶✱ ❤á✿ ✏♣r♦tós ❛r✐t❤♠ós ❡st✐♥ ♠♦♥❛❞✐ ♠♦♥❡ ♠❡tr♦②♠❡♥♦s✑ ✱ q✉❡ s✐❣♥✐✜❝❛ ✏◆ú♠❡r♦ ♣r✐♠♦ é t♦❞♦ ❛q✉❡❧❡ q✉❡ só ♣♦❞❡ s❡r ♠❡❞✐❞♦ ❛tr❛✈és ❞❛ ✉♥✐❞❛❞❡✑✳ ❉❡✜♥✐çã♦ ✶✳✸✶✳ ❉✐③✲s❡ q✉❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ✉♠ ♣r✐♠♦✱ q✉❛♥❞♦ ❞✐❢❡r❡♥t❡ ❞❡ 1 |p| > 1✱ ❡ 1 ❡ |p| p é ✉♠ ◆ú♠❡r♦ Pr✐♠♦✱ ♦✉ ❛♣❡♥❛s sã♦ s❡✉s ú♥✐❝♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s✳ ❯♠ ✐♥t❡✐r♦ ❡ q✉❡ ♥ã♦ é ♣r✐♠♦ ❞✐③✲s❡ ❝♦♠♣♦st♦✳ ❚❡♦r❡♠❛ ✶✳✸✷✳ p ❙❡ ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ a✱ ♥ã♦ ❞✐✈✐❞❡ ✉♠ ✐♥t❡✐r♦ ❡♥tã♦ a ❡ p sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ✭♣r✐♠♦s ❡♥tr❡ s✐✮✳ d = mdc(a, p)✳ ❊♥tã♦ d|a ❡ d|p✳ ❉❛ r❡❧❛çã♦ d|p✱ r❡s✉❧t❛ q✉❡ d = 1 ♦✉ d = |p|✱ ♣♦rq✉❡ p é ♣r✐♠♦✳ ❈♦♠♦ d = |p| é ✐♠♣♦ssí✈❡❧✱ ♣♦rq✉❡ p ♥ã♦ ❞✐✈✐❞❡ a✱ s❡❣✉❡✲s❡ q✉❡ d = 1✱ ✐st♦ é✱ ♦ mdc(a, p) = 1✳ ▲♦❣♦✱ a ❡ p sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ ♣r✐♠♦s✳ ❈♦r♦❧ár✐♦ ✶✳✸✸✳ ❞❡ ❛♠❜♦s✱ a ❡ ❙❡ p é ✉♠ ♣r✐♠♦ t❛❧ q✉❡ p|ab✱ ❡♥tã♦ p|a ♦✉ p|b ✭♣♦❞❡♥❞♦ s❡r ❢❛t♦r b✮✳ ❉❡♠♦♥str❛çã♦✳ ❙❡ p|a✱ ♥ã♦ ❤á ♦ q✉❡ ❞❡♠♦♥str❛r✳ ❊♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ t❡♠♦s ❙✉♣♦♥❤❛ q✉❡ mdc(p, a) = 1✳ ✶✳✷✶✳ ✶✺ ▲♦❣♦✱ p|b✱ p ♥ã♦ ❞✐✈✐❞❡ a✳ ♣❡❧♦ ❈♦r♦❧ár✐♦

(27) ❈♦r♦❧ár✐♦ ✶✳✸✹✳ ❙❡ k✱ ❝♦♠ 1≤k≤n p é ✉♠ ♣r✐♠♦ t❛❧ q✉❡ t❛❧ q✉❡ p(|a1 a2 a3 . . . an )✱ ❡♥tã♦ ❡①✐st❡ ✉♠ í♥❞✐❝❡ p|ak ✳ ❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ■♥❞✉çã♦✱ ❛ ♣r♦♣♦s✐çã♦ é ✈❡r❞❛❞❡✐r❛ ♣❛r❛ ❡ ♣❛r❛ n=2 ✭♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✵✮✳ ❆ss✐♠✱ s✉♣♦♥❤❛♠♦s ✉♠ ♣r♦❞✉t♦ ❝♦♠ ♠❡♥♦s ❞❡ n ❢❛t♦r❡s✱ ❡♥tã♦ p n>2 n=1 ✭✐♠❡❞✐❛t♦✮ ❡ q✉❡✿ s❡ p ❞✐✈✐❞❡ ❞✐✈✐❞❡ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞♦s ❢❛t♦r❡s p|(a1 a2 . . . an−1 )✱ ❡♥tã♦ p|an ♦✉ p|a1 a2 . . . an−1 ✱ ❡♥tã♦ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ❛ss❡❣✉r❛ q✉❡ p|ak ✱ ❝♦♠ ❛❧❣✉♠ 1 ≤ k ≤ n − 1✳ ❊♠ q✉❛❧q✉❡r ❞♦s ❝❛s♦s✱ p ❞✐✈✐❞❡ ✉♠ ❞♦s ✐♥t❡✐r♦s a1 , a2 , a3 , . . . , an ✳ ✭❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✮✳ ▲♦❣♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✸✸✱ s❡ (p|a1 a2 . . . an−1 )✳ ❙❡ p|an ✱ ❛ ♣r♦♣♦s✐çã♦ ❡stá ❞❡♠♦♥str❛❞❛✳ P♦ré♠✱ s❡ p, q1 , q2 , . . . , qn sã♦ t♦❞♦s 1 ≤ k ≤ n✱ t❛❧ q✉❡ p = qk ✳ ❈♦r♦❧ár✐♦ ✶✳✸✺✳ ❙❡ ♦s ✐♥t❡✐r♦s ❡♥tã♦ ❡①✐st❡ ✉♠ í♥❞✐❝❡ k✱ ❝♦♠ ♣r✐♠♦s ❡ p|(q1 q2 . . . qn )✱ k ✱ ❝♦♠ 1 ≤ k ≤ n✱ t❛❧ q✉❡ p|qk ✳ ❈♦♠♦ ♦s ú♥✐❝♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ qk sã♦ 1 ❡ |qk |✱ ♣♦rq✉❡ qk é ♣r✐♠♦✱ s❡❣✉❡✲s❡ q✉❡ p = 1 ♦✉ p = qk ✳ ▼❛s✱ |p| > 1✱ ♣♦rq✉❡ p é ♣r✐♠♦✳ ▲♦❣♦✱ p = qk ✳ ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✸✹✱ ❡①✐st❡ ✉♠ í♥❞✐❝❡ ❚❡♦r❡♠❛ ✶✳✸✻✳ ❚♦❞♦ ✐♥t❡✐r♦ ❝♦♠♣♦st♦ ♣♦ss✉✐ ✉♠ ❞✐✈✐s♦r ♣r✐♠♦✳ ❉❡♠♦♥str❛çã♦✳ ❙❡❥❛ a ✉♠ ✐♥t❡✐r♦ ❝♦♠♣♦st♦✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝♦♥❥✉♥t♦ ♦s ❞✐✈✐s♦r❡s ♣♦s✐t✐✈♦s ❞❡ a✱ ❡①❝❡t♦ ♦s ❞✐✈✐s♦r❡s A = {x : x|a 1 ❡ a✱ A ❞❡ t♦❞♦s ✐st♦ é✿ 1 < x < a}. ❡ P❡❧♦ ✏Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✑ ❡①✐st❡ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ p ❞❡ A✱ q✉❡ ✈❛♠♦s p ❢♦ss❡ ❝♦♠♣♦st♦ ❛❞♠✐t✐r✐❛ ♣❡❧♦ ♠❡♥♦s ✉♠ ❞✐✈✐s♦r d 1 < d < p✱ ❡ ❡♥tã♦ d|p ❡ p|a✱ ♦ q✉❡ ✐♠♣❧✐❝❛ d|a✱ ✐st♦ é✱ p ♥ã♦ s❡r✐❛ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ ❞❡ A✱ s❡ ❢♦ss❡ ❝♦♠♣♦st♦✳ ▲♦❣♦✱ p é ♣r✐♠♦✳ ♠♦str❛r s❡r ♣r✐♠♦✳ ❉❡ ❢❛t♦✱ s❡ t❛❧ q✉❡ ❚❡♦r❡♠❛ ✶✳✸✼ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮✳ ❚♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n>1 é ✐❣✉❛❧ ❛ ✉♠ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s✳ ❉❡♠♦♥str❛çã♦✳ ▼♦str❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❛ ❢❛t♦r❛çã♦ ♣♦r ✐♥❞✉çã♦✳ ❙❡ n é ♣r✐♠♦ m = 1, p1 = n✮✳ ❙❡ n é ❝♦♠♣♦st♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r n = ab✱ a, b ∈ N, 1 < a < n, 1 < b < n✳ P♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ a ❡ b s❡ ❞❡❝♦♠♣õ❡♠ ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ♣r✐♠♦s✳ ❏✉♥t❛♥❞♦ ❛s ❢❛t♦r❛çõ❡s ❞❡ a ❡ b ✭❡ r❡♦r❞❡♥❛♥❞♦ ♦s ❢❛t♦r❡s✮ ♦❜t❡♠♦s ✉♠❛ ❢❛t♦r❛çã♦ ❞❡ n✳ ♥ã♦ ❤á ♦ q✉❡ ♣r♦✈❛r ✭❡s❝r❡✈❡♠♦s ❈♦r♦❧ár✐♦ ✶✳✸✽✳ ❆ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n > 1 ❝♦♠♦ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s é ú♥✐❝❛✱ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛ q✉❡ n = p 1 . . . p m = q 1 . . . qr p1 ≤ . . . ≤ pm ✱ q1 ≤ . . . ≤ qr ✳ ❈♦♠♦ p1 |q1 . . . qr t❡♠♦s p1 |qi ♣❛r❛ ❛❧❣✉♠ ✈❛❧♦r ❞❡ i✳ ❈♦♠♦ qi é ♣r✐♠♦✱ p1 = qi ❡ p1 ≥ q1 ✳ ❆♥❛❧♦❣❛♠❡♥t❡ t❡♠♦s q1 ≤ p1 ✱ ❞♦♥❞❡ p1 = q1 ✳ ❝♦♠✱ ▼❛s ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ n = p 2 . . . p m = q2 . . . qr p1 ❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ ❢❛t♦r❛çã♦✱ ❞♦♥❞❡ m=r ✶✻ ❡ p i = qi ♣❛r❛ t♦❞♦ i✳

(28) ❈♦r♦❧ár✐♦ ✶✳✸✾✳ ❚♦❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n > 1 ❛❞♠✐t❡ ✉♠❛ ú♥✐❝❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❛ ❢♦r♠❛ n = pk11 pk22 . . . pkr r ♦♥❞❡✱ ♣❛r❛ i = 1, 2, . . . , r ❝❛❞❛ ki é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ ❝❛❞❛ pi é ✉♠ ♣r✐♠♦✱ ❝♦♠ p1 < p2 < . . . < pr , ❞❡♥♦♠✐♥❛❞❛ ❞❡❝♦♠♣♦s✐çã♦ ❝❛♥ô♥✐❝❛ ❞♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ n > 1✳ ❉❡♠♦♥str❛çã♦✳ ❝♦♠ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✸✽✱ q1 ≤ q2 ≤ . . . ≤ qm (m ≥ 1)✳ n é ✉♠ ♣r♦❞✉t♦ ❞❡ ❢❛t♦r❡s ♣r✐♠♦s q 1 q 2 . . . qm ✱ ❆❣r✉♣❛♥❞♦✲s❡ ♦s ❢❛t♦r❡s ♣r✐♠♦s r❡♣❡t✐❞♦s ♥❛ ❢♦r♠❛ ❞❡ ♣♦tê♥❝✐❛s ❞❡ ♣r✐♠♦s✱ t❡♠♦s ❛ r❡♣r❡s❡♥t❛çã♦ ❡♥✉♥❝✐❛❞❛ ❡ t❛❧ r❡♣r❡s❡♥t❛çã♦ é ú♥✐❝❛✳ ✶✳✻ ❈♦♥❣r✉ê♥❝✐❛ ❉❡✜♥✐çã♦ ✶✳✹✵✳ ❉❛❞♦s a ❡ b ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❞✐③❡♠♦s q✉❡ a é ❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ m(m > 0) q✉❛♥❞♦ m|(a − b)✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r a ≡ b(mod m). ❙❡ m ∤ (a − b)✱ ❞✐③❡♠♦s q✉❡ a é ✐♥❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ m ❡ ❞❡♥♦t❛♠♦s a 6≡ b(mod m). ❊①❡♠♣❧♦ ✶✳✹✶✳ 11 ≡ 3(mod 2) ♣♦✐s 2|(11 − 3)✳ ❈♦♠♦ 5 ∤ 6 ❡ 6 = 17 − 11 t❡♠♦s q✉❡ 17 6≡ 11(mod 5)✳ ❖❜s❡r✈❛çã♦ ✶✳✹✷✳ ✶✳ ❉♦✐s ✐♥t❡✐r♦s q✉❛✐sq✉❡r sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ 1✳ ✷✳ ❉♦✐s ✐♥t❡✐r♦s sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ 2✱ s❡ ❛♠❜♦s sã♦ ♣❛r❡s ♦✉ ❛♠❜♦s sã♦ í♠♣❛r❡s✳ ✸✳ a ≡ 0(mod − m) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ m|a✳ Pr♦♣♦s✐çã♦ ✶✳✹✸✳ ❙❡ a ❡ b sã♦ ✐♥t❡✐r♦s✱ t❡♠♦s q✉❡ a ≡ b(mod m) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ a = b + km✳ ❉❡♠♦♥str❛çã♦✳ ✉♠ ✐♥t❡✐r♦ k t❛❧ a ≡ b(mod m)✱ ❡♥tã♦ m|(a − b) q✉❡ a − b = km✱ ✐st♦ é✱ ❙❡ ♦ q✉❡ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ a = b + km. k s❛t✐s❢❛③❡♥❞♦ a = b + km✱ a ≡ b(mod m)✳ ❆ r❡❝í♣r♦❝❛ t❛♠❜é♠ é s✐♠♣❧❡s ♣♦✐s s❡ ❡①✐st❡ km = a − b✱ ❡♥tã♦ m|(a − b) ❡✱ ♣♦rt❛♥t♦✱ t❡♠♦s Pr♦♣♦s✐çã♦ ✶✳✹✹✳ ❙❡ a, b, m ❡ d sã♦ ✐♥t❡✐r♦s✱ m > 0✱ ❛s s❡❣✉✐♥t❡s s❡♥t❡♥ç❛s sã♦ ✈❡r❞❛❞❡✐r❛s✿ ✭✐✮ a ≡ a(mod m) ✭Pr♦♣r✐❡❞❛❞❡ r❡✢❡①✐✈❛✮✱ ✶✼

(29) ✭✐✐✮ s❡ a ≡ b(mod m) ✱ ❡♥tã♦ b ≡ a(mod m) ✭ Pr♦♣r✐❡❞❛❞❡ s✐♠étr✐❝❛✮✱ ✭✐✐✐✮ s❡ a ≡ b(mod m) ❡ tr❛♥s✐t✐✈❛✮✳ ❉❡♠♦♥str❛çã♦✳ ✭✐✮ b ≡ d(mod m)✱ ❡♥tã♦ a ≡ d(mod m) ✭Pr♦♣r✐❡❞❛❞❡ ❈♦♠♦ m|0✱ ❡♥tã♦ m|(a − a)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ a ≡ a(mod m)✳ ✭✐✐✮ ❙❡ a ≡ b(mod m) ✱ ❡♥tã♦ a = b + k1 m ♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ k1 ✳ ▲♦❣♦ b = a − k1 m✱ ♦ q✉❡ ✐♠♣❧✐❝❛✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹✸✱ b ≡ a(mod m✮✳ ✭✐✐✐✮ ❙❡ a ≡ b(mod m) ❡ b ≡ d(mod m)✱ ❡♥tã♦ ❡①✐st❡♠ ✐♥t❡✐r♦s k1 ❡ k2 t❛✐s q✉❡ a − b = k1 m ❡ b − d = k2 m✳ ❙♦♠❛♥❞♦✲s❡✱ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ ❡st❛s ú❧t✐♠❛s ❡q✉❛çõ❡s✱ ♦❜t❡♠♦s a − d = (k1 + k2 )m✱ ♦ q✉❡ ✐♠♣❧✐❝❛ a ≡ d(mod m)✳ ❊st❛ ♣r♦♣♦s✐çã♦ ♥♦s ❞✐③ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ❞❡✜♥✐❞❛ ♥♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s✱ é ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✱ ♣♦✐s ❛❝❛❜❛♠♦s ❞❡ ♣r♦✈❛r q✉❡ ❡❧❛ é r❡✢❡①✐✈❛✱ s✐♠étr✐❝❛ ❡ tr❛♥s✐t✐✈❛✳ ❚❡♦r❡♠❛ ✶✳✹✺✳ ❙❡ a, b, c, d ❡ m sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ a ≡ c ≡ d(mod m)✱ b(mod m) ❡ ❡♥tã♦ ✭✐✮ a + c ≡ b + d(mod m)✳ ✭✐✐✮ a − c ≡ b − d(mod m) ✭✐✐✐✮ ac ≡ bd(mod m)✳ ❉❡ a ≡ b(mod m) ❡ c ≡ d(mod m) t❡♠♦s a − b = km ❡ c − d = k1 m✳ ❙♦♠❛♥❞♦✲s❡ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ♦❜t❡♠♦s ❉❡♠♦♥str❛çã♦✳ ✭✐✮ (a + c) − (b + d) = (k + k1 )m ❡ ✐ss♦ ✐♠♣❧✐❝❛ a + c ≡ b + d(mod m)✳ ✭✐✐✮ ❇❛st❛ s✉❜tr❛✐r ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ a − b = km ❡ c − d = k1 m ♦❜t❡♠♦s (a − b) − (c − d) = (a − c) − (b − d) = (k − k1 )m ♦ q✉❡ ✐♠♣❧✐❝❛ a − c ≡ b − d(mod m)✳ ✭✐✐✐✮ ▼✉❧t✐♣❧✐❝❛♠♦s ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ a − b = km ♣♦r c ❡ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ c − d = k1 m ♣♦r b✱ ♦❜t❡♥❞♦ ❡ ac − bc = ckm bc − bd = bk1 m. ❇❛st❛✱ ❛❣♦r❛✱ s♦♠❛r ♠❡♠❜r♦ ❛ ♠❡♠❜r♦ ❛s ú❧t✐♠❛s ✐❣✉❛❧❞❛❞❡s ♦❜t❡♥❞♦ ac − bc + bc − bd = ac − bd = (ck + bk1 )m, ♦ q✉❡ ✐♠♣❧✐❝❛ ac ≡ bd(mod m)✳ ✶✽

(30) ❚❡♦r❡♠❛ ✶✳✹✻✳ ❙❡ a, b, c ❡ m sã♦ ✐♥t❡✐r♦s ❡ ac ≡ a ≡ b(mod m/d) ♦♥❞❡ d = mdc(c, m) 6= 0✳ ❉❡♠♦♥str❛çã♦✳ ❉❛ ❤✐♣ót❡s❡ ac ≡ bc(mod m)✱ bc(mod m)✱ ❡♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ ac − bc = c(a − b) = km. ❙❡ ❞✐✈✐❞✐r♠♦s ♦s ❞♦✐s ♠❡♠❜r♦s ♣♦r d✱ t❡r❡♠♦s (c/d)(a − b) = k(m/d)✳ ▲♦❣♦ (m/d)|(c/d)(a − b). P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✵✱ t❡♠✲s❡ (m/d)|(a − b)✱ mdc(m/d, c/d) = 1✳ ❊♥tã♦✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✶✱ ♦ q✉❡ ✐♠♣❧✐❝❛ a ≡ b(mod m/d). ❉❡✜♥✐çã♦ ✶✳✹✼ ✭■♥✈❡rs♦ ❛❞✐t✐✈♦✮✳ ❉❛❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a, b ❡ m > 0✱ ❞✐③✲s❡ q✉❡ b é ✉♠ ■♥✈❡rs♦ ❆❞✐t✐✈♦ ❞❡ a ♠ó❞✉❧♦ m q✉❛♥❞♦ a + b ≡ 0 mod m✳ ❉❡✜♥✐çã♦ ✶✳✹✽ ✳ ❉❛❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a, b ❡ m > 0✱ ✭■♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮ ❞✐③✲s❡ q✉❡ b é ✉♠ ■♥✈❡rs♦ ▼✉❧t✐♣❧✐❝❛t✐✈♦ ❞❡ a ♠ó❞✉❧♦ m q✉❛♥❞♦ ab ≡ 1 mod m✳ ❖❜s❡r✈❛çã♦ ✶✳✹✾✳ ◆♦t❡ q✉❡✱ s❡ mdc(a, m) = 1✱ ❡♥t❛♦ a ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♠ó❞✉❧♦ m✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ ab ≡ 1 mod m ❡ ac ≡ 1 mod m. ❊♥tã♦✱ ❡①✐t❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s k1 ❡ k2 t❛✐s q✉❡ ab − 1 = k1 m ❡ ac − 1 = k2 m. ▲♦❣♦ ab − ac = (k1 + k2 )m, ❞❡ ♠♦❞♦ q✉❡ m|a(b − c)✳ ❆❣♦r❛✱ ❝♦♠♦ ♣♦r ❤✐♣ót❡s❡ mdc(a, m) = 1✱ ❛♣❧✐❝❛♥❞♦ ♦ ❈♦r♦❧ár✐♦ ✶✳✷✶✱ ♦❜t❡♠♦s q✉❡ m|(b − c)✱ q✉❡ s✐❣♥✐✜❝❛ b ≡ c mod m ✭✐st♦ é✱ b ❡ c sã♦ ✐❣✉❛✐s ♠ó❞✉❧♦ m✮✳ ❉❡✜♥✐çã♦ ✶✳✺✵✳ ❙❡ h ❡ k sã♦ ❞♦✐s ✐♥t❡✐r♦s ❝♦♠ h ≡ k(mod m)✱ ❞✐③❡♠♦s q✉❡ k é ✉♠ r❡sí❞✉♦ ❞❡ h ♠ó❞✉❧♦ m✳ ❉❡✜♥✐çã♦ ✶✳✺✶✳ ❯♠ ❝♦♥❥✉♥t♦ ❞❡ ✐♥t❡✐r♦s {r1 , r2 , . . . , rs } é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m q✉❛♥❞♦✿ ✶✳ ri 6≡ rj (mod m) ♣❛r❛ i 6= j ✱ ✷✳ ♣❛r❛ t♦❞♦ ✐♥t❡✐r♦ n✱ ❡①✐st❡ ✉♠ ri t❛❧ q✉❡ n ≡ ri (mod m)✳ ❊①❡♠♣❧♦ ✶✳✺✷✳ ❖ ❝♦♥❥✉♥t♦ {0, 1, 2, . . . , m − 1} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ ❙♦❧✉çã♦✿ ❈♦♠ ❡❢❡✐t♦✿ ✶✾

(31) ✶✳ ❙❡❥❛♠ i, j ∈ {0, 1, 2, . . . , m − 1}✱ ❞✐❣❛♠♦s q✉❡ ❝♦♠ i < j ✳ ❚❡♠♦s q✉❡ j 6≡ i (mod m) ♣♦✐s✱ s❡ j ≡ i (mod m)✱ ❡♥tã♦ ♣❡❧❛ ❞❡✜♥✐çã♦ ❡①✐st✐r✐❛ ✉♠ ♥ú♠❡r♦ k t❛❧ q✉❡ j − i = km ✭♥♦t❡ q✉❡ k > 0✮✱ q✉❡ ✐♠♣❧✐❝❛r✐❛ ❞✐③❡r q✉❡ j ≥ m✳ ✷✳ ❙❡❥❛ n ∈ Z✳ P❡❧♦ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s ✭❈♦r♦❧ár✐♦ ✶✳✾✮✱ ❡①✐st❡♠ q, r ∈ Z✱ ❝♦♠ 0 ≤ r < m t❛❧ q✉❡ n = qm + r✳ ▲♦❣♦✱ n ≡ r (mod m)✳ P♦rt❛♥t♦✱ {0, 1, 2, . . . , m − 1} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ ❚❡♦r❡♠❛ ✶✳✺✸✳ ❙❡ k ✐♥t❡✐r♦s r1 , r2 , . . . , rk ❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m ❡♥tã♦ k = m✳ ❉❡♠♦♥str❛çã♦✳ P❡❧♦ ❊①❡♠♣❧♦ ✶✳✺✷✱ ♦ ❝♦♥❥✉♥t♦ {0, 1, 2, . . . , m − 1} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♥❝❧✉í♠♦s q✉❡ ❝❛❞❛ ri é ❝♦♥❣r✉❡♥t❡ ❛ ❡①❛t❛♠❡♥t❡ ✉♠ ❞♦s i✱ ♦ q✉❡ ♥♦s ❣❛r❛♥t❡ k ≤ m✳ ❈♦♠♦ ♦ ❝♦♥❥✉♥t♦ r1 , r2 , . . . , rk ❢♦r♠❛✱ ♣♦r ❤✐♣ót❡s❡✱ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✱ ❝❛❞❛ i é ❝♦♥❣r✉❡♥t❡ ❛ ❡①❛t❛♠❡♥t❡ ✉♠ ❞♦s ri ❡ ♣♦rt❛♥t♦ m ≤ k ✳ ❆ss✐♠✱ k = m✳ ❙❡ r1 , r2 , . . . , rm é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m ❡ a ❡ b sã♦ ✐♥t❡✐r♦s ❝♦♠ mdc(a, m) = 1✱ ❡♥tã♦ ❚❡♦r❡♠❛ ✶✳✺✹✳ ar1 + b, ar2 + b, . . . , arm + b t❛♠❜é♠ é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ♦ r❡s✉❧t❛❞♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✱ s❡rá s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ q✉❛✐sq✉❡r ❞♦✐s ✐♥t❡✐r♦s ❞♦ ❝♦♥❥✉♥t♦ ar1 + b, ar2 + b, . . . , arm + b✱ sã♦ ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳ P❛r❛ t❛♥t♦✱ s✉♣♦♥❤❛ q✉❡ ari + b ≡ arj + b (mod m)✳ ❆ss✐♠✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✺✱ t❡♠♦s ari ≡ arj (mod m)✳ ▼❛s✱ ❝♦♠♦ mdc(a, m) = 1✱ ♦ ❚❡♦r❡♠❛ ✶✳✹✻ ♥♦s ❞✐③ ri ≡ rj (mod m)✳ ❖ ❢❛t♦ ❞❡ ri ≡ rj (mod m) ✐♠♣❧✐❝❛ i = j ✱ ✉♠❛ ✈❡③ q✉❡✱ r1 , r2 , . . . , rm ❢♦r♠❛♠ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✱ ♦ q✉❡ ❝♦♠♣❧❡t❛ ❛ ❞❡♠♦♥str❛çã♦✳ Pr♦♣♦s✐çã♦ ✶✳✺✺✳ a ≡ b (mod m)✳ k k ❙❡ a✱ b✱ k ❡ m sã♦ ✐♥t❡✐r♦s ❝♦♠ k > 0 ❡ a ≡ b (mod m)✱ ❡♥tã♦ ❉❡♠♦♥str❛çã♦✳ ❙❡ a ≡ b (mod m)✱ ❡♥tã♦ m|(a − b)✳ ❈♦♠♦ ak − bk = (a − b)(ak−1 + ak−2 b + ak−3 b2 + . . . + abk−2 + bk−1 ), t❡♠✲s❡ q✉❡ m|(ak − bk )✱ ❡ ♣♦rt❛♥t♦ ak ≡ bk (mod m)✳ Pr♦♣♦s✐çã♦ ✶✳✺✻✳ ✭✐✮ ❙❡❥❛♠ a, b ∈ Z✱ m, n, m1 , . . . , mr ∈ N \ {0, 1}✳ ❚❡♠♦s q✉❡✿ s❡ a ≡ b (mod m) ❡ n|m✱ ❡♥tã♦ a ≡ b (mod n)❀ ✭✐✐✮ a ≡ b (mod mi ), i = 1, . . . , r ⇐⇒ a ≡ b (mod mmc(m1 , . . . , mr ))❀ ✭✐✐✐✮ s❡ a ≡ b (mod m)✱ ❡♥tã♦ mdc(a, m) = mdc(b, m)✳ ❉❡♠♦♥str❛çã♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ b ≥ a❀ ❝❛s♦ ❢♦ss❡ a ≥ b ❛r❣✉♠❡♥t❛♠♦s ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦✳ ✷✵

(32) ✭✐✮ ❙❡ a ≡ b (mod m)✱ ❡♥tã♦ m|(b − a)✳ ❈♦♠♦ n|m✱ s❡❣✉❡✲s❡ q✉❡ n|(b − a)✳ ▲♦❣♦✱ a ≡ b (mod n)✳ ✭✐✐✮ ❙❡ a ≡ b (mod mi ), i = 1, . . . , r✱ ❡♥tã♦ mi |(b − a)✱ ♣❛r❛ t♦❞♦ i✳ ❙❡♥❞♦ b − a ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ❝❛❞❛ mi ✱ s❡❣✉❡✲s❡ q✉❡ mmc(m1 , . . . , mr )|(b − a) ♦ q✉❡ ♣r♦✈❛ q✉❡ a ≡ b (mod mmc(m1 , . . . , mr ))✳ ❆ r❡❝í♣r♦❝❛ ❞❡❝♦rr❡ ❞♦ ít❡♠ ✭✐✮✳ ✭✐✐✐✮ ❙❡ a ≡ b (mod m)✱ ❡♥tã♦ m|(b − a) ❡✱ ♣♦rt❛♥t♦✱ b = a + tm ❝♦♠ t ∈ N✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✶✹✱ t❡♠♦s q✉❡✸ mdc(b, m) = mdc(a + tm, m) = mdc(a, m)✳ ❚❡♦r❡♠❛ ✶✳✺✼ ✭P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✮✳ ❙❡❥❛ p ♣r✐♠♦✳ ❙❡ p∤a ❡♥tã♦ ap−1 ≡ 1 mod p. ❙❛❜❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s p ♥ú♠❡r♦s 0, 1, 2, . . . , p − 1 ❝♦♥st✐t✉✐ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ p✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❝♦♥t❡♥❞♦ ♥♦ ♠á①✐♠♦ p ❡❧❡♠❡♥t♦s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p ♣♦❞❡ s❡r ❝♦❧♦❝❛❞♦ ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❜✐✉♥í✈♦❝❛ ❝♦♠ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ 0, 1, 2, . . . , p − 1✳ ❱❛♠♦s✱ ❛❣♦r❛✱ ❝♦♥s✐❞❡r❛r ♦s ♥ú♠❡r♦s a, 2a, 3a, . . . , (p − 1)a✳ ❈♦♠♦ mdc(a, p) = 1✱ ♥❡♥❤✉♠ ❞❡st❡s ♥ú♠❡r♦s ia✱ 1 ≤ i ≤ p−1 é ❞✐✈✐sí✈❡❧ ♣♦r p✱ ♦✉ s❡❥❛✱ ♥❡♥❤✉♠ é ❝♦♥❣r✉❡♥t❡ ❛ ③❡r♦ ♠ó❞✉❧♦ p✳ ◗✉❛✐sq✉❡r ❞♦✐s ❞❡❧❡s sã♦ ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p✱ ♣♦✐s aj ≡ ak (mod p) ✐♠♣❧✐❝❛ j ≡ k (mod p) ❡ ✐st♦ só é ♣♦ssí✈❡❧ s❡ j = k ✱ ✉♠❛ ✈❡③ q✉❡ ❛♠❜♦s j ❡ k sã♦ ♣♦s✐t✐✈♦s ❡ ♠❡♥♦r❡s ❞♦ q✉❡ p✳ ❚❡♠♦s✱ ♣♦rt❛♥t♦✱ ✉♠ ❝♦♥❥✉♥t♦ {a, 2a, 3a, . . . , (p−1)a} ❞❡ p − 1 ❡❧❡♠❡♥t♦s ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ p ❡ ♥ã♦✲❞✐✈✐sí✈❡✐s ♣♦r p✳ ▲♦❣♦✱ ❝❛❞❛ ✉♠ ❞❡❧❡s é ❝♦♥❣r✉❡♥t❡ ❛ ❡①❛t❛♠❡♥t❡ ✉♠ ❞❡♥tr❡ ♦s ❡❧❡♠❡♥t♦s 1, 2, 3, . . . , p − 1✳ ❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❡st❛s ❝♦♥❣r✉ê♥❝✐❛s✱ ♠❡♠❜r♦ ❛ ♠❡♠❜r♦✱ t❡r❡♠♦s ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✺ ❉❡♠♦♥str❛çã♦✳ (1a)(2a)(3a) · · · (p − 1)a ≡ 1 · 2 · 3 · · · (p − 1)(mod p), ♦✉ s❡❥❛ ap−1 (p − 1)! ≡ (p − 1)! (mod p). ▼❛s✱ ❝♦♠♦ mdc((p − 1)!, p) = 1✱ ♣♦❞❡♠♦s ❝❛♥❝❡❧❛r ♦ ❢❛t♦r (p − 1)! ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s✱ ♦❜t❡♥❞♦ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳ ❈♦r♦❧ár✐♦ ✶✳✺✽✳ ❙❡ p ap−1 ≡ 1 (mod p), é ✉♠ ♣r✐♠♦ ❡ a é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦ ap ≡ a (mod p). ❉❡♠♦♥str❛çã♦✳ ❚❡♠♦s q✉❡ ❛♥❛❧✐s❛r ❞♦✐s ❝❛s♦s✱ s❡ p|a ❡ s❡ p ∤ a✳ ❙❡ p|a✱ ❡♥tã♦ p|(a(ap−1 − 1)) ✱ ♣♦rt❛♥t♦ ap ≡ a(mod p)✳ ❙❡ p ∤ a✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✺✼ p|(ap−1 − 1) ❡✱ ♣♦rt❛♥t♦✱ p|(ap − a)✳ ▲♦❣♦✱ ❡♠ ❛♠❜♦s ♦s ❝❛s♦s✱ ap ≡ a (mod − p)✳ ✸❖ r❡s✉❧t❛❞♦ t❛♠❜é♠ s❡❣✉❡ ❞✐r❡t♦ ❞♦ ▲❊▼❆ a < na < b✳ ❙❡ ❡①✐st❡ mdc(a, b − na)✱ ❡♥tã♦ ✷✶ a, b, n ∈ N mdc(a, b) = mdc(a, b − na)✳ ❉❊ ❊❯❈▲■❉❊❙✿ ❙❡❥❛♠ mdc(a, b) ❡①✐st❡ ❡ ❝♦♠

(33) ❊①❡♠♣❧♦ ✶✳✺✾✳ ❯s❛♥❞♦ ♦ P❡q✉❡♥♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ❡♥❝♦♥tr❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 2100000 ♣♦r 17✳ p−1 ❙♦❧✉çã♦✿ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t t❡♠♦s a ≡ 1(mod p) q✉❛♥❞♦ p é ♣r✐♠♦ ❡ p ∤ a✳ ▲♦❣♦✱ ❝♦♠♦ 17 é ♣r✐♠♦ ❡ 17 ∤ 2✱ t❡♠♦s 216 ≡ 1(mod 17)✳ ▼❛s 100000 = 6250 × 16 ❡✱ ♣♦rt❛♥t♦✱ 2100000 = (216 )6250 ≡ 16250 ≡ 1 (mod 17). ▲♦❣♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♣♦r 17 ❞❡ 2100000 é 1✳ ❉❡✜♥✐çã♦ ✶✳✻✵✳ ❙❡ n é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ ❛ ❢✉♥çã♦ φ ❞❡ ❊✉❧❡r✱ ❞❡♥♦t❛❞❛ ♣♦r φ(n)✱ é ❞❡✜♥✐❞❛ ❝♦♠♦ s❡♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ♠❡♥♦r❡s ❞♦ q✉❡ ♦✉ ✐❣✉❛✐s ❛ n q✉❡ sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ n✳ ❊①❡♠♣❧♦ ✶✳✻✶✳ P❡❧❛ ❞❡✜♥✐çã♦ t❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ φ(8) = 4 ❡ φ(p) = p (p − ♣r✐♠♦), ♣♦✐s 1, 3, 5, 7 sã♦ r❡❛❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ 8 ❡ 1, 2, . . . , p − 1 sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ p✳ ❉❡✜♥✐çã♦ ✶✳✻✷✳ ❯♠ ❙✐st❡♠❛ ❘❡❞✉③✐❞♦ ❞❡ ❘❡sí❞✉♦s ♠ó❞✉❧♦ m é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ φ(m) ✐♥t❡✐r♦s r1 , r2 , . . . , rφ(m) ✱ t❛✐s q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞♦ ❝♦♥❥✉♥t♦ é r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦ ❝♦♠ m✱ ❡ s❡ i 6= j ✱ ❡♥tã♦ ri 6≡ rj (mod m)✳ ❊①❡♠♣❧♦ ✶✳✻✸✳ ❖ ❝♦♥❥✉♥t♦ {0, 1, 2, 3, 4, 5, 6, 7} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ 8✱ ♣♦rt❛♥t♦ {1, 3, 5, 7} é ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ 8✳ ❆ ✜♠ ❞❡ s❡ ♦❜t❡r ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ♠ó❞✉❧♦ m✱ ❜❛st❛ r❡t✐r❛r ♦s ❡❧❡♠❡♥t♦s ❞♦ s✐st❡♠❛ ❝♦♠♣❧❡t♦ q✉❡ ♥ã♦ sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ m✳ ❚❡♦r❡♠❛ ✶✳✻✹✳ ❙❡❥❛ a ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ mdc(a, m) = 1✳ ❙❡ r1 , r2 , . . . , rφ(m) é ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✱ ❡♥tã♦ ar1 , ar2 , . . . , arφ(m) é✱ t❛♠❜é♠✱ ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ ❉❡♠♦♥str❛çã♦✳ ar1 , ar2 , . . . , arφ(m) ✱ φ(m) ❡❧❡♠❡♥t♦s✳ ❉❡✈❡♠♦s ♠♦str❛r m ❡✱ ❞♦✐s ❛ ❞♦✐s✱ ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳ P♦r ❤✐♣ót❡s❡ mdc(a, m) = 1 ❡ mdc(ri , m) = 1✳ ❙❡❥❛ d = mdc(ari , m)✳ ❆ss✐♠✱ d|ari ❡ d|m✳ ❈♦♠♦ mdc(a, m) = 1✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✶✱ t❡♠✲s❡ q✉❡ d|ri ✳ ▼❛s ❛ss✐♠✱ d|m ❡ d|ri ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ d = 1✱ ✐st♦ é✱ mdc(ari , m) = 1✳ ▲♦❣♦✱ ♥♦s r❡st❛ ♠♦str❛r q✉❡ ari 6≡ arj (mod m) s❡ i 6= j ✳ ▼❛s✱ ❝♦♠♦ mdc(a, m) = 1✱ s❡ ari ≡ arj (mod m)✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✻✱ t❡♠♦s ri ≡ rj (mod m)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ i = j ✱ ✉♠❛ ✈❡③ q✉❡ r1 , r2 , . . . , rφ(m) ✱ é ✉♠ s✐st❡♠❛ r❡❞✉③✐❞♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳ P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s ❈♦♥s✐❞❡r❡ q✉❡ t♦❞♦s ❡❧❡s sã♦ r❡❧❛t✐✈❛♠❡♥t❡ ♣r✐♠♦s ❝♦♠ ❛ ❞❡♠♦♥str❛çã♦✳ ❚❡♦r❡♠❛ ✶✳✻✺ ✭❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✮✳ ❙❡ m é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ a ✉♠ ✐♥t❡✐r♦ ❝♦♠ mdc(a, m) = 1✱ ❡♥tã♦ ❉❡♠♦♥str❛çã♦✳ aϕ(m) ≡ 1 mod m. ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ t❡♦r❡♠❛ r❡q✉❡r ❛r❣✉♠❡♥t♦s ♠❛✐s ❡❧❛❜♦r❛❞❛s ❡ ♣r❡❢❡r✐♠♦s ♥ã♦ ❛♣r❡s❡♥tá✲❧❛ ❛q✉✐✳ ❯♠❛ ❞❡♠♦♥str❛çã♦ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ♣♦r ❡①❡♠♣❧♦ ❡♠ ❬✶✹✱ ♣á❣✐♥❛ ✹✸❪✳ ✷✷

(34) ❖❜s❡r✈❡ q✉❡ ♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ❜❛st❛ s✉♣♦r q✉❡ m = p✳ P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s q✉❡ ϕ(p) = p − 1✳ P♦rt❛♥t♦✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♥♦s ❢♦r♥❡❝❡ ❛ ♠❡s♠❛ ❛✜r♠❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t✱ ♥❡ss❡ ❝❛s♦✳ ❊①❡♠♣❧♦ ✶✳✻✻✳ ❙♦❧✉çã♦✿ ❈♦♠♦ ▼♦str❛r q✉❡ 2✱ 3✱ 45✱ 7 13 n − n = n(n12 − 1)✱ ❡ 13 sã♦ ❞✐✈✐s♦r❡s ❞❡ n13 − n ♣❛r❛ t♦❞♦ n✳ n12 − 1 = (n − 1)(n11 + n10 + . . . + n + 1)✱ n12 − 1 = (n2 − 1)(n10 + n8 + . . . + n2 + 1)✱ n12 − 1 = (n4 − 1)(n8 + n4 + 1) ❡ n12 − 1 = (n6 − 1)(n6 + 1)✱ n, (n − 1), (n2 − 1), (n4 − 1), (n6 − 1) ❡ (n12 − 1) sã♦ 2|(n13 − n) ♣♦✐s n(n − 1) é ♣❛r✳ ❆❣♦r❛✱ ❝❛s♦ n ♥ã♦ s❡❥❛ n13 − n✳ ♣♦r 3, 5, 7 ❡ t❡♠♦s q✉❡ ❞✐✈✐s♦r❡s ❞❡ ▲♦❣♦✱ ❞✐✈✐sí✈❡❧ 13 t❡r❡♠♦s q✉❡✿ 3|(n13 − n) 5|(n13 − n) 7|(n13 − n) ♣♦✐s ♣♦✐s ♣♦✐s n2 ≡ 1(mod 3) n4 ≡ 1(mod 5) n6 ≡ 1(mod 7) ✭❊✉❧❡r✮✱ ✭❊✉❧❡r✮✱ ✭❊✉❧❡r✮ ❡ 13|(n13 − n) ✶✳✼ ♣♦✐s n12 ≡ 1(mod 13) ✭❊✉❧❡r✮✳ ❈♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s ❉❡✜♥✐çã♦ ✶✳✻✼✳ ❯♠❛ ❝♦♥❣r✉ê♥❝✐❛ ❛❧❣é❜r✐❝❛ ❞♦ t✐♣♦ ax ≡ b(mod m) ♦♥❞❡ a✱ b✱ m ∈ Z✱ a 6= 0 ❡ m > 0✱ ❡ x é ✉♠❛ ✈❛r✐á✈❡❧ ❡♠ Z✱ r❡❝❡❜❡ ♦ ♥♦♠❡ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r ♦✉ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉✳ ❯♠❛ s♦❧✉çã♦ ❞❡ ax ≡ b(mod m) é ✉♠ u ✐♥t❡✐r♦ t❛❧ q✉❡ au ≡ b(mod m). ❆♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♣❛r❛ u ❡ m✱ t❡♠♦s q✉❡ ❡①✐st❡♠ ✐♥t❡✐r♦s q ❡ x0 (0 ≤ x0 < m) t❛✐s q✉❡ u = mq + x0 . ❆ss✐♠✱ au = amq + ax0 ♦✉ au − ax0 = amq ✳ ❉❡s❞❡ q✉❡ amq ≡ 0(mod m)✱ s❡❣✉❡ q✉❡ (au − ax0 ) ≡ 0(mod m)✱ ❡♥tã♦ ax0 ≡ au(mod m)✳ ▲♦❣♦✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✹✹✱ t❡♠✲s❡ ax0 ≡ b(mod m), ♦ q✉❡ ♠♦str❛ q✉❡ x0 t❛♠❜é♠ é s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❝♦♥s✐❞❡r❛❞❛✳ ❈♦♥✈❡♥❝✐♦♥❛r❡♠♦s q✉❡ t♦❞♦s ♦s x ∈ Z t❛✐s q✉❡ x ≡ x0 (mod m) ❝♦♥st✐t✉❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ ❞❡ ax ≡ b(mod m)✳ ✷✸

(35) ❊①❡♠♣❧♦ ✶✳✻✽✳ P♦r ❡①❡♠♣❧♦✱ ❝♦♠♦ 4 é s♦❧✉çã♦ ❞❡ ❡❧❡♠❡♥t♦s ❞❡ 2x ≡ 3(mod 5)✱ ❡♥tã♦ t♦❞♦s ♦s {4 + 5t : t ∈ Z} = {. . . − 6, −1, 4, 9, . . .} sã♦ ❛♣❡♥❛s r❡♣r❡s❡♥t❛çõ❡s ❞❛ ♠❡s♠❛ s♦❧✉çã♦✳ ax ≡ b(mod m)✱ ♦♥❞❡ a 6= 0✱ ❛❞♠✐t❡ s♦❧✉çõ❡s ❡♠ Z s❡✱ ❡ s♦♠❡♥t❡ s❡✱ b é ❞✐✈✐sí✈❡❧ ♣♦r d = mdc(a, m)✳ ◆❡st❡ ❝❛s♦✱ s❡ x0 é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦❧✉çõ❡s t❡♠ d ❡❧❡♠❡♥t♦s✱ ❛ Pr♦♣♦s✐çã♦ ✶✳✻✾✳ ❯♠❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r s❛❜❡r✿ x0 , x0 + m m m , x0 + 2 , . . . , x0 + (d − 1) . d d d (⇒) ❙✉♣♦♥❤❛ q✉❡ x0 s❡❥❛ ✉♠❛ s♦❧✉çã♦ ❞❡ ax ≡ b(mod m)✳ ❊♥tã♦ ax0 − my0 = b✱ ♣❛r❛ ❛❧❣✉♠ y0 ∈ Z✳ ❉❡s❞❡ q✉❡ d|ax0 ❡ d|my0 ✱ ❧♦❣♦ d|b✳ (⇐) ❆❣♦r❛ ✈❛♠♦s d|b✳ ❊♥tã♦✱ ❡①✐st❡ t ∈ Z t❛❧ q✉❡ b = tb✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡ ✭✶✳✸✮✱ s❛❜❡♠♦s q✉❡ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r x1 ❡ y1 t❛✐s q✉❡ d = ax1 + my1 . ❆ss✐♠✱ ❉❡♠♦♥str❛çã♦✳ b = td = a(tx1 ) + m(ty1 ). ❊♥tã♦✱ m|(a(tx1 ) − b)✱ ♦✉ s❡❥❛✱ tx1 é s♦❧✉çã♦ ❞❡ ax ≡ b(mod m)✳ ❈♦♥s✐❞❡r❛♥❞♦ ❣❛r❛♥t✐❞❛ ❛ ❡①✐stê♥❝✐❛ ❞❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r x0 ❞❡ ax ≡ b(mod m)✱ ✈❛♠♦s s✉♣♦r q✉❡ x s❡❥❛ ✉♠❛ ♦✉tr❛ s♦❧✉çã♦ q✉❛❧q✉❡r✳ ❊♥tã♦✱ ax0 ≡ b(mod m) ❡ ax ≡ b(mod m). ▲♦❣♦✱ a(x − x0 ) ≡ 0(mod m)✳ ❊♥tã♦✱ ❡①✐t❡ l ∈ Z t❛❧ q✉❡ a(x − x0 ) = lm✳ ❈♦♥s✐❞❡r❛♥❞♦ ❛❣♦r❛ r = a/d ❡ s = m/d✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✵✱ t❡♠♦s mdc(r, s) = 1✳ ▲♦❣♦✱ a(x − x0 ) = lm ⇒ rd(x − x0 ) = lsd ⇒ r(x − x0 ) = ls. ❈♦♠♦ mdc(r, s) = 1✱ ♣❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✶✱ ❝♦♥❝❧✉í♠♦s q✉❡ s|(x − x0 )✱ q✉❡ ♣♦r ❞❡✜♥✐çã♦✱ ❡①✐st❡ t ∈ Z t❛❧ q✉❡ x − x0 = ts✱ ♦✉ s❡❥❛✱ x = x0 + t m . d ❆♣❧✐❝❛♥❞♦ ♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♣❛r❛ t ❡ d✱ t❡♠✲s❡ t = dq + r (0 ≤ r < d)✳ ❆ss✐♠✱ x0 + ♦♥❞❡ 0 ≤ t1 < t2 < d✱ ❡♥tã♦ m m t1 ≡ x0 + t2 (mod m) d d m m t1 ≡ t2 (mod m) d d ❡ ❝♦♠♦ mdc(m/d, m) = m/d✱ ❧❡✈❛ ❛ ❝♦♥❝❧✉✐r q✉❡ t1 ≡ t2 (mod m) ♦ q✉❡ é ✐♠♣♦ssí✈❡❧✳ P♦rt❛♥t♦✱ ❛s s♦❧✉çõ❡s ❞♦ ❡♥✉♥❝✐❛❞♦✱ s❡♥❞♦ ✐♥❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✱ sã♦ t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ ax ≡ b(mod m)✱ ❝♦♥❢♦r♠❡ ❝♦♥✈❡♥çõ❡s ❢❡✐t❛ ❛♣ós ❛ ❉❡✜♥✐çã♦ ✶✳✻✼✳ ✷✹

(36) ❊①❡♠♣❧♦ ✶✳✼✵✳ ❙❡ ❡♠ ax ≡ b(mod m) s❡ t❡♠ mdc(a, m) = 1✱ ❡♥tã♦ ❡ss❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r só ❛❞♠✐t❡ ✉♠❛ s♦❧✉çã♦✳ ➱ ♦ ❝❛s♦ ❞❡ 3x ≡ 1(mod 5) ❝✉❥♦ ❝♦♥❥✉♥t♦ s♦❧✉çã♦ é {2}✳ 6x ≡ 15(mod 21) ❛❞♠✐t❡ 6 ❝♦♠♦ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r✳ 21 21 , 6 + 2 } = {6, 13, 20}✳ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s é {6, 6 + 3 3 ❊①❡♠♣❧♦ ✶✳✼✶✳ ❆ ❝♦♥❣r✉ê♥❝✐❛ ❈♦♠♦ ✶✳✽ mdc(6, 21) = 3✱ ♦ ❙✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❯♠❛ ✈❡③ ❡st✉❞❛❞❛s ❛s ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s✱ ♣♦❞❡♠♦s ♣❡♥s❛r ❛❣♦r❛ ❡♠ r❡s♦❧✈❡r s✐st❡♠❛s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s s✐♠✉❧tâ♥❡❛s✳ ❚❛✐s s✐st❡♠❛s s❡ ❛♣r❡s❡♥t❛♠ ❣❡♥❡r✐❝❛♠❡♥t❡ ❛ss✐♠✿   a1 x ≡ b1 (mod m1 )    a2 x ≡ b2 (mod m2 ) ✳✳  ✳    a x ≡ b (mod m ) r r r ♦♥❞❡ ♦s ai (i = 1, 2, . . . , r) sã♦ s✉♣♦st♦s ♥ã♦ ♥✉❧♦s✳ ❯♠❛ ❙♦❧✉çã♦ ❞♦ s✐st❡♠❛ é ✉♠ ✐♥t❡✐r♦ x0 q✉❡ é s♦❧✉çã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❝♦♥❣r✉ê♥❝✐❛s q✉❡ ❞❡❧❡ ❢❛③❡♠ ♣❛rt❡✳ ❆ss✐♠✱ s❡ ✉♠❛ ❞❡ s✉❛s ❝♦♥❣r✉ê♥❝✐❛s ♥ã♦ ❛❞♠✐t❡ s♦❧✉çã♦✱ ♦ ♠❡s♠♦ ♦❝♦rr❡ ❝♦♠ ♦ s✐st❡♠❛✳ P❛r❛ ✐♥tr♦❞✉③✐r ❛s ✐❞é✐❛s✱ ❝♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❡①❡♠♣❧♦✿  3x ≡ 1(mod 5) 2x ≡ 3(mod 9 ❯♠❛ ❞❛s s♦❧✉çõ❡s ❞❛ ♣r✐♠❡✐r❛ ❝♦♥❣r✉ê♥❝✐❛ é 2 ❡ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❛ s❡❣✉♥❞❛ é 6✳ ▲♦❣♦✱ ❛s s♦❧✉çõ❡s ❣❡r❛✐s sã♦ ❞❛❞❛s ♣♦r ❡ x = 2 + 5t, t ∈ Z ✭♣❛r❛ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦✮ x = 6 + 9s, s ∈ Z ✭♣❛r❛ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦✮ q✉❡ ♣♦❞❡♠ s❡r tr❛❞✉③✐❞❛s✱ ❡♠ t❡r♠♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✱ ♣♦r✿ x ≡ 2(mod 5) ❡ x ≡ 6(mod 9). ❈♦♠♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ ♣r✐♠❡✐r❛ ❞❡ss❛s ❝♦♥❣r✉ê♥❝✐❛s ♣♦r 3 ❧❡✈❛ ❛ 3x ≡ 1(mod 5) ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛ s❡❣✉♥❞❛ ♣♦r 2 ❧❡✈❛ ❛ 2x ≡ 3(mod 9)✱ ❡♥tã♦ ♦ s✐st❡♠❛ ❞❛❞♦ ❡q✉✐✈❛❧❡ ❛✿  x ≡ 2(mod 5) x ≡ 6(mod 9). ❉❛í ♣♦rq✉❡✱ ❞♦r❛✈❛♥t❡✱ ♥♦s ❛t❡r❡♠♦s ❛♣❡♥❛s ❛ ❡st❡ t✐♣♦ ❞❡ s✐st❡♠❛ ✭❝♦❡✜❝✐❡♥t❡s ❞❡ x ✐❣✉❛✐s ❛ 1✮✳ ❆❧✐ás ❛ r❡s♦❧✉çã♦ ❞❡st❡ ú❧t✐♠♦✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ❛❝❤❛r ❛ ✐♥t❡rs❡❝çã♦ ❞♦s ❝♦♥❥✉♥t♦s s♦❧✉çõ❡s ❞❡ ❝❛❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦ s✐st❡♠❛✱ ♣♦❞❡ s❡r ❡♥❝❛♠✐♥❤❛❞♦ ❞❛ ♠❛♥❡✐r❛ ❤❛❜✐t✉❛❧ ♥❡st❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳ ❱❡❥❛♠♦s ❝♦♠♦✿ s✉❜st✐t✉✐♥❞♦✲s❡ ❛ s♦❧✉çã♦ ❣❡r❛❧ x = 2 + 5t ❞❛ ♣r✐♠❡✐r❛ ❝♦♥❣r✉ê♥❝✐❛ ♥❛ s❡❣✉♥❞❛ ♦❜té♠✲s❡ ✷✺

(37) 2 + 5t ≡ 6(mod 9) q✉❡ ❡q✉✐✈❛❧❡ ❛ 5t ≡ 4(mod 9) ❙❡♥❞♦ t0 = 8 ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❡st❛ ú❧t✐♠❛✱ ❡♥tã♦ t = 8 + 9k é s✉❛ s♦❧✉çã♦ ❣❡r❛❧✳ ❆ss✐♠✱ x = 2 + 5t = 2 + 5(8 + 9k) = 42 + 45k (k ∈ Z) ♦✉ x ≡ 42(mod 45) é ❛ s♦❧✉çã♦ ❞♦ s✐st❡♠❛✳ Pr♦♣♦s✐çã♦ ✶✳✼✷✳ ❯♠ s✐st❡♠❛  x ≡ a1 (mod m1 ) x ≡ a2 (mod m2 ) a1 − a2 é ❞✐✈✐sí✈❡❧ ♣♦r d = mdc(m1 , m2 )✳ x0 é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ s✐st❡♠❛ ❡ s❡ m = mmc(m1 , m2 )✱ x ≡ x0 (mod m) é s✉❛ s♦❧✉çã♦ ❣❡r❛❧✳ ❛❞♠✐t❡ s♦❧✉çã♦✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ◆❡st❡ ❝❛s♦✱ s❡ ❡♥tã♦ ❉❡♠♦♥str❛çã♦✳ (⇒) ❙❡ x0 é s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ s✐st❡♠❛✱ ❡♥tã♦ t ∈ Z t❛❧ q✉❡ x0 = a1 + m1 t ❉❛í✱ ❡ a1 + m1 t ≡ a2 (mod m2 ). m1 t ≡ a2 − a1 (mod m2 ) ❡✱ ♣❡❧❛ Pr♦♣♦s✐çã♦ ✶✳✻✾✱ d|(a2 − a1 )✳ (⇐) ❈♦♠♦ d|(a2 − a1 )✱ ♣♦r ❤✐♣ót❡s❡✱ ❡♥tã♦ m1 y ≡ a2 − a1 (mod m2 ) ❛❞♠✐t❡ ✉♠❛ s♦❧✉çã♦ y0 ✳ ▲♦❣♦✱ ❈♦♠♦✱ ♦❜✈✐❛♠❡♥t❡✱ a1 + m1 y0 ≡ a2 (mod m2 ). a1 + m1 y0 ≡ a1 (mod m1 ), ❡♥tã♦ a1 + m1 y + 0 é s♦❧✉çã♦ ❞♦ s✐st❡♠❛✳ ❙❡ x0 ✐♥❞✐❝❛ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ s✐st❡♠❛ ❡ x ✐♥❞✐❝❛ ❣❡♥❡r✐❝❛♠❡♥t❡ s✉❛s s♦❧✉çõ❡s✱ ❡♥tã♦ x0 ≡ a1 (mod m1 ) ❡ x ≡ a1 (mod m1 )✱ ❡ s❡❣✉❡ q✉❡ x ≡ x0 (mod m1 ), ♦✉ s❡❥❛✱ m1 |(x − x0 )✳ ❆♥❛❧♦❣❛♠❡♥t❡ s❡ ❝❤❡❣❛ q✉❡ m2 |(x − x0 )✳ ❊♥tã♦✱ m|(x − x0 )✱ ♦ q✉❡ t❡♠ ♦ s❡❣✉✐♥t❡ s✐❣♥✐✜❝❛❞♦✿ x ≡ x0 (mod m)✳ ✷✻

(38) ❯♠ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❈♦r♦❧ár✐♦ ✶✳✼✸✳   x ≡ a1 (mod m1 )    x ≡ a2 (mod m2 ) ✳✳  ✳    x ≡ a (mod m ) r r ❛❞♠✐t❡ s♦❧✉çõ❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ai − aj é ❞✐✈✐sí✈❡❧ ♣♦r dij = mdc(mi , mj )✱ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ í♥❞✐❝❡s i, j (i 6= j)✳ ◆❡st❡ ❝❛s♦✱ s❡ x0 é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r✱ ❡♥tã♦ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞♦ s✐st❡♠❛ é ❞❛❞❛ ♣♦r✿ ♦♥❞❡ m = mmc(m1 , m2 , . . . , mr )✳ ❊①❡♠♣❧♦ ✶✳✼✹✳ x ≡ x0 (mod m) ❈♦♥s✐❞❡r❡♠♦s ♦ s✐st❡♠❛   x ≡ 2(mod 5) x ≡ 3(mod 4)  x ≡ 9(mod 6) ❱❡r✐✜❝❛✲s❡ q✉❡ ❡❧❡ s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❝♦r♦❧ár✐♦ ❡ ♣♦rt❛♥t♦ ❛❞♠✐t❡ s♦❧✉çõ❡s✳ ❯♠❛ ❞❡❧❛s é ♦ ♥ú♠❡r♦ 27✳ ❈♦♠♦ mmc(5, 4, 6) = mmc(mmc(5, 4), 6) = mmc(20, 6) = 60 ❡♥tã♦ x ≡ 27(mod 60) é ❛ s♦❧✉çã♦ ❣❡r❛❧✳ Pr♦♣♦s✐çã♦ ✶✳✼✺ ✭❚❡♦r❡♠❛ ❞♦ ❘❡st♦ ❈❤✐♥ês✮✳ ❙❡❥❛♠ m1 , m2 . . . , mr ♥ú♠❡r♦s ✐♥t❡✐r♦s ♠❛✐♦r❡s q✉❡ ③❡r♦ ❡ t❛✐s q✉❡ mdc(mi , mj ) = 1✱ s❡♠♣r❡ q✉❡ i 6= j ✳ ❋❛ç❛♠♦s m = m1 m2 . . . mr ❡ s❡❥❛♠ b1 , b2 , . . . , br ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s♦❧✉çõ❡s ❞❛s ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s m y ≡ 1(mod mj ) mj ❊♥tã♦ ♦ s✐st❡♠❛ (j = 1, 2, . . . , r)   x ≡ a1 (mod m1 )    x ≡ a2 (mod m2 ) ✳✳  ✳    x ≡ a (mod m ) r r é ♣♦ssí✈❡❧ ✭❛❞♠✐t❡ s♦❧✉çõ❡s✮ ♣❛r❛ q✉❛✐sq✉❡r a1 , a2 , . . . , ar ∈ Z s❡ s✉❛ s♦❧✉çã♦ ❣❡r❛❧ é ❞❛❞❛ ♣♦r✿ x ≡ a1 b1 m + . . . + ar br mmr (mod m) m1 ✷✼

(39) ❉❡♠♦♥str❛çã♦✳ ◗✉❡ ♦ s✐st❡♠❛ é ♣♦ssí✈❡❧ ❞❡❝♦rr❡ ❞♦ ❝♦r♦❧ár✐♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳ ◆♦t❡♠♦s q✉❡✱ ❝♦♠♦ mdc(mj , mi ) = 1✱ i 6= j ✱ ❡♥tã♦ m ) = 1, mdc(mj , mj ♣❛r❛ q✉❡ ✐♠♣❧✐❝❛ ❛ ❡①✐stê♥❝✐❛ ❞❡ s♦❧✉çõ❡s ♣❛r❛ ❝❛❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r m y ≡ 1(mod mj )✱ mj ❛s q✉❛✐s ❡st❛♠♦s ✐♥❞✐❝❛♥❞♦ ♣♦r bj (j = 1, 2, . . . , r). ❆ss✐♠✱ m bj ≡ 1(mod mj ) mj ❡ ♣♦rt❛♥t♦✿ aj bj P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ m ≡ aj (mod mj ) (j = 1, 2, . . . , r)✳ mj i 6= j ✱ m ≡ 0(mod mj ) mi ❡ ❡♥tã♦ ai bi m ≡ 0(mod mj )✳ mi ▲♦❣♦✱ a1 b1 ♣❛r❛ t♦❞♦ m m m + . . . + aj bj + . . . + ar br ≡ aj (mod mj )✱ m1 mj mr j, 1 ≤ j ≤ r✳ ❆ss✐♠✱ ❞❡ ❢❛t♦ x0 Pr é ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞♦ s✐st❡♠❛✳ i=1 ai bi m mi ❖ ❝♦r♦❧ár✐♦ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❣❛r❛♥t❡ ❡♥tã♦ q✉❡ x ≡ x0 (mod m) mdc(mi , mj ) = 1✱ mmc(m1 , m2 , . . . , mr ) = m1 m2 . . . mr = m✳ é ❛ s♦❧✉çã♦ ❣❡r❛❧ ♣♦st♦ q✉❡✱ ❝♦♠♦ s❡♠♣r❡ q✉❡ i 6= j ✱ ❡♥tã♦ ❊①❡♠♣❧♦ ✶✳✼✻✳ ❱❛♠♦s r❡s♦❧✈❡r ♦ s✐st❡♠❛   x ≡ 1(mod 2) x ≡ 2(mod 3)  x ≡ 3(mod 5) ✉s❛♥❞♦ ♦ ❚♦❡r❡♠❛ ❞♦ ❘❡st♦ ❈❤✐♥ês✳ ◆❡st❡ ❝❛s♦ m = 30 ❡ ❛s ❝♦♥❣r✉ê♥❝✐❛s ❛ r❡s♦❧✈❡r sã♦✿ 15y ≡ 1(mod 2), ❞❛s q✉❛✐s b1 = 1, b2 = 1 ❡ 10y ≡ 1(mod 3) b3 = 1 ❡ 6y ≡ 1(mod 5), sã♦ s♦❧✉çõ❡s ♣❛rt✐❝✉❧❛r❡s✳ ❆ss✐♠✱ ❛ s♦❧✉çã♦ ❣❡r❛❧ ❞♦ s✐st❡♠❛ é ❞❛❞❛ ♣♦r x ≡ 1.1.15 + 2.1.10 + 3.1.6 ≡ 23(mod 30)✳ ✷✽

(40) ✶✳✾ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ❖ ♦❜❥❡t✐✈♦ ❞❡ss❡ ♠ét♦❞♦ é ❝❛❧❝✉❧❛r ❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❡ br ♠ó❞✉❧♦ n✱ s❡♥❞♦ b✱ ♥ú♠❡r♦s ♥❛t✉r❛✐s ❣r❛♥❞❡s✳ P❛r❛ ❢❛③❡r ❡ss❡ ❝á❧❝✉❧♦✱ é ♥❡❝❡ssár✐♦ ❝♦♥✈❡rt❡r♠♦s r r ❡ n ❡♠ ♥ú♠❡r♦ ❜✐♥ár✐♦✳ P❛r❛ t❛♥t♦✱ s✉♣♦♥❤❛♠♦s r= k X aj 2j , j=0 s❡♥❞♦ aj = 0 ♦✉ 1✳ P♦r ❡①❡♠♣❧♦✱ s❡ r = 106✱ ♣❛ss❛♠♦s✲♦ ♣❛r❛ ❛ ❜❛s❡ ❜✐♥ár✐❛ ❢❛③❡♥❞♦ ❛ ❝♦♥t❛ s✐♠♣❧❡s 106 = 0 · 20 + 1 · 21 + 0 · 22 + 1 · 23 + 0 · 24 + 1 · 25 + 1 · 26 , ❞❡ ❢♦r♠❛ q✉❡ k = 6✱ ❡ a0 = 0✱ a1 = 1✱ a2 = 0✱ a3 = 1✱ a4 = 0✱ a5 = 1 ❡ a6 = 1✳ ❆❧❣♦r✐t♠♦✿ ❙❡❥❛♠ c✱ P❛ss♦ P❛ss♦ P❛ss♦ 1) 2) 3) d ❡ bj ❀ j = 0, . . . , k ♥ú♠❡r♦s ♥❛t✉r❛✐s ✭❛✉①✐❧✐❛r❡s✮✳ a0 = 1✱ ❡♥tã♦ ❢❛ç❛ c = b✳ ❙❡♥ã♦✱ ❢❛ç❛ c = 1✳ ❙❡❥❛ b0 = b✳ P❛r❛ ❝❛❞❛ j = 1, . . . , k ❢❛ç❛✿ ❝❛❧❝✉❧❡ ❙❡ bj ≡ b2j−1 (mod n). ❙❡ aj = 1✱ ❝❛❧❝✉❧❡ d ≡ cbj (mod n) c = d✳ ❙❡♥ã♦ ❞❡✐①❡ c ✐♥❛❧t❡r❛❞♦✳ r P❛ss♦ 4) ❖ ♥ú♠❡r♦ c é ❝♦♥❣r✉❡♥t❡ ❛ b ❡ ❢❛ç❛ n✱ ♠ó❞✉❧♦ ♦✉ s❡❥❛✱ c ≡ br (mod n). P❡r❝❡❜❡♠♦s q✉❡ ♥❛ ❡t❛♣❛ i ❞♦ P❛ss♦ 3✱ t❡♠♦s Pi c ≡ b0 j=0 ❆ss✐♠✱ ❛♦ tér♠✐♥♦ ❞♦ ❛❧❣♦r✐t♠♦✱ t❡♠♦s ❊①❡♠♣❧♦ ✶✳✼✼✳ ❊♥❝♦♥tr❡♠♦s a (mod n). c ≡ br (mod n)✳ t❛❧ q✉❡ n = 451✳ aj 2j a ≡ br (mod n)✱ ❙♦❧✉çã♦✿ ❈♦♥❢♦r♠❡ ❞❡st❛❝❛♠♦s ❛♥t❡s✱ ♣❛ss❛♥❞♦ s❡♥❞♦ b = 227✱ r = 106 ❡ r = 106 ♣❛r❛ ❛ ❜❛s❡ ❜✐♥ár✐❛✱ t❡♠♦s✿ 106 = 0 · 20 + 1 · 21 + 0 · 22 + 1 · 23 + 0 · 24 + 1 · 25 + 1 · 26 = 11010102 , ♦♥❞❡ k=6 ❡ a0 = 0✱ a1 = 1✱ a2 = 0✱ a3 = 1✱ a4 = 0✱ a5 = 1 ❛❧❣♦r✐t♠♦✿ P❛ss♦ P❛ss♦ P❛ss♦ 1) ❈♦♠♦ a0 6= 1✱ 2) b0 = 227✳ 3) ❡♥tã♦ c = 1✳ ✷✾ ❡ a6 = 1✳ ❙❡❣✉✐♥❞♦ ♦

(41) • P❛r❛ j = 1 b1 ≡ 2272 (mod 451) ⇒ b1 = 115 a1 = 1✱ ❡♥tã♦ d ≡ 1.115(mod 451) ⇒ d = 115 ⇒ c = 115✳ • j=2 b2 ≡ 1152 (mod 451) ⇒ b2 = 146 a2 = 0 ⇒ c = 115✳ • j=3 b3 ≡ 1462 (mod 451) ⇒ b3 = 119 a3 = 1✱ ❡♥tã♦ d ≡ 115.119(mod 451) ⇒ d = 20 ⇒ c = 20✳ • P❛r❛ j = 4 b4 ≡ 1192 (mod 451) ⇒ b4 = 180 a4 = 0 ⇒ c = 20✳ • j=5 b5 ≡ 1802 (mod 451) ⇒ b5 = 379 a5 = 1✱ ❡♥tã♦ d ≡ 20.379(mod 451) ⇒ d = 364 ⇒ c = 364✳ • j=6 b6 ≡ 3792 (mod 451) ⇒ b6 = 223 a6 = 1✱ ❡♥tã♦ d ≡ 364.223(mod 451) ⇒ d = 443 ⇒ c = 443✳ P❛r❛ P❛r❛ P❛r❛ P❛r❛ P❛ss♦ 4) ▲♦❣♦✱ a ≡ br (mod n) ⇒ 443 ≡ 227106 (mod 451). ✶✳✶✵ ▼❛tr✐③❡s ❯♠ té❝♥✐❝♦ ❞❡ ❜❛sq✉❡t❡❜♦❧✱ q✉❡r❡♥❞♦ ❛♥❛❧✐s❛r ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦s t✐t✉❧❛r❡s ❞❡ s✉❛ ❡q✉✐♣❡✱ ❝♦❧♦❝♦✉ ❡♠ ✉♠❛ t❛❜❡❧❛ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♠❛r❝❛❞♦s ♣♦r ❝❛❞❛ t✐t✉❧❛r ❡♠ s❡t❡ ❥♦❣♦s✿ ❚✐t✉❧❛r❡s ✭✐✮❏♦❣♦s ✭❥✮ ✶ ✷ ✸ ✹ ✺ ✶ ✶✽ ✶✺ ✷✵ ✶✽ ✶✾ ✷ ✶✼ ✶✻ ✶✾ ✷✷ ✶✽ ✸ ✶✽ ✶✽ ✷✵ ✷✵ ✶✷ ✹ ✶✼ ✶✽ ✷✶ ✷✵ ✶✹ ✺ ✷✶ ✷✷ ✶✹ ✶✽ ✷✵ ✻ ✶✽ ✷✶ ✶✹ ✷✷ ✶✼ ✼ ✷✵ ✶✽ ✷✷ ✷✸ ✶✽ ♥❛ q✉❛❧ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❛ ❧✐♥❤❛ ✐ ❡ ❝♦❧✉♥❛ ❥ é ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♠❛r❝❛❞♦s ♣♦r ❝❛❞❛ t✐t✉❧❛r ✐ ❡♠ ❝❛❞❛ ❥♦❣♦ ❥✳ ◆♦t❡ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ss❛ t❛❜❡❧❛✳ ❙❡ q✉✐s❡r♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ s❛❜❡r q✉❛❧ ♦ ♥ú♠❡r♦ ❞❡ ♣♦♥t♦s ♠❛r❝❛❞♦ ♣❡❧♦ t✐t✉❧❛r ❞❡ ♥ú♠❡r♦ 2 ♥♦ 5♦ ❥♦❣♦✱ ❜❛st❛ ♦❧❤❛r♠♦s ♣❛r❛ ♦ ❝r✉③❛♠❡♥t♦ ❞❛ ❧✐♥❤❛ 2 ❝♦♠ ❛ ❝♦❧✉♥❛ 5 ❡ ❡♥❝♦♥t❛r 22✳ ❚❛❜❡❧❛s ❝♦♠♦ ❡ss❛ sã♦ ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③❡s✳ ❱❛♠♦s ❢♦r♠❛❧✐③❛r ♦ q✉❡ é ✉♠❛ ♠❛tr✐③✱ ♦✉ s❡❥❛✱ ❞❡✜♥✐r❡♠♦s ✉♠❛ ♠❛tr✐③ ❡ s✉❛s ♦♣❡r❛çõ❡s✳ ✸✵

(42) ❉❡✜♥✐çã♦ ✶✳✼✽✳ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s m ❡ n ♥❛t✉r❛✐s✱ ♥ã♦ ♥✉❧♦s✱ ❝❤❛♠❛✲s❡ ♠❛tr✐③ n (✐♥❞✐❝❛✲s❡ m × n) t♦❞❛ t❛❜❡❧❛ M ❢♦r♠❛❞❛ ♣♦r ♥ú♠❡r♦s r❡❛✐s ❞✐str✐❜✉í❞♦s ❡♠ m ❧✐♥❤❛s ❡ n ❝♦❧✉♥❛s✳ ❯♠❛ ♠❛tr✐③ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❡♥tr❡ ♣❛rê♥t❡s❡s ( ) ♦✉ ❡♥tr❡ ❝♦❧❝❤❡t❡s [ ]✳ m ♣♦r ❊①❡♠♣❧♦ ✶✳✼✾✳ ✶✳ ✷✳  3 5 √ −1 4 2 0 5  0 9 −1 7  é ✉♠❛ ♠❛tr✐③  2 × 3. é ✉♠❛ ♠❛tr✐③ 1 × 4. ❊♠ ✉♠❛ ♠❛tr✐③ q✉❛❧q✉❡r M ✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ é ✐♥❞✐❝❛❞♦ ♣♦r aij ✱ ❡ é ❝❤❛♠❛❞♦ ❞❡ M ✳ ❖ í♥❞✐❝❡ i ✐♥❞✐❝❛ ❛ ❧✐♥❤❛ ❡ ♦ í♥❞✐❝❡ j ❛ ❝♦❧✉♥❛ às q✉❛✐s ♦ ❡❧❡♠❡♥t♦ ♣❡rt❡♥❝❡✳ ❈♦♠♦ ❛s ❧✐♥❤❛s sã♦ ❡♥✉♠❡r❛❞❛s ❞❡ ❝✐♠❛ ♣❛r❛ ❜❛✐①♦ ✭❞❡ 1 ❛té m✮ ❡ ❛s ❝♦❧✉♥❛s ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛ ✭❞❡ 1 ❛té n✮✱ ❡♥tã♦ ✉♠❛ ♠❛tr✐③ m × n é r❡♣r❡s❡♥t❛❞❛ ♣♦r✿   ❡♥tr❛❞❛ ❞❛ ♠❛tr✐③ a11 a12  a21 a22  M =  ✳✳ ✳✳  ✳ ✳ am1 am2 . . . a1n . . . a2n   . ✳ ✳ ✳ ✳✳  ✳  . . . amn ▼❛tr✐③❡s ❡s♣❡❝✐❛✐s ❍á ♠❛tr✐③❡s q✉❡✱ ♣♦r ❛♣r❡s❡♥t❛r❡♠ ✉♠❛ ✉t✐❧✐❞❛❞❡ ♠❛✐♦r ♥❛ t❡♦r✐❛ ❞❛s ♠❛tr✐③❡s✱ r❡❝❡❜❡♠ ♥♦♠❡s ❡s♣❡❝✐❛✐s✳ ✶✳ ▼❛tr✐③ ❧✐♥❤❛✿ ➱ ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ 1 × n✱ ✐st♦ é✱ é ✉♠❛ ♠❛tr✐③ q✉❡ t❡♠ ✉♠❛ ú♥✐❝❛ ❧✐♥❤❛✳ ❊①❡♠♣❧♦✳ 0 9 −1 7  ✷✳ ▼❛tr✐③ ❝♦❧✉♥❛✿ ➱ ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ m × 1✱ ✐st♦ é✱ é ✉♠❛ ♠❛tr✐③ q✉❡ t❡♠ ✉♠❛ ú♥✐❝❛ ❝♦❧✉♥❛✳ ❊①❡♠♣❧♦✳  5  1  −3  ✸✳ ▼❛tr✐③ ♥✉❧❛✿ ➱ ✉♠❛ ♠❛tr✐③ q✉❡ t❡♠ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ✐❣✉❛✐s ❛ ③❡r♦✳ ■♥❞✐❝❛♠♦s ♣♦r 0m×n ✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡✱ s❡ 0m×n = (aij )m×n ✱ ❞❡✈❡✲s❡ t❡r aij = 0, ∀i, j, 1 ≤ i ≤ m ❡ 1 ≤ j ≤ n. ❊①❡♠♣❧♦✳  0 0 0 0 0 0 ✸✶ 

(43) ✹✳ ▼❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n✿ ➱ ✉♠❛ ♠❛tr✐③ ❞♦ t✐♣♦ ♠❛tr✐③ q✉❡ t❡♠ ✐❣✉❛❧ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s✳ n × n✱ ♦✉ s❡❥❛✱ é ✉♠❛ ❊①❡♠♣❧♦✳   a11 a12 . . . a1n  a21 a22 . . . a2n    . . . . . . ✳✳✳ . . . an1 an2 . . . ann     ❈❤❛♠❛✲s❡ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s q✉❡ tê♠ ♦s í♥❞✐❝❡s ✐❣✉❛✐s✱ ✐st♦ é✿ {aij |i = j} = {a11 , a22 , a33 , . . . , ann }✳ ❈❤❛♠❛✲s❡ ❞✐❛❣♦♥❛❧ s❡❝✉♥❞ár✐❛ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ ❞♦s ❡❧❡♠❡♥t♦s q✉❡ t❡♠ s♦♠❛ ❞♦✐s í♥❞✐❝❡s ✐❣✉❛❧ ❛ n + 1✱ n ♦ ❝♦♥❥✉♥t♦ ✐st♦ é✿ {aij |i + j = n + 1} = {a1n , a2,n−1 , a3,n−2 , . . . , an1 }✳ ❊①❡♠♣❧♦ ✶✳✽✵✳ ❆ ♠❛tr✐③      M =    é q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ ✸✳ s❡❝✉♥❞ár✐❛ é ✺✳ ▼❛tr✐③ 8 9 6 4 −7     −5     3 ~ ~ −1 2 ❙✉❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ é {−7, 4, −1}✳ ❞✐❛❣♦♥❛❧✿ ➱ ✉♠❛ ♠❛tr✐③  q✉❛❞r❛❞❛ {8, 4, 3} ❝✉❥♦s ❡ s✉❛ ❞✐❛❣♦♥❛❧ ❡❧❡♠❡♥t♦s q✉❡ ♥ã♦ ♣❡rt❡♥❝❡♠ à ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ sã♦ ✐❣✉❛✐s ❛ ③❡r♦✳ ❊①❡♠♣❧♦✳  ✻✳ ▼❛tr✐③ ✉♥✐❞❛❞❡ (♦✉ 3 0 0 −2  ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡) ❞❡ ♦r❞❡♠ n (✐♥❞✐❝❛✲s❡ In )✿ ➱ ✉♠❛ ♠❛tr✐③ ❞✐❛❣♦♥❛❧ ❡♠ q✉❡ ♦s ❡❧❡♠❡♥t♦s ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ sã♦ ✐❣✉❛✐s ❛ ✶✱ ♦✉ s❡❥❛✱ In = (aij )n×n t❛❧ q✉❡ aij =  1, 0, i=j . i 6= j s❡ s❡ ❊①❡♠♣❧♦✳  1  0 I4 =   0 0 ✸✷ 0 1 0 0 0 0 1 0  0 0  . 0  1

(44) ✼✳ ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ❙✉♣❡r✐♦r✿ ➱ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ♦♥❞❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❛❜❛✐①♦ ❞❛ ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧ sã♦ ♥✉❧♦s✱ ✐st♦ é✱ ♣❛r❛ aij = 0✱  2 −1 0  0 −1 4  . 0 0 3  ▼❛tr✐③ ❚r✐❛♥❣✉❧❛r ■♥❢❡r✐♦r✿ aij = 0✱ ♣❛r❛ ➱ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❡♠ q✉❡ m = n ❡ i < j✳ ❊①❡♠♣❧♦✳ ✾✳ ❡ i > j✳ ❊①❡♠♣❧♦✳ ✽✳ m = n  ▼❛tr✐③ ❙✐♠étr✐❝❛✿ ❊①❡♠♣❧♦✳  2 0 0 0  1 −1 0 0     1 2 2 0 . 1 0 5 4 ➱ ✉♠❛ ♠❛tr✐③ ♦♥❞❡ m = n ❡ aij = aji ✱ ∀i, j, 1 ≤ i, j ≤ n✳  4 3 −1  3 2 0 . −1 0 5  ❖❜s❡r✈❡ q✉❡✱ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ♠❛tr✐③ s✐♠étr✐❝❛✱ ❛ ♣❛rt❡ s✉♣❡r✐♦r é ✉♠❛ ✏r❡✢❡①ã♦✑ ❞❛ ♣❛rt❡ ✐♥❢❡r✐♦r✱ ❡♠ r❡❧❛çã♦ à ❞✐❛❣♦♥❛❧ ♣r✐♥❝✐♣❛❧✳ ❖♣❡r❛çõ❡s ❝♦♠ ▼❛tr✐③❡s ■❣✉❛❧❞❛❞❡ ❞❡ ▼❛tr✐③❡s ❉❡✜♥✐çã♦ ✶✳✽✶✳ ❉✉❛s ♠❛tr✐③❡s sã♦ ❞✐t❛s ✐❣✉❛✐s q✉❛♥❞♦ ♣♦ss✉❡♠ ❛ ♠❡s♠❛ ♦r❞❡♠ ❡ ❛s ❡♥tr❛❞❛s ❝♦rr❡s♣♦♥❞❡♥t❡s sã♦ ✐❣✉❛✐s✳ ❊①❡♠♣❧♦ ✶✳✽✷✳ ❈♦♥s✐❞❡r❡ ❛s ♠❛tr✐③❡s  1 −1 A =  4 0 , 2 5   0 4 B =  −2 5  1 0   1 −1 ❡ C =  4 0 . 2 5  ❚❡♠♦s A = C ❡ A ♥ã♦ ✐❣✉❛❧ ✭❞✐❢❡r❡♥t❡✮ ❛ C ✭♥♦t❛çã♦ ✉s✉❛❧✿ A 6= B ✮✳ ❆❞✐çã♦ ❉❡✜♥✐çã♦ ✶✳✽✸✳ ❆ s♦♠❛ ❞❡ ❞✉❛s ♠❛tr✐③❡s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ Am×n = [aij ] ❡ Bm×n = [bij ]✱ é ❞❡✜♥✐❞❛ ♣♦r [aij + bij ] ❡ ❞❡♥♦t❛❞❛ ♣♦r A + B ✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✱ A + B = [aij + bij ]m×n . ❊①❡♠♣❧♦ ✶✳✽✹✳      1 3 0 4 1 −1  4 0  +  −2 5  =  2 5  3 5 1 0 2 5  ✸✸

(45) Pr♦♣r✐❡❞❛❞❡s✿ ❉❛❞❛s ❛s ♠❛tr✐③❡s A✱ B ❡ C ❞❡ ♠❡s♠❛ ♦r❞❡♠ m × n✱ t❡♠♦s✿ ✐✮ A + B = B + A ✭❝♦♠✉t❛t✐✈✐❞❛❞❡✮ ✐✐✮ A + (B + C) = (A + B) + C ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡✮ ✐✐✐✮ A + 0 = A✱ ♦♥❞❡ 0 ❞❡♥♦t❛ ❛ ♠❛tr✐③ ♥✉❧❛ m × n. ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s é s✐♠♣❧❡s✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛s ❡♠ ❬✷✱ ✸❪✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ♣♦r ♠❛tr✐③ ❉❡✜♥✐çã♦ ✶✳✽✺✳ ❖ ♣r♦❞✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ k ♣♦r ✉♠❛ ♠❛tr✐③ A = (aij )m×n ✱ é ❞❡✜♥✐❞♦ ♣♦r kA = [kaij ]✳ ◆❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❝❛❞❛ ❡❧❡♠❡♥t♦ ❞❛ ♠❛tr✐③ kA é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ❡♥tr❛❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡♠ A✱ ♣❡❧♦ ♥ú♠❡r♦ k ✳ ❊①❡♠♣❧♦ ✶✳✽✻✳    8 −20 2 −5 4  3 0  =  12 0  4 24 1 6  Pr♦♣r✐❡❞❛❞❡s✿ ❙❡♥❞♦ A ❡ B ♠❛tr✐③❡s ❞♦ ♠❡s♠♦ t✐♣♦ ❡ s❡♥❞♦ r ❡ s ♥ú♠❡r♦s✱ t❡♠✲s❡ q✉❡✿ ✐✮ r(sA) = s(rA) = (rs)A❀ ✐✐✮ r(A + B) = rA + rB ❀ ✐✐✐✮ (r + s)A = rA + sA❀ ✐✈✮ 1A = A✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s é s✐♠♣❧❡s✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛s ❡♠ ❬✷✱ ✸❪✳ ❚r❛♥s♣♦s✐çã♦ ❉❡✜♥✐çã♦ ✶✳✽✼✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ A = [aij ]m×n ✱ ♣♦❞❡♠♦s ♦❜t❡r ✉♠❛ ♦✉tr❛ ♠❛tr✐③ At = [bij ]n×m ✱ ❝✉❥❛s ❧✐♥❤❛s sã♦ ❛s ❝♦❧✉♥❛s ❞❡ A✱ ✐st♦ é✱ bij = aji . At é ❝❤❛♠❛❞❛ tr❛♥s♣♦st❛ ❞❡ A✳ ❊①❡♠♣❧♦ ✶✳✽✽✳  2 1 ⇒ A= 0 3  −1 4 3×2  t A = ✸✹  2 0 −1 1 3 4  . 2×3

(46) Pr♦♣r✐❡❞❛❞❡s✿ ✐✮ ❯♠❛ ♠❛tr✐③ é s✐♠étr✐❝❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡❧❛ é ✐❣✉❛❧ à s✉❛ tr❛♥s♣♦st❛✱ ✐st♦ é✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A = At ✳ ❊①❡♠♣❧♦✳ B=  1 3 3 2  ⇒ t B =   1 3 3 2 . 2×3 ✐✐✮ (At )t = A✳ ■st♦ é✱ ❛ tr❛♥s♣♦st❛ ❞❛ tr❛♥s♣♦st❛ ❞❡ ✉♠❛ ♠❛tr✐③ é ❡❧❛ ♠❡s♠❛✳ ✐✐✐✮ (A + B)t = At + B t ✳ ❊♠ ♣❛❧❛✈r❛s✱ ❛ tr❛♥s♣♦st❛ ❞❡ ✉♠❛ s♦♠❛ é ✐❣✉❛❧ à s♦♠❛ ❞❛s tr❛♥s♣♦st❛s✳ ✐✈✮ (kA)t = kAt ✱ ♦♥❞❡ k é q✉❛❧q✉❡r ❡s❝❛❧❛r✳ ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s é s✐♠♣❧❡s✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛s ❡♠ ❬✷✱ ✸❪✳ ▼✉❧t✐♣❧✐❝❛çã♦ ❞❡ ▼❛tr✐③❡s ❆♥t❡s ❞❡ ❞❡✜♥✐r♠♦s ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❡ ♠❛tr✐③❡s✱ ✈❛♠♦s ❞❡✜♥✐r ♣r♦❞✉t♦ ❞❡ ❧✐♥❤❛ ♣♦r ❝♦❧✉♥❛✳ ❉❡✜♥✐çã♦ ✶✳✽✾✳ ❙❡❥❛♠ ❛s ♠❛tr✐③❡s A = (aij )m×k ❡ B = (bij )k×n ✳ ❈♦♥s✐❞❡r❡♠♦s ❛ ❧✐♥❤❛ ✐ ❞❡ ❆ ❡ ❛ ❝♦❧✉♥❛ ❥ ❞❡ ❇✱ ✐st♦ é✿  ai1 ai2 ai3 · · · aik  e       b1j b2j b3j ✳ ✳ ✳ bkj     .   ❖ ♣r♦❞✉t♦ ❞❛ ❧✐♥❤❛ ♣❡❧❛ ❝♦❧✉♥❛ é✿ ai1 b1j + ai2 b2j + ai3 b3j + . . . + aik b1k . ❖✉ s❡❥❛✱ ♠✉❧t✐♣❧✐❝❛♠♦s✱ ♦r❞❡♥❛❞❛♠❡♥t❡✱ ♦s ❡❧❡♠❡♥t♦s ❞❛ ❧✐♥❤❛ ❝♦❧✉♥❛ j ❉❡✜♥✐çã♦ ✶✳✾✵✳ s❡ ✐♥❞✐❝❛ ♣♦r AB A = (aij )m×k ♣❡❧❛ ♠❛tr✐③ B = (bij )k×n q✉❡ A × B ✱ é ❛ ♠❛tr✐③ C = (cij )m×n t❛❧ q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ cij ❧✐♥❤❛ i ❞❡ A ♣❡❧❛ ❝♦❧✉♥❛ j ❞❡ B ✳ ❖ ♣r♦❞✉t♦ ❞❛ ♠❛tr✐③ ❊①❡♠♣❧♦ ✶✳✾✶✳  = ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ♦✉ ♣♦r é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛  i ❡ s♦♠❛♠♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳ 2 5 8 1 4 3   4 3 4  3 6 1 = 1 2 0  2×4+5×3+8×1 2×3+5×6+8×2 2×4+5×1+8×0 1×4+4×3+3×1 1×3+4×6+3×2 1×4+4×1+3×0   31 52 13 = . 13 21 8 ✸✺  =

(47) ❖❜s❡r✈❛çã♦ ✶✳✾✷✳ ❙❡ A ❡ B sã♦ ♠❛tr✐③❡s✱ ❡♥tã♦✿ ✶✳ ♦ ♣r♦❞✉t♦ AB é ❞❡✜♥✐❞♦ ❛♣❡♥❛s q✉❛♥❞♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❡ A ❢♦r ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡ B ✳ ❊①❡♠♣❧♦✳       2 1 2 2 1 −1  4 2  = 4 4  . 0 4 2×2 5 3 3×2 5 7 3×2 ❈♦♠♦ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③ ❢♦✐ ♣♦ssí✈❡❧ ❢❛③❡r ❛ ♠✉❧t✐♣❧✐❝❛çã♦✳ ❊①❡♠♣❧♦✳  1 −1 0 4   2 1  4 2  . 2×2 5 3 3×2  ◆ã♦ é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ❡st❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ♣♦rq✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❛ ♣r✐♠❡✐r❛ ♠❛tr✐③ é ❞✐❢❡r❡♥t❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❛ s❡❣✉♥❞❛ ♠❛tr✐③✳ ✷✳ ❛ ♠❛tr✐③ C t❛❧ q✉❡ C = AB ♣♦ss✉✐ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡ A ❡ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s ❞❡ B ✱ ✐st♦ é✿ Am×k Bk×n = Cm×n ✳ ✸✳ ❊♠ ❣❡r❛❧ AB 6= BA✳ ❊①❡♠♣❧♦✳    1 2 3 1 −1 1 ❙❡❥❛♠ A =  −3 2 −1  ❡ B =  2 4 6 ✳ ❊♥tã♦ 1 2 3 −2 1 0     −11 6 −1 0 0 0 AB =  0 0 0  e BA =  −22 12 −2  . −11 6 −1 0 0 0  ◆♦t❡ ❛✐♥❞❛ q✉❡ AB = 0✱ s❡♠ q✉❡ A = 0 ♦✉ B = 0✳ Pr♦♣r✐❡❞❛❞❡s✿ ❉❡s❞❡ q✉❡ s❡❥❛♠ ♣♦ssí✈❡✐s ❛s ♦♣❡r❛çõ❡s✱ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ ✈á❧✐❞❛s✿ ✶✳ AI = IA = A ✭■st♦ ❥✉st✐✜❝❛ ♦ ♥♦♠❡ ❞❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✳✮ ✷✳ A(B + C) = AB + AC ✭❞✐str✐❜✉t✐✈✐❞❛❞❡ à ❡sq✉❡r❞❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❡♠ r❡❧❛çã♦ à s♦♠❛✮ ✸✳ (A + B)C = AC + BC ✭❞✐str✐❜✉t✐✈✐❞❛❞❡ à ❞✐r❡✐t❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦✱ ❡♠ r❡❧❛çã♦ à s♦♠❛✮ ✹✳ (AB)C = A(BC) ✭❛ss♦❝✐❛t✐✈✐❞❛❞❡✮ ✺✳ (AB)t = B t At ✻✳ 0A = 0 ❡ A0 = 0 ❆ ✈❡r✐✜❝❛çã♦ ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s é s✐♠♣❧❡s✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛s ❡♠ ❬✷✱ ✸❪✳ ✸✻

(48) ✶✳✶✶ ❉❡t❡r♠✐♥❛♥t❡ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ✭q✉❛❞r❛❞❛✮ ❉❡ ❢♦r♠❛ ❤❡✉ríst✐❝❛✱ ♦ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A = [aij ] ❞❡ ♦r❞❡♠ n é ✉♠ ♥ú♠❡r♦ r❡❛❧ ❛ ❡❧❛ ❛ss♦❝✐❛❞♦✳ P❛r❛ ❛♣r❡s❡♥t❛r ♦ ❝♦♥❝❡✐t♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ♠♦❞♦ ♠✐♥✐♠❛♠❡♥t❡ ❢✉♥❞❛♠❡♥t❛❞♦✱ ❞❡✜♥✐r❡♠♦s ❞❡t❡r♠✐♥❛♥t❡ ✐♥❞✉t✐✈❛♠❡♥t❡ ❛tr❛✈és ❞❛ ♦r❞❡♠ ❞❛ ♠❛tr✐③✳ ❈❛❞❛ ♠❛tr✐③ t❡♠ ✉♠ ú♥✐❝♦ ❞❡t❡r♠✐♥❛♥t❡✳ ■♥❞✐❝❛r❡♠♦s ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ss❛ ♠❛tr✐③ ♣♦r✿ det(A) ♦✉

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63) . ✳✳✳ . . .

(64)

(65) ... ... an1 an2 . . . ann

(66) a11 a12 . . . a1n a21 a22 . . . a2n ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ✶❛ ♦r❞❡♠ ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ A = (a11 )✱ ❞❡ ✶❛ ♦r❞❡♠✱ é ♦ ✈❛❧♦r ❞♦ s❡✉ ú♥✐❝♦ ❡❧❡♠❡♥t♦ a11 ✱ ♦✉ s❡❥❛✿ det(A) = |a11 | = a11 . ❊①❡♠♣❧♦ ✶✳✾✸✳ ❙❡ M = (4)✱ ❡♥tã♦ det(M ) = 4✳ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ✷❛ ♦r❞❡♠ ❛ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ✷ ♦r❞❡♠ A = A é ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ♥ú♠❡r♦ r❡❛❧  a11 a12 a21 a22  ✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ a11 a22 − a12 a21 . ◆♦t❡ q✉❡

(67)

(68) a11

(69)

(70) det(A) =

(71)

(72)

(73) a21| ❊①❡♠♣❧♦ ✶✳✾✹✳

(74) a12

(75)

(76)

(77)

(78) = a11 det([a22 ]) − a12 det([a21 ]).

(79) " a22

(80) ❈❛❧❝✉❧❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ M=  2 −3 1 5  ✳ ❙♦❧✉çã♦✿

(81)

(82) 2

(83)

(84) det(M ) =

(85)

(86) ~

(87) 1 ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ M

(88) −3

(89)

(90)

(91)

(92) = 2 · 5 − (−3) · 1 = 10 + 3 = 13.

(93) 5

(94) é ✶✸✳ ✸✼

(95) ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ✸ ❛ ♦r❞❡♠ ❈♦♠ ❝❡rt❛ ❛♥❛❧♦❣✐❛ ❛♦ ❝❛s♦ ❞❡ ✷❛ ♦r❞❡♠✱ ❞❛❞❛ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ n = 3  a11 a12 a13 A =  a21 a22 a23  , a31 a32 a33  ❞❡✜♥✐♠♦s det(A) = a11 det  a22 a23 a32 a33  − a12 det  a21 a23 a31 a33  + a13 det  a21 a22 a31 a32  . ❖❜s❡r✈❡ q✉❡ ❛s ♠❛tr✐③❡s A11 =  a22 a23 a32 a33  , A12 =  a21 a23 a31 a33  ❡ A13 =  a21 a22 a31 a32  , sã♦ ♦❜t✐❞❛s ❞❛ ♠❛tr✐③ A ❛♦ s❡ r❡t✐r❛r ❛ ✶❛ ❧✐♥❤❛ ❡ ❛ ✶❛ ❝♦❧✉♥❛✱ ❛ ✶❛ ❧✐♥❤❛ ❡ ❛ ✷❛ ❝♦❧✉♥❛ ❡ ❛ ✶❛ ❧✐♥❤❛ ❡ ❛ ✸❛ ❝♦❧✉♥❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r det(A) = a11 det(A11 ) − a12 det(A12 ) + a13 det(A13 ). ❆ ♣❛rt✐r ❞❛ s✉❜♠❛tr✐③ Aij ♦❜t✐❞❛ ❞❛ ♠❛tr✐③ A q✉❛♥❞♦ s❡ r❡t✐r❛ ❛ i−és✐♠❛ ❧✐♥❤❛ ❡ ❛ j−és✐♠❛ ❝♦❧✉♥❛✱ t❡♠♦s ♦s ❝❤❛♠❛❞♦s ❈♦❢❛t♦r❡s ∆ij = (−1)i+j det(Aij ). ▲♦❣♦✱ det(A) = a11 ∆11 + a12 ∆11 + a13 ∆11 . ✭✶✳✺✮ ❉❡s❡♥✈♦❧✈❡♥❞♦ t♦❞❛s ❛s ❝♦♥t❛s✱ t❛♠❜é♠ s❡ t❡♠ |A| = a11 .a22 .a33 + a12 .a23 .a31 + a13 .a21 .a32 − a13 .a22 .a31 − a11 .a23 .a32 − a12 .a21 .a33 , ❝♦♥❤❡❝✐❞❛ ❝♦♠ ❘❡❣r❛ ❞❡ ❙❛rr✉s✳ ❊①❡♠♣❧♦ ✶✳✾✺✳ ❈❛❧❝✉❧❡ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ ❙♦❧✉çã♦✿ ❚❡♠♦s✿  1 3 4 A =  5 2 −3 ✳ 1 4 2  det(A) = 1.2.2+1.(−3).3+4.4.5−1.2.4−2.3.5−1.4.(−3) = 4−9+80−8−30+12 = 49. ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❛ ♠❛tr✐③ A é 49✳ ❉❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ ♠❛✐♦r q✉❡ ✸ ❖ ♠ét♦❞♦ ❞❡ ❝á❧❝✉❧♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❛♣r❡s❡♥t❛❞♦ ❛q✉✐ é ❝❤❛♠❛❞♦ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ▲❛♣❧❛❝❡ ❡ ♥♦s ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❞❡t❡r♠✐♥❛♥t❡s ❞❡ ♠❛tr✐③❡s ❝♦♠ ♦r❞❡♠ n✱ ♣❛r❛ n ≥ 4✳ P❛r❛ t❛♥t♦✱ ✐♥s♣✐r❛❞♦ ♥❛ ❢ór♠✉❧❛ ✭✶✳✺✮✱ ❞❛❞❛ ✉♠❛ ♠❛tr✐③ An×n  a11 a12  a21 a22  =  ... ... an1 an2 ✸✽  . . . a1n . . . a2n   , ✳✳✳ ...  . . . ann

(96) ❞❡✜♥✐♠♦s det(An×n ) = a11 ∆11 + a12 ∆12 + . . . + a1n ∆1n ✭✶✳✻✮ ❉❡✈❡✲s❡ ❞❡st❛❝❛r ❢♦rt❡♠❡♥t❡ q✉❡ ♥❛ ❢ór♠✉❧❛ ✭✶✳✻✮ ❞❛❞❛✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❢♦✐ ❛ ✏❞❡s❡♥✈♦❧✈✐❞♦✑ ❛tr❛✈és ❞❛ ✶ ❞❛ i−és✐♠❛ ❧✐♥❤❛✳ ❖ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ ❛tr❛✈és ❧✐♥❤❛✱ ♦✉ ❛té ♠❡s♠♦ ❛tr❛✈és ❞❛ j−és✐♠❛ ❝♦❧✉♥❛✱ q✉❡ ♦ r❡s✉❧t❛❞♦ ❞♦ ❞❡t❡r♠✐♥❛♥t❡ s❡rá ♦ ♠❡s♠♦✳ ❛ ❝♦❧✉♥❛✳ ❚❡♠♦s ❊①❡♠♣❧♦ ✶✳✾✻✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❝á❧❝✉❧♦ ♣❡❧❛ ✷

(97)

(98)

(99) 1 −2 3

(100)

(101)

(102) 1 −1

(103)

(104) = (−2)∆12 + 1∆22 + (−1)∆32 , |B| =

(105)

(106) 2

(107) −2 −1 2

(108) ♦♥❞❡ ∆12 = ∆12 = ∆22 = ∆32 = P♦rt❛♥t♦

(109)

(110) (−1)

(111)

(112)

(113)

(114) 1+2

(115) (−1)

(116)

(117)

(118) 2+2

(119) (−1)

(120)

(121)

(122) 3+2

(123) (−1)

(124) 1+2

(125)

(126)

(127) 2 −1

(128)

(129) = −

(130)

(131)

(132) −2 2

(133)

(134)

(135) 2 −1

(136)

(137) = −

(138)

(139)

(140) −2 2

(141) 1 3

(142)

(143) =8 −2 2

(144)

(145) 1 3

(146)

(147) = 7. 2 −1

(148)

(149) 2 −1

(150)

(151) = −2 −2 2

(152)

(153) 2 −1

(154)

(155) = −2 −2 2

(156) |B| = (−2) · (−2) + 1 · 8 + (−1) · 7 = 5. ❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ ▲❛♣❧❛❝❡ é ✉♠❛ ❢ór♠✉❧❛ ❞❡ r❡❝♦rrê♥❝✐❛ q✉❡ ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❞❡ ♦r❞❡♠ s✉❜♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♦r❞❡♠ n − 1✳ n✱ ❛ ♣❛rt✐r ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❛s ❊♠ ❣r❛♥❞❡ ♣❛rt❡ ❞♦s ❝❛s♦s ❡❧❡ s✐♠♣❧✐✜❝❛ ♠✉✐t♦ ♦ ❝á❧❝✉❧♦ ❞❡ ❞❡t❡r♠✐♥❛♥t❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ s❡ ❢♦r ✉t✐❧✐③❛❞♦ ❡♠ ❝♦♥❥✉♥t♦ ❝♦♠ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❞❡t❡r♠✐♥❛♥t❡✳ Pr♦♣r✐❡❞❛❞❡s✿ ✶✳ ❙❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ❞❡ ✉♠❛ ♠❛tr✐③ ❡♥tã♦ ✷✳ A sã♦ ♥✉❧♦s✱ det(A) = 0❀ det(A) = det(At )❀ ✸✳ ❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ✉♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ❞❛ ♠❛tr✐③ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ✜❝❛ ♠✉❧t✐♣❧✐❝❛❞♦ ♣♦r ❡st❛ ❝♦♥st❛♥t❡❀ ✹✳ ❯♠❛ ✈❡③ tr♦❝❛❞❛ ❛ ♣♦s✐çã♦ ❞❡ ❞✉❛s ❧✐♥❤❛s ✭♦✉ ❝♦❧✉♥❛s✮✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ tr♦❝❛ ❞❡ s✐♥❛❧❀ ✺✳ ❖ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ q✉❡ t❡♠ ❞✉❛s ❧✐♥❤❛s ✭♦✉ ❝♦❧✉♥❛s✮ ✐❣✉❛✐s é ③❡r♦❀ ✸✾

(157)

(158)

(159) a11 ... a1n

(160) ✳ ✳

(161) ✳ ✳ ✳ ✳

(162)

(163)

(164) bi1 + ci1 . . . bin + cin

(165) ✳ ✳

(166) ✳ ✳ ✳ ✳

(167)

(168) a . . . a n1 nn ✻✳

(169)

(170)

(171)

(172) a11 . . . a1n

(173)

(174) ✳ ✳

(175)

(176) ✳ ✳ ✳

(177)

(178)

(179)

(180)

(181) =

(182) bi1 . . . bin

(183)

(184) ✳ ✳

(185)

(186) ✳✳ ✳ ✳

(187)

(188)

(189)

(190) a . . . a n1 nn

(191)

(192)

(193)

(194) a11 . . . a1n

(195)

(196) ✳ ✳

(197)

(198) ✳ ✳ ✳

(199)

(200)

(201)

(202)

(203) +

(204) ci1 . . . cin

(205)

(206) ✳ ✳

(207)

(208) ✳✳ ✳ ✳

(209)

(210)

(211)

(212) a . . . a n1 nn

(213)

(214)

(215)

(216)

(217)

(218)

(219) .

(220)

(221)

(222)

(223) ❈✉✐❞❛❞♦✦ ❖❜s❡r✈❡ q✉❡ ❛q✉✐ t❡♠♦s ❛ s♦♠❛ ♥✉♠❛ ❧✐♥❤❛✱ ❡ ♥ã♦ ✉♠❛ s♦♠❛ ❞❡ ♠❛tr✐③❡s✳ ❉❡ ♠♦❞♦ ❣❡r❛❧✱ ♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ s♦♠❛ ❞❡ ❞✉❛s ♠❛tr✐③❡s ♥ã♦ é ✐❣✉❛❧ à s♦♠❛ ❞♦s ❞❡t❡r♠✐♥❛♥t❡s ❞❛s ♠❛tr✐③❡s✳ ❖✉ s❡❥❛✱ ♣♦❞❡ ❛❝♦♥t❡❝❡r ❞❡ det(A + B) 6= det(A) + det(B)❀ ✼✳ ❖ ❞❡t❡r♠✐♥❛♥t❡ ♥ã♦ s❡ ❛❧t❡r❛ s❡ s♦♠❛r♠♦s ❛ ✉♠❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ♦✉tr❛ ❧✐♥❤❛ ✭♦✉ ❝♦❧✉♥❛✮ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r ✉♠❛ ❝♦♥st❛♥t❡❀ ❊①❡♠♣❧♦✳

(224)

(225) 3 −2 1

(226)

(227) 2 5 0

(228)

(229) 2 4 −2

(230)

(231)

(232)

(233)

(234) 3 −2 1

(235)

(236)

(237)

(238)

(239) =

(240) 2 5 0

(241) .

(242)

(243)

(244)

(245)

(246) 8 0 0

(247) ❆q✉✐✱ à t❡r❝❡✐r❛ ❧✐♥❤❛✱ s♦♠❛♠♦s ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r 2✳ det(AB) = det(A) det(B). ✽✳ ▼❛✐s ❞❡t❛❧❤❡s s♦❜r❡ ❡ss❛s ♣r♦♣r✐❡❞❛❞❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♠ ❬✷✱ ✸❪✳ ✶✳✶✷ ▼❛tr✐③ ❆❞❥✉♥t❛ ✲ ▼❛tr✐③ ■♥✈❡rs❛ ❈♦♥s✐❞❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ♠❛tr✐③ q✉❛❞r❛❞❛✿ a11 a12  a21 a22   a31 a32  ✳ ✳  ✳ ✳ ✳  ✳   ai1 ai2  ✳ ✳  ✳✳ ✳ ✳  an1 an2  . . . a1j . . . a1n . . . a2j . . . a2n   . . . a3j . . . a3n   ✳ ✳ ✳✳  ✳ ✳ . . . ✳✳  . ✳  . . . aij . . . ain   ✳ ✳ ✳✳  ✳ ✳ ✳ . . . ✳✳ . . . anj . . . ann ❈♦♠♦ ❞❡st❛❝❛❞♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ❛ ♣❛rt✐r ❞❡ ❝❛❞❛ ❡♥tr❛❞❛ ✏❝♦❢❛t♦r ❞♦ ❡❧❡♠❡♥t♦ ❡❧❡♠❡♥t♦ aij ✑✮✱ aij ✑✱ q✉❡ é ♦ ♥ú♠❡r♦ q✉❡ ✐♥❞✐❝❛♠♦s ♣♦r aij ❞❛ ♠❛tr✐③ t❡♠♦s ♦ ∆ij ✭✏❧ê✲s❡ ❝♦❢❛t♦r ❞♦ ❞❡✜♥✐❞♦ ♣♦r✿ ∆ij = (−1)i+j det(Aij ), ♦♥❞❡ Aij é ❛ ♠❛tr✐③ q✉❡ s❡ ♦❜té♠ ❡❧❡♠✐♥❛♥❞♦ ❛ ❧✐♥❤❛ i ❈♦♠ ❡ss❡s ❝♦❢❛t♦r❡s ♣♦❞❡♠♦s ❢♦r♠❛r ✉♠❛ ♥♦✈❛ ♠❛tr✐③ ❝♦❢❛t♦r❡s ❞❡ A✱ ♦✉ s❡❥❛✱ A = [∆ij ]. ✹✵ ❡ ❛ ❝♦❧✉♥❛ A✱ j ❞❛ ♠❛tr✐③ A✳ ❞❡♥♦♠✐♥❛❞❛ ♠❛tr✐③ ❞♦s

(248) ❊①❡♠♣❧♦ ✶✳✾✼✳ ❉❛❞❛ ❛ ♠❛tr✐③  2 1 0 A =  −3 1 4  . 1 6 5  ❉❡t❡r♠✐♥❡ ❛ ♠❛tr✐③ ❞♦s ❝♦❢❛t♦r❡s ❞❡ A✳ ❙♦❧✉çã♦✿ ❯s❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❡ ❞❡s❡♥✈♦❧✈❡♥❞♦ ♦s ❝á❧❝✉❧♦s ♦❜t❡♠♦s ∆11 = (−1) 1+1 ∆12 = (−1) 1+2 ∆13 = (−1) 1+3  det  1 4 6 5 det  −3 4 1 5 det  −3 1 1 6 = −19;   = 19; = −19; ❡ ❛♥❛❧♦❣❛♠❡♥t❡ ♦s ❞❡♠❛✐s ❝♦❢❛t♦r❡s✳ ❊♥tã♦✱  −19 19 −19 A =  −5 10 −11  . 4 −8 5  ❉❡✜♥✐çã♦ ✶✳✾✽✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A✱ ❝❤❛♠❛r❡♠♦s ❞❡ ♠❛tr✐③ ❛❞❥✉♥t❛ ❞❡ A à tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❞♦s ❝♦❢❛t♦r❡s ❞❡ A✳ ❙✐♠❜♦❧✐❝❛♠❡♥t❡✿ t adj(A) = A . ❊①❡♠♣❧♦ ✶✳✾✾✳ ❯s❛♥❞♦ ❛s ✐♥❢♦r♠❛çõ❡s ❞♦ ❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ t❡♠♦s  −19 −5 4 t 10 −8  . adj(A) = A =  19 −19 −11 5  ❉❡✈❡♠♦s ♥♦t❛r q✉❡ ✉s❛♥❞♦ ❛s ♠❛tr✐③❡s ❞♦s ❞♦✐s ❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✱ t❡♠♦s  −19 2 1 0 A · adj(A) =  −3 1 4   19 −19 1 6 5  1 0 = −19  0 1 0 0     −19 0 0 −5 4 10 −8  =  0 −19 0  0 0 −19 −11 5  0 0  = −19I3 . 1 ❊ss❡ ❝❛s♦ ♣❛rt✐❝✉❧❛r ♥ã♦ é ✉♠❛ ♠❡r❛ ❝♦✐♥❝✐❞ê♥❝✐❛✱ ♣♦✐s ♥❛ ✈❡r❞❛❞❡ t❡♠♦s ❡♠ ❣❡r❛❧ ♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳ ❚❡♦r❡♠❛ ✶✳✶✵✵✳ ❙❡ A é ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n ❡ In é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n✱ ❡♥tã♦ A · adj(A) = adj(A) · A = det(A)In . ✹✶

(249) ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛✱ ❛♣❡s❛r ❞❡ s✐♠♣❧❡s✱ ♥ã♦ ❛ ❛♣r❡s❡♥t❛r❡♠♦s ❛q✉✐✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛ ❬✸✱ ♣á❣✐♥❛ ✼✸❪✳ ❉❡✜♥✐çã♦ ✶✳✶✵✶✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ✐♥✈❡rsí✈❡❧ q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ In é ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ ✐♥✈❡rs❛ ❞❡ A ❡ ❞❡♥♦t❛♠♦s ❊①❡♠♣❧♦ ✶✳✶✵✷✳ B=A n✳ −1 ❞❡ ♦r❞❡♠ B A= ❞❡ ♦r❞❡♠   B ❡①✐st❡✱ ❞✐③✲s❡ q✉❡ ❡❧❛ é ❛ é ❛ ✐♥✈❡rs❛ ❞❡ A −1 6 2 11 4 n✱ ❞✐③✲s❡ q✉❡ A é AB = BA = In ✱ ♦♥❞❡ t❛❧ q✉❡ ◗✉❛♥❞♦ ❛ ♠❛tr✐③ ❡ ❞✐③❡♠♦s ❙❡❥❛ A n A✳ . ❆ss✉♠✐♥❞♦ ❛ ♣r✐♥❝í♣✐♦ q✉❡ ❛ ✐♥✈❡rs❛ ❡①✐st❡✱ ♣♦❞❡♠♦s ♣r♦❝✉r❛r B= t❛❧ q✉❡ AB = I2 ❡ BA = I2 .   P♦rt❛♥t♦✱   a b c d ■♠♣♦♥❞♦ ❛ ♣r✐♠❡✐r❛ ❝♦♥❞✐çã♦✱  6 2 11 4 t❡♠♦s  a b c d   6a + 2c 6b + 2d 11a + 4c 11b + 4d 6a + 2c = 1 11a + 4c = 0 =  e  = 1 0 0 1    1 0 0 1 . 6b + 2d = 0 . 11b + 4d = 1 a = 2, b = −1, c = − 11 ❡ d = 3✳ 2      2 −1 6 2 1 0 , = − 11 11 4 0 1 3 2 ❘❡s♦❧✈❡♥❞♦ ♦s s✐st❡♠❛s✱ t❡♠♦s ♦✉ s❡❥❛✱ AB = I ✳ ❚❛♠❜é♠  ♦✉ s❡❥❛✱ BA = I 2 −1 3 − 11 2  6 2 11 4 A é ❛ ✐♥✈❡rs❛ ❞❛ ♠❛tr✐③ A ❖❜s❡r✈❛çã♦ ✶✳✶✵✸✳ =B=  =  1 0 0 1  , ✶✳ ❙❡ =B A −1 −1 2 −1 − 11 3 2  ❡ ❞❡♥♦t❛♠♦s✳ A ❡ B sã♦ ♠❛tr✐③❡s q✉❛❞r❛❞❛s ❞❡ ♠❡s♠❛ ♦r❞❡♠✱ ❛♠❜❛s ✐♥✈❡rsí✈❡✐s ✭✐st♦ é✱ ❡①✐st❡♠ (AB)  ❡✱ ♣♦rt❛♥t♦✱ −1 −1 ❊♥tã♦✱ A−1 ❡ B −1 ✮✱ ❡♥tã♦ AB é ✐♥✈❡rsí✈❡❧ ❡ ✳ ❉❡ ❢❛t♦✱ ❜❛st❛ ♦❜s❡r✈❛r q✉❡ (AB)(B −1 A−1 ) = A(BB −1 )A−1 = AIA−1 = AA−1 = I ❡ q✉❡✱ ❛♥❛❧♦❣❛♠❡♥t❡✱ (B −1 A−1 )(AB) = I ✳ ✹✷

(250) ✷✳ ❙❡ A B t❛❧ q✉❡ BA = I ✱ B = A−1 ✳ é ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❡ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ é ✐♥✈❡rsí✈❡❧✱ ♦✉ s❡❥❛✱ A−1 ❡①✐st❡ ❡✱ ❛❧é♠ ❞✐ss♦✱ A ❚❡♦r❡♠❛ ✶✳✶✵✹✳ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n ❡♥tã♦ A é ✐♥✈❡rsí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ det(A) 6= 0✳ s❡✱ (⇒) −1 = A A = In ✳ A ❉❡♠♦♥str❛çã♦✳ ❙❡ AA ▲♦❣♦✱ t❡♠♦s −1 A−1 é ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦ ❡①✐st❡ ♠❛tr✐③ q✉❛❞r❛❞❛ t❛❧ q✉❡ det(AA−1 ) = det(A−1 A) = det(In ). P❡❧❛ Pr♦♣r✐❡❞❛❞❡ ✽ ❞❡ ❞❡t❡r♠✐♥❛♥t❡✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❢❛t♦ q✉❡ det(In ) = 1✱ t❡♠♦s q✉❡ det(A) det(A−1 ) = det(A−1 ) det(A) = 1. ❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ det(A) 6= 0 ✭♣♦✐s ❝❛s♦ ❝♦♥trár✐♦✱ ✐st♦ é✱ s❡ det(A) = 0✱ t❡rí❛♠♦s det(A) det(A ) = 0✮✳ (⇐) ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ det(A) = 6 0✳ ❙❡♥❞♦ A ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❞❡ ♦r❞❡♠ n✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✵✵✱ t❡♠♦s q✉❡ A · adj(A) = adj(A) · A = det(A)In ✳ ❆ss✐♠✱ s❡♥❞♦ det(A) 6= 0✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡     1 1 A adj(A) = adj(A) A = In . det(A) det(A) −1 ▲♦❣♦✱ A é ✐♥✈❡rsí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é A−1 = 1 adj(A) det(A) ✭✶✳✼✮ ✳ ❖❜s❡r✈❛çã♦ ✶✳✶✵✺✳ ❈♦♠ ♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r é ♣♦ssí✈❡❧ ❞❛r ❡①❡♠♣❧♦s ❞❡ ♠❛tr✐③❡s q✉❛❞r❛❞❛s q✉❡ ♥ã♦ sã♦ ✐♥✈❡rsí✈❡✐s✳ P♦r ❡①❡♠♣❧♦✱ A= ♥ã♦ é ✐♥✈❡rsí✈❡❧ ♣♦✐s 1 1 1 1  6 2 11 4   det(A) = 0✳ ❊①❡♠♣❧♦ ✶✳✶✵✻✳ ❈♦♥s✐❞❡r❡ ❛ ♠❛tr✐③ A= det(A) = 24 − 22 = 2 6= 0 ♣❡❧❛ ❢ór♠✉❧❛ ✭✶✳✼✮✳  . ❡✱ ♣♦rt❛♥t♦✱ ❡①✐st❡ ❛ ✐♥✈❡rs❛ ❞❡ A✳ ❈❛❧❝✉❧❡♠♦s ❛ ✐♥✈❡rs❛ ❉❡s❡♥✈♦❧✈❡♥❞♦ ♦s ❝á❧❝✉❧♦s✱ ♦❜t❡♠♦s A=  4 −11 −2 6  ❡ adj(A) =  4 −2 −11 6  . ❊♥tã♦✱ A −1 1 1 = adj(A) = det(A) 2  4 −2 −11 6 ✹✸  =  2 −1 3 − 11 2  .

(251) ✶✳✶✸ ▼❛tr✐③❡s ❊❧❡♠❡♥t❛r❡s ❉❡✜♥✐çã♦ ✶✳✶✵✼✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ A✱ ❡♥t❡♥❞❡✲s❡ ♣♦r ❖♣❡r❛çõ❡s ❊❧❡♠❡♥t❛r❡s s♦❜r❡ ❛s ▲✐♥❤❛s ❞❡ A✱ q✉❛❧q✉❡r ✉♠❛ ❞❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s✿ A❀ ✶✳ ♣❡r♠✉t❛r ❞✉❛s ❧✐♥❤❛s ❞❡ ✷✳ ♠✉❧t✐♣❧✐❝❛r t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡ ✉♠❛ ❧✐♥❤❛ ♣♦r ✉♠❛ ♥ú♠❡r♦ ♥ã♦ ♥✉❧♦❀ ✸✳ s✉❜st✐t✉✐r ✉♠❛ ❧✐♥❤❛ ♣❡❧❛ s♦♠❛ ❞❡❧❛ ♠❡s♠❛ ❝♦♠ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ♦✉tr❛✳ ❖❜s❡r✈❛çã♦ ✶✳✶✵✽✳ ◆♦t❡ q✉❡ t♦❞❛s ❛s ✸ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❧✐♥❤❛s sã♦ r❡✈❡rsí✈❡✐s✱ ♦✉ s❡❥❛✱ ❛♦ ❛♣❧✐❝❛r ✉♠❛ ♦♣❡r❛çã♦ ❡ ♦❜t❡r ♦✉tr❛ ♠❛tr✐③✱ ♣♦❞❡✲s❡ ✈♦❧t❛r ♣❛r❛ ❛ ♠❛tr✐③ ♦r✐❣✐♥❛❧ ❝♦♠ ❛ ❛♣❧✐❝❛çã♦ ❞❛ ♠❡s♠❛ ♦♣❡r❛çã♦ ❛❞❡q✉❛♥❞❛♠❡♥t❡✳ ❊①❡♠♣❧♦ ✶✳✶✵✾✳  A=  1 2 1 0 B=  2 2 3 0 1 0 ❚❡♠♦s ♣♦r ❡①❡♠♣❧♦✿ : ❖♣❡r❛çã♦ ✶  : ❖♣❡r❛çã♦ ✷  7−→ 7−→  1 0 1 2  2 2 3 0 5 0 :  ❖♣❡r❛çã♦ ✶ : ❖♣❡r❛çã♦ ✷  7−→  7−→ 1 2 1 0  2 2 3 0 1 0 ,  ❡   2 1 C= 0 1 : 3 1 ❖♣❡r❛çã♦ ✸ ❉❡✜♥✐çã♦ ✶✳✶✶✵✳ ✐❞❡♥t✐❞❛❞❡ ❞❡ In ✱   2 1  0 1 : 7 3 7−→ ❖♣❡r❛çã♦ ✸   2 1  0 1 . 3 1 7−→ ❯♠❛ ▼❛tr✐③ ❊❧❡♠❡♥t❛r é ✉♠❛ ♠❛tr✐③ ♦❜t✐❞❛ ❛ ♣❛rt✐r ❞❛ ♠❛tr✐③ ❛tr❛✈és ❞❛ ❡①❡❝✉sã♦ ❞❡ ✉♠❛ ú♥✐❝❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r s♦❜r❡ ❛s ❧✐♥❤❛s In ✳ ❊①❡♠♣❧♦ ✶✳✶✶✶✳ ❚❡♠♦s ♣♦r ❡①❡♠♣❧♦✿ I2 =  1 0 0 1  :  1 0 0 I3 =  0 1 0  : 0 0 1   1 0 0 I3 =  0 1 0  : 0 0 1  ❡ ❖♣❡r❛çã♦ ✶ ❖♣❡r❛çã♦ ✷ ❖♣❡r❛çã♦ ✸ 7−→ 7−→ 7−→  0 1 1 0  ,  1 0 0  0 1 0  0 0 3   1 0 0  2 1 0 . 0 0 1  ❖❜s❡r✈❛çã♦ ✶✳✶✶✷✳ ❯♠❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r E1 é ✐♥✈❡rsí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é ❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r E2 ✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ♠❛tr✐③ ♦❜t✐❞❛ ♣♦r ❡❢❡t✉❛r ❛ ♦♣❡r❛çã♦ ✐♥✈❡rs❛ ❝♦♠ ❧✐♥❤❛s ❞❛ ♦♣❡r❛çã♦ ❡❢❡t✉❛❞❛ ♣❛r❛ ♦❜t❡r ❚❡♦r❡♠❛ ✶✳✶✶✸✳ m × n✱ ❡♥tã♦ ❙❡❥❛ EA E E1 ✳ ✉♠❛ ♠❛tr✐③ ❡❧❡♠❡♥t❛r ♦❜t✐❞❛ ❞❡ é ✐❣✉❛❧ ❛ ♠❛tr✐③ ♦❜t✐❞❛ ❞❡ ❡❧❡♠❡♥t❡r ♣❛r❛ s❡ ♦❜t❡r E✳ ✹✹ A In ✳ ❙❡ A é ✉♠❛ ♠❛tr✐③ ❡❢❡t✉❛♥❞♦✲s❡ ❛ ♠❡s♠❛ ♦♣❡r❛çã♦

(252) ❉❡♠♦♥str❛çã♦✳ ❋❛r❡♠♦s ❛ ♣r♦✈❛ ❞❡ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r q✉❡ ♣♦❞❡ s❡r ❢❛❧❝✐❧♠❡♥t❡ I3 ✱   a11 a12 a13 a14 A =  a21 a22 a23 a24  , a31 a32 a33 a34 ❛❞❛♣t❛❞♦ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ❝❛s♦✳ ❈♦♥s✐❞❡r❡  1 0 0 E= 0 1 0  k 0 1  ♦♥❞❡ E é ♦❜t✐❞❛ ❞❡ I3 ❡ ♣❡❧❛ ♦♣❡r❛çã♦ ❡❧❡♠❡♥t❛r 3✱ ❝♦♠ k ✉♠ ♥ú♠❡r♦ r❡❛❧✳ ❊❢❡t✉❛♥❞♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s ♠❛tr✐③❡s✱ ♦❜t❡♠♦s  a11 a12 a13 a14 , a21 a22 a23 a24 EA =  ka11 + a31 ka12 + a32 ka13 + a33 ka14 + a34  q✉❡ é ❡①❛t❛♠❡♥t❡ ❛ ♠❛tr✐③ ♦❜t✐❞❛ q✉❛♥❞♦ ❡❢❡t✉❛✲s❡ ❛ ♦♣❡r❛çã♦ ✸ ♥❛ ♠❛tr✐③ ❚❡♦r❡♠❛ ✶✳✶✶✹✳ ❙❡❥❛ A ✉♠❛ ♠❛tr✐③ n × n✳ A A✳ é ✐♥✈❡rsí✈❡❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A = Ek−1 · · · E2−1 E1−1 , ♦♥❞❡ E 1 , E2 , . . . , E k sã♦ ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ t❡♦r❡♠❛✱ ❛♣❡s❛r ❞❡ s✐♠♣❧❡s✱ ♥ã♦ ❛ ❛♣r❡s❡♥t❛r❡♠♦s ❛q✉✐✳ ❖ ❧❡✐t♦r ♣♦❞❡ ❡♥❝♦♥trá✲❧❛ ❬✷✱ ♣á❣✐♥❛ ✺✽❪✳ ◆❛ ♣rát✐❝❛✱ ♦♣❡r❛♠♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❝♦♠ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✱ ❛té ❝❤❡❣❛r♠♦s à ♠❛tr✐③ A✳ I ❛s ♠❛tr✐③❡s A ❡ I✱ ❛tr❛✈és ❞❡ ♥❛ ♣♦s✐çã♦ ❝♦rr❡s♣♦♥❞❡t❡ à ♠❛tr✐③ ❆ ♠❛tr✐③ ♦❜t✐❞❛ ♥♦ ❧✉❣❛r ❝♦rr❡s♣♦♥❞❡♥t❡ à ♠❛tr✐③ I s❡rá ❛ ✐♥✈❡rs❛ ❞❡ A✿ (A|I) −→ (I|A−1 ). ❊①❡♠♣❧♦ ✶✳✶✶✺✳ ❙❡❥❛  1 2 7 A =  0 3 1 . 0 5 2  ❈♦❧♦q✉❡♠♦s ❛ ♠❛tr✐③ ❥✉♥t♦ ❝♦♠ ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❡ ❛♣❧✐q✉❡♠♦s ❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❧✐♥❤❛s✱ ♣❛r❛ r❡❞✉③✐r ❛ ♣❛rt❡ ❡sq✉❡r❞❛ ✭q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ A✮ à ❢♦r♠❛ ❞❛ ✐❞❡♥t✐❞❛❞❡✳ ❈❛❞❛ ♦♣❡r❛çã♦ ❞❡✈❡ s❡r ❡❢❡t✉❛❞❛ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ♥❛ ♣❛rt❡ ❞✐r❡✐t❛ ❞❛ ♠❛tr✐③✿  1 2 7 1 0 0  0 3 1 0 1 0 . 0 5 2 0 0 1  ❚r♦❝❛♥❞♦ ❛ s❡❣✉♥❞❛ ❡ t❡r❝❡✐r❛ ❧✐♥❤❛s✱ ♦❜t❡♠♦s✿   1 2 7 1 0 0  0 5 2 0 0 1 . 0 3 1 0 1 0 ❙♦♠❛♠♦s à s❡❣✉♥❞❛ ❛ t❡r❝❡✐r❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r −2✿   1 2 7 1 0 0  0 −1 0 0 −2 1  . 0 3 1 0 1 0 ✹✺

(253) ▼✉❧t✐♣❧✐❝❛♠♦s ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣♦r −1✿   1 2 7 1 0 0  0 1 0 0 2 −1  . 0 3 1 0 1 0 ❙♦♠❛♠♦s à ♣r✐♠❡✐r❛ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛❞s ♣♦r −2✿  1 0 7 1 −4 2  0 1 0 0 2 −1  . 0 3 1 0 1 0  −3✿  ❙♦♠❛♠♦s à t❡r❝❡✐r❛ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r  1 0 7 1 −4 2  0 1 0 0 2 −1  . 0 0 1 0 −5 3 −7✿  ❙♦♠❛♠♦s à ♣r✐♠❡✐r❛ ❛ t❡r❝❡✐r❛ ❧✐♥❤❛ ♠✉❧t✐♣❧✐❝❛❞❛ ♣♦r  1 0 0 1 31 −19  0 1 0 0 2 −1  . 0 0 1 0 −5 3 ❋✐♥❛❧♠❡♥t❡✱ ♦❜t❡♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ à ❡sq✉❡r❞❛ ❡ ❛ ✐♥✈❡rs❛ ❞❡ A à ❞✐r❡✐t❛✳ P♦rt❛♥t♦✱  1 31 −19 A−1  0 2 −1  . 0 −5 3  ✶✳✶✹ ▼❛tr✐③❡s ❡ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❯♥✐♥❞♦ ♦s ❡st✉❞♦s q✉❡ ❥á ❞❡s❡♥✈♦❧✈❡♠♦s✱ ✈❛♠♦s ❛❣♦r❛ ❛♣r❡s❡♥t❛r ❛❧❣✉♥s ❜r❡✈❡s r❡s✉❧t❛❞♦s q✉❡ r❡❧❛❝✐♦♥❛♠ ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠❛tr✐③❡s ❝♦♥s✐❞❡r❛❞❛s ❝♦♠ ❛rt✐♠ét✐❝❛ ♠♦❞✉❧❛r✳ ❉❡✜♥✐çã♦ ✶✳✶✶✻✳ ❉❛❞♦s  a11 a12  a21 a22  A =  ✳✳ ✳ ✳  ✳ ✳ ak1 ak2 ❝♦♠ m ∈ Z, m > 0, ❡  . . . a1l . . . a2l   ❡ ✳  ✳✳ ✳ ✳ ✳  . . . akl ❛s ♠❛tr✐③❡s  b11 b12 . . . b1l  b21 b22 . . . b2l  B =  ✳✳ ✳ ✳ ✳✳ ✳ ✳  ✳ ✳ ✳ ✳ bk1 bk2 . . . bkl    ,  aij , bij ∈ Z, 1 ≤ i ≤ k ❡ 1 ≤ j ≤ l✱ ❞✐③❡♠♦s q✉❡ ❛ ♠❛tr✐③ A é ❈♦♥❣r✉❡♥t❡ ❛ B q✉❛♥❞♦ aij ≡ bij (mod m)✱ ♣❛r❛ 1 ≤ i ≤ k ❡ 1 ≤ j ≤ l✱ ❡ ❞❡♥♦t❛♠♦s ♣♦r ♠❛tr✐③ A ≡ B (mod m). ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ t❡♠♦s ❛ Pr♦♣r✐❡❞❛❞❡ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r✳ ✹✻ 8 ❡ ♦ ❚❡♦r❡♠❛ ✶✳✶✵✵ ♣❛r❛ ♠❛tr✐③❡s ❝♦♠

(254) ❚❡♦r❡♠❛ ✶✳✶✶✼✳ ❙❡ A B ❡ sã♦ ♠❛tr✐③❡s q✉❛❞r❛❞❛ ❝♦♠ ❡♥tr❛❞❛s ❞❡ Z ❡ ♦r❞❡♥s n✱ ❡♥tã♦ det(AB) ≡ det(A) det(B) (mod m). n = 2✳  b12 , b22 ❉❡♠♦♥str❛çã♦✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ✈❛♠♦s ❞❡♠♦♥str❛r ♦ ❝❛s♦ A= ❝♦♠  a11 a12 a21 a22 aij , bij ∈ Z, 1 ≤ i ≤ 2 AB =  ❡ 1 ≤ j ≤ 2✱ ❡  B=  b11 b21 ❈♦♥s✐❞❡r❡ ❞❡ ♠♦❞♦ q✉❡  a11 b11 + a12 b21 a11 b12 + a12 b22 a21 b11 + a22 a21 a21 b12 + a22 b22 . ❊♥tã♦✱ det(AB) ≡ (a11 b11 + a12 b21 )(a21 b12 + a22 b22 ) − (a11 b12 + a12 b22 )(a21 b11 + a22 a21 ) ≡ (a11 a22 − a12 a21 )(b11 b22 − b12 b21 ) ≡ det(A) det(B) (mod m). ❚❡♦r❡♠❛ ✶✳✶✶✽✳ ❙❡ A é ♠❛tr✐③ q✉❛❞r❛❞❛ ❝♦♠ ❡♥tr❛❞❛s ❞❡ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n✱ Z ❡ ♦r❞❡♠ n ❡ In é ❛ ❡♥tã♦ A · adj(A) ≡ adj(A) · A ≡ det(A)In (mod m). ❉❡♠♦♥str❛çã♦✳ P♦r s✐♠♣❧✐❝✐❞❛❞❡✱ ✈❛♠♦s ❞❡♠♦♥str❛r ♦ ❝❛s♦ A= ❝♦♠ aij ∈ Z, 1 ≤ i ≤ 2 ❡ 1 ≤ j ≤ 2✱  a11 a12 a21 a22 n = 2✳ ❈♦♥s✐❞❡r❡  ❞❡ ♠♦❞♦ q✉❡ adj(A) =  a22 −a12 −a21 a11  . ❊♥tã♦✱ A · adj(A) =  a11 a22 − a12 a21 0 0 a11 a22 − a12 a21  ≡ det(A)I2 (mod m). ❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛✱ ♦❜t❡♠♦s adj(A) · A ≡ det(A)I2 (mod m). ❈♦♠ ♦s ❞♦✐s r❡s✉❧t❛❞♦s ❛♥t❡✐♦r❡s ♣♦❞❡♠♦s ❞❡♠♦♥str❛r ♦ ❛♥á❧♦❣♦ ❡♠ ❛r✐t✐♠ét✐❝❛ ♠♦❞✉❧❛r ❞♦ ❚❡♦r❡♠❛ ✶✳✶✵✹✳ ✹✼

(255) ❉❡✜♥✐çã♦ ✶✳✶✶✾✳ ❉❛❞❛ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A ❝♦♠ ❡♥tr❛❞❛s ❡♠ Z ❡ ❞❡ ♦r❞❡♠ n✱ A é ✐♥✈❡rsí✈❡❧ (mod m) q✉❛♥❞♦ ❡①✐st❡ ✉♠❛ ♠❛tr✐③ B ❞❡ ♦r❞❡♠ n t❛❧ q✉❡ AB ≡ BA ≡ In (mod m)✳ ◗✉❛♥❞♦ ❛ ♠❛tr✐③ B ❡①✐st❡✱ ❞✐③✲s❡ q✉❡ ❡❧❛ é ❛ ✐♥✈❡rs❛ ❞❡ A (mod m) ❡ ❞❡♥♦t❛♠♦s B = A−1 ❡ ❞✐③❡♠♦s A−1 é ❛ ✐♥✈❡rs❛ ❞❡ A (mod m)✳ ❞✐③✲s❡ q✉❡ ❚❡♦r❡♠❛ ✶✳✶✷✵✳ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❆ ❝♦♠ ❡♥tr❛❞❛s ❡♠ ♠ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ r❡sí❞✉♦ ❞❡ det(A) Z é ✐♥✈❡rsí✈❡❧ ♠ó❞✉❧♦ ♠ó❞✉❧♦ ♠ t❡♠ ✉♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ ♠ó❞✉❧♦ ♠✳ ❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ é ❛♥á❧♦❣❛ ❛♦ ❝❛s♦ r❡❛❧✳ (⇒) ❙❡ A é ✐♥✈❡rsí✈❡❧ ♠ó❞✉❧♦ m✱ AA−1 ≡ A−1 A ≡ In (mod m)✳ ▲♦❣♦✱ ❡♥tã♦ ❡①✐st❡ ♠❛tr✐③ q✉❛❞r❛❞❛ A−1 t❛❧ q✉❡ t❡♠♦s det(AA−1 ) ≡ det(A−1 A) ≡ det(In ) (mod m). P❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✶✼✒ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❢❛t♦ q✉❡ det(In ) ≡ 1 (mod m)✱ t❡♠♦s q✉❡ det(A) det(A−1 ) ≡ det(A−1 ) det(A) ≡ 1 (mod m). ❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ (⇐) det(A) t❡♠ ✉♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ (mod m)✳ det(A) t❡♠ ✉♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ (mod m)✳ ❞❡ ♦r❞❡♠ n✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶✶✽✱ t❡♠♦s q✉❡ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ ✉♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ ❙❡♥❞♦ A A · adj(A) ≡ adj(A) · A ≡ det(A)In (mod m). det(A) ✉♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ (mod m)✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r     1 1 adj(A) ≡ adj(A) A ≡ In (mod m). A det(A) det(A) ❆ss✐♠✱ t❡♥❞♦ ▲♦❣♦✱ A q✉❡ é ✐♥✈❡rsí✈❡❧ ❡ s✉❛ ✐♥✈❡rs❛ é A−1 = 1 adj(A) (mod m). det(A) ✭✶✳✽✮ ✳ ❙❛❜❡♠♦s q✉❡ ♦ r❡sí❞✉♦ ❞♦ ♠ó❞✉❧♦ det(A) ♠ó❞✉❧♦ m só t❡rá ✉♠ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ m s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡st❡ r❡sí❞✉♦ ❡ m ♥ã♦ t✐✈❡r❡♠ ❢❛t♦r ♣r✐♠♦ ❝♦♠✉♠✳ ❚❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦✳ ❈♦r♦❧ár✐♦ ✶✳✶✷✶✳ m A ❝♦♠ ❡♥tr❛❞❛s ❡♠ Z é ✐♥✈❡rtí✈❡❧ ♠ó❞✉❧♦ det(A) ♠ó❞✉❧♦ m ♥ã♦ tê♠ ❢❛t♦r❡s ♣r✐♠♦s ❡♠ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ m ❡ ♦ r❡sí❞✉♦ ❞❡ ❝♦♠✉♥s✳ ❈♦♠♦ ♦s ú♥✐❝♦s ❢❛t♦r❡s ♣r✐♠♦s ❞❡ m = 26 sã♦ 2 ❡ 13✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦r♦❧ár✐♦✳ ❈♦r♦❧ár✐♦ ✶✳✶✷✷✳ ❯♠❛ ♠❛tr✐③ q✉❛❞r❛❞❛ A ❝♦♠ ❡♥tr❛❞❛s ❡♠ Z26 é ✐♥✈❡rtí✈❡❧ ♠ó❞✉❧♦ 26 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ♦ r❡sí❞✉♦ ❞❡ det(A) ♠ó❞✉❧♦ ✹✽ 26 ♥ã♦ é ❞✐✈✐sí✈❡❧ ♣♦r 2 ♦✉ 13✳

(256) ❈❛s♦ ♣❛rt✐❝✉❧❛r✿ ❙❡  a b c d  Z ❡ ♦ det(A) = ad − bc (mod m) é r❡❧❛t✐✈❛♠❡♥t❡ det(A)(mod m) é ❞❛❞❛ ♣♦r✿   d −b −1 −1 (mod m) A = (ad − bc) −c a t❡♠ ❡♥tr❛❞❛s ❡♠ ♣r✐♠♦ ❝♦♠ m✱ ❡♥tã♦ ❛ ✐♥✈❡rs❛ ❞❡ ♦♥❞❡ (ad − bc)−1 é ♦ ✐♥✈❡rs♦ ❞❡ ❊①❡♠♣❧♦ ✶✳✶✷✸✳ ad − bc (mod m)✳ ❊♥❝♦♥tr❡ ❛ ✐♥✈❡rs❛ ❞❡ A= ♠ó❞✉❧♦  5 3 8 5  26✳ ❙♦❧✉çã♦✿ ❖ −1 1 ✭✶✳✾✮ det(A) = ad − bc = 5 · 5 − 3 · 8 = 25 − 24 = 1✳ ▲♦❣♦✱ (ad − bc)−1 = ≡ 1(mod 26) ✭✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮✳ ❆ss✐♠ ♣♦r ✭✶✳✾✮✱ t❡♠♦s✿     5 23 5 −3 −1 (mod 26). ≡ A =1 18 5 −8 5 ✹✾

(257) ❈❛♣ít✉❧♦ ✷ ❈ó❞✐❣♦s ❊❧❡♠❡♥t❛r❡s ❡ ❈r✐♣t♦❣r❛✜❛ ❆ ❝r✐♣t♦❣r❛✜❛ r❡♣r❡s❡♥t❛ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ t❡①t♦s ❝♦♠✉♥s ❡♠ ♠❡♥s❛❣❡♥s ❝♦❞✐✜❝❛❞❛s✱ q✉❡ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ♦❝✉❧t❛r ❛ ✐♥❢♦r♠❛çã♦ ♣❛r❛ q✉❡ ♣❡ss♦❛s ♥ã♦ ❛✉t♦r✐③❛❞❛s ♥ã♦ t❡♥❤❛♠ ❛❝❡ss♦✱ ❣❛r❛♥t✐♥❞♦ ♣r✐✈❛❝✐❞❛❞❡✳ ❆ ♣❛❧❛✈r❛ ❝r✐♣t♦❣r❛✜❛ t❡♠ ♦r✐❣❡♠ ❣r❡❣❛ ✭❦r✐♣t♦s ❂ ❡s❝♦♥❞✐❞♦✱ ♦❝✉❧t♦ ❡ ❣r✐❢♦ ❂ ❣r❛✜❛✮ ❡ r❡♣r❡s❡♥t❛ ❛ ❝✐ê♥❝✐❛ ❞♦s ❝ó❞✐❣♦s✱ ♥❛ q✉❛❧ s❡ ✉t✐❧✐③❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ té❝♥✐❝❛s q✉❡ tr❛♥s❢♦r♠❛♠ ✉♠❛ ♠❡♥s❛❣❡♠ ❡♠ ❝ó❞✐❣♦s ❛tr❛✈és ❞❡ ✉♠ ♣r♦❝❡ss♦ ❝❤❛♠❛❞♦ ❝♦❞✐✜❝❛çã♦✱ ♣❡r♠✐t✐♥❞♦ q✉❡ ❛♣❡♥❛s ♦ ❞❡st✐♥❛tár✐♦ ❞❡s❡❥❛❞♦ ❝♦♥s✐❣❛ ❞❡❝♦❞✐✜❝❛r ❡ ❧❡r ❛ ♠❡♥s❛❣❡♠✱ ❛ ♣❛rt✐r ❞♦ ♣r♦❝❡ss♦ ✐♥✈❡rs♦✱ ❛ ❞❡❝♦❞✐✜❝❛çã♦✳ ▼❛s ❛✜♥❛❧✱ ♦ q✉❡ é ✉♠ ❝ó❞✐❣♦❄ ❙❡❣✉♥❞♦ ♦ ♠✐♥✐❞✐❝✐♦♥ár✐♦ ✭✈❡r ❬✻❪✮✱ ❝ó❞✐❣♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✐♥❛✐s ❝♦♥✈❡♥❝✐♦♥❛✐s ♦✉ s❡❝r❡t♦s ✉t✐❧✐③❛❞♦s ❡♠ ❝♦rr❡s♣♦♥❞ê♥❝✐❛s ❡ ❝♦♠✉♥✐❝❛çõ❡s✱ ♦✉ s❡❥❛✱ ❝ó❞✐❣♦ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ s✉❜st✐t✉t♦s ♣❛r❛ ✉♠❛ ❞❡t❡r♠✐♥❛❞❛ ✐♥❢♦r♠❛çã♦✳ ❉♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ♠❛t❡♠át✐❝♦✱ ❜❛s❡❛❞♦ ❡♠ ❬✶❪✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ❝ó❞✐❣♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❉❡✜♥✐çã♦ ✷✳✶✳ ❯♠ ❈ó❞✐❣♦ é ✉♠ s✐st❡♠❛ ❢♦r♠❛❞♦ ♣♦r ✉♠ q✉í♥t✉♣❧♦ (T, C, K, S, D)✱ ♦♥❞❡ • T é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❝❛r❛❝t❡r❡s ❞♦ t❡①t♦ ❝♦♠✉♠ ✭♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✮✳ • C é ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s ❝❛r❛❝t❡r❡s ❞❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✳ • K é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❤❛✈❡s q✉❡ ♦❜❡❞❡❝❡♠ ❞❡t❡r♠✐♥❛❞❛s r❡❣r❛s✳ • S é ♦ ❝♦♥❥✉♥t♦ ❞❡ r❡❣r❛s ❞❡ ❝♦❞✐✜❝❛çã♦✳ • D é ♦ ❝♦♥❥✉♥t♦ ❞❡ r❡❣r❛s ❞❡ ❞❡❝♦❞✐✜❝❛çã♦✳ q✉❡ s❛t✐s❢❛③ ❛ s❡❣✉✐♥t❡ ❝♦♥❞✐çã♦✿ ♣❛r❛ ❝❛❞❛ k ∈ K ❡①✐st❡ ✉♠❛ r❡❣r❛ ♣❛r❛ ❝♦❞✐✜❝❛r✱ sk ∈ S ✱ ❡ ✉♠❛ r❡❣r❛ ❝♦rr❡s♣♦♥❞❡♥t❡ ♣❛r❛ ❞❡❝♦❞✐✜❝❛r✱ dk ∈ D✱ t❛✐s q✉❡ sk : T −→ C ✱ dk : C −→ T ❡ dk (sk (t)) = t✱ ♣❛r❛ q✉❛❧q✉❡r t ∈ T ✳ ❆ ❋✐❣✉r❛ ✷✳✶ ✐❧✉str❛ ❛ ❉❡✜♥✐çã♦ ✷✳✶✳ ❊①✐st❡♠ ❞✉❛s ♠❛♥❡✐r❛s s✐♠♣❧❡s ❞❡ tr❛♥s❢♦r♠❛r ✉♠❛ ♠❡♥s❛❣❡♥s ❡♠ ❝ó❞✐❣♦s✱ ♦✉ s❡❥❛✱ ❞❡ ❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s✳ ❆ ♣r✐♠❡✐r❛ ❞❡❧❛s ♣r♦❝✉r❛ ❡s❝♦♥❞❡r ♦ ❝♦♥t❡ú❞♦ ❞❛ ♠❡♥s❛❣❡♠ ❛tr❛✈és ❞❡ ❝ó❞✐❣♦s ✺✵

(258) ❋✐❣✉r❛ ✷✳✶✿ ■❧✉str❛çã♦ ♣r❡❞❡✜♥✐❞♦s ❡♥tr❡ ❛s ♣❛rt❡s ❡♥✈♦❧✈✐❞❛s ♥❛ tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s✳ ■♠❛❣✐♥❡ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ ✉♠❛ r❡❜❡❧✐ã♦ ❡♠ ✉♠ ♣r❡sí❞✐♦ ❡ ♦ ❝♦♠❛♥❞❛♥t❡ ❞❛ ♦♣❡r❛çã♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s ♣♦❧✐❝✐❛✐s ❞❡✈❡♠ ❞❡❝✐❞✐r s❡ ✈ã♦ ✐♥✈❛❞✐r ♦✉ ♥ã♦ ♦ ♣r❡sí❞✐♦✳ P❛r❛ ✐ss♦✱ ♦ ❝♦♠❛♥❞❛♥t❡ ❞✐③ q✉❡ ✈❛✐ ❛♥❛❧✐s❛r ❛ s✐t✉❛çã♦ ❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❛♥á❧✐s❡✱ ❞❡❝✐❞✐r ♦ q✉❡ ❢❛③❡r✳ ❙❡ ❡❧❡ ❣r✐t❛r ❛ ♣❛❧❛✈r❛ ✏❚❘❆◆◗❯❊✑ é ♣❛r❛ ✐♥✈❛❞✐r ♦ ♣r❡sí❞✐♦✱ ♠❛s s❡ ❡❧❡ ❣r✐t❛r ✏P❘❊▼❆✑✱ ♥ã♦ ❞❡✈❡ ✐♥✈❛❞✐r✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ s❡ ❛ ♠❡♥s❛❣❡♠ ❝❛✐r ❡♠ ♠ã♦s ❡rr❛❞❛s ♥❛❞❛ ❛❝♦♥t❡❝❡rá ❥á q✉❡ ♥ã♦ t❡rá s✐❣♥✐✜❝❛❞♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛♦ t♦♠❛r ✉♠❛ ❞❡❝✐sã♦ q✉❡ ♥ã♦ s❡❥❛ ♥❡♥❤✉♠❛ ❞❡ss❛s ❞✉❛s✱ ❡❧❡ ♥ã♦ t❡rá ❝♦♠♦ ❛✈✐s❛r ❛♦s ♣♦❧✐❝✐❛✐s✱ ✉♠❛ ✈❡③ q✉❡ ❡ss❡ t✐♣♦ ❞❡ tr♦❝❛ ❞❡ ♠❡♥s❛❣❡♥s só ❞á ❝❡rt♦ s❡ ❡ss❛s ❢♦r❡♠ ♣r❡❞❡✜♥✐❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ♦ q✉❡ ❢❛③ ❝♦♠ q✉❡ ❡ss❛ ♠❛♥❡✐r❛ ❞❡ tr♦❝❛r ♠❡♥s❛❣❡♥s s❡ t♦r♥❡ ❢rá❣✐❧✳ ❆ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❝♦❞✐✜❝❛r ♠❡♥s❛❣❡♠ é ✉s❛♥❞♦ ❛s té❝♥✐❝❛s ❞❡ ❝r✐♣t♦❣r❛✜❛✳ ◆❡st❡ ❝❛s♦✱ ♣♦❞❡♠♦s ❞❡st❛❝❛r ♦s s❡❣✉✐♥t❡s t✐♣♦s ❞❡ ❝♦❞✐✜❝❛çõ❡s✿ • ❈ó❞✐❣♦ ❞❡ ❏ú❧✐♦ ❈és❛r✿ • ❈ó❞✐❣♦ ❆✜♠✿ • ❈ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡✿ ❈♦♥s✐st❡ ❡♠ tr♦❝❛r ❝❛❞❛ ❧❡tr❛ ♣❡❧❛ t❡r❝❡✐r❛ ❧❡tr❛ s✉❜s❡q✉❡♥t❡ ❞♦ ❛❧❢❛❜❡t♦✳ ➱ ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝ó❞✐❣♦ ❞❡ ❏ú❧✐♦ ❈és❛r✱ ❜❛s❡❛❞♦ ♥❛ s✉❜st✐t✉✐çã♦ ❝í❝❧✐❝❛ ❞♦ ❛❧❢❛❜❡t♦✳ ➱ ✉♠ ♠ét♦❞♦ ❞❡ ❝♦❞✐✜❝❛çã♦ q✉❡ ✉s❛ ✉♠❛ sér✐❡ ❞❡ ❞✐❢❡r❡♥t❡s ❝ó❞✐❣♦s ❞❡ ❏ú❧✐♦ ❈és❛r ❣❡♥❡r❛❧✐③❛❞♦ ❝♦♠ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❜❛s❡❛❞♦ ❡♠ ❧❡tr❛s ❞❡ ✉♠❛ ❝❤❛✈❡✳ • ❈ó❞✐❣♦ ❞❡ ❍✐❧❧✿ • ❙✐st❡♠❛ ❘❙❆✿ ❇❛s❡✐❛✲s❡ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ❞❡s❝♦❜r✐r ♦s ❢❛t♦r❡s ♣r✐♠♦s • ❖ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✿ ❇❛s❡✐❛✲s❡ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❢❛t♦r❛r ✐♥t❡✐r♦s✱ ❛ss✐♠ ❝♦♠♦ ♦ • ❖ ▼ét♦❞♦ ▼❍ ✭▼❡r❦❧❡ ❡ ❍❡❧❧♠❛♥✮✿ • ◆❡ss❡ t✐♣♦ ❞❡ ❝ó❞✐❣♦✱ ❛ ♠❡♥s❛❣❡♠ é ❞✐✈✐❞✐❞❛ ❡♠ ❜❧♦❝♦s ❡ ❝♦❞✐✜❝❛❞❛ ❛tr❛✈és ❞❡ ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s✳ ❡①✐st❡♥t❡s ❡♠ ♥ú♠❡r♦s ♠✉✐t♦ ❣r❛♥❞❡s✳ ❙✐st❡♠❛ ❘❙❆✳ ▼♦❝❤✐❧❛ ❍❡❧❧♠❛♥✱ ❡♠ 1978✱ Pr♦❜❧❡♠❛ ❞❛ ❊st❡ ♠ét♦❞♦ ❢♦✐ ❝r✐❛❞♦ ♣♦r ▼❡r❦❧❡ ❡ ❜❛s❡❛♥❞♦✲s❡ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞♦ ❝❤❛♠❛❞♦ ✳ ❈ó❞✐❣♦ ❊❧●❛♠❛❧✿ ➱ ✉♠ s✐st❡♠❛ ❝♦♠ ♦ ✉s♦ ❞❡ ❝❤❛✈❡s ❛ss✐♠étr✐❝❛s✱ ❝r✐❛❞♦ ♣❡❧♦ ❡st✉❞✐♦s♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡❣í♣❝✐♦ ❚❛❤❡r ❊❧❣❛♠❛❧✱ ❡♠ 1984✳ ❙✉❛ s❡❣✉r❛♥ç❛ s❡ ❜❛s❡✐❛ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s♦❧✉çã♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ♣♦❞❡ ❛♣r❡s❡♥t❛r✳ ✺✶

(259) ❆ ♣r✐♥❝✐♣❛❧ ✈❛♥t❛❣❡♠ ♥❛ ✉t✐❧✐③❛çã♦ ❞❡ss❡s ❝ó❞✐❣♦s é ❛ ♥ã♦ ❧✐♠✐t❛çã♦ ❞❛s ♣♦ssí✈❡✐s ♠❡♥s❛❣❡♥s ❛ s❡r❡♠ ❡♥✈✐❛❞❛s✱ ❛❧é♠ ❞❡ s❡ t♦r♥❛r❡♠ ♠❛✐s ❞✐❢í❝❡✐s ❞❡ s❡r❡♠ ❞❡❝♦❞✐✜❝❛❞❛s✳ ❚♦❞♦ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❛❞q✉✐r✐❞♦ ❡ ❡①♣♦st♦ ♥❡st❡ ❝❛♣ít✉❧♦ é ❝♦♠ ❜❛s❡ ❡♠ ❬✶❪✱ ❬✷❪ ❡ ❬✶✺❪✳ ✷✳✶ ❈ó❞✐❣♦ ❞❡ ❈és❛r ❖ ❝ó❞✐❣♦ ❞❡ ❈és❛r✱ ❝♦♥❤❡❝✐❞♦ t❛♠❜é♠ ❝♦♠♦ ❝ó❞✐❣♦ ❞❡ s✉❜st✐t✉✐çã♦✱ é ✉♠❛ ❞❛s ♠❛✐s s✐♠♣❧❡s ❡ ❝♦♥❤❡❝✐❞❛s té❝♥✐❝❛s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❛❧é♠ ❞❡ s❡r ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❝ó❞✐❣♦ ❆✜♠✱ ❝♦♠♦ ✈❡r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✳ ❖ ♥♦♠❡ ❞❡ss❡ ❝ó❞✐❣♦ é ❡♠ ❤♦♠❡♥❛❣❡♠ ❛ ❏ú❧✐♦ ❈és❛r✱ ♦ ✐♠♣❡r❛❞♦r r♦♠❛♥♦✱ q✉❡ ♦ ✉s❛✈❛ ♣❛r❛ ❡♥✈✐❛r ♦r❞❡♥s s❡❝r❡t❛s ❛♦s s❡✉s ❣❡♥❡r❛✐s✳ ❈♦♥s✐❞❡r❛♥❞♦✲s❡ ♦ ❛❧❢❛❜❡t♦✱ ♦ ❝ó❞✐❣♦ ❞❡ ❈és❛r ❝♦♥s✐st❡ ❡♠ tr♦❝❛r ❝❛❞❛ ❧❡tr❛ ♣❡❧❛ t❡r❝❡✐r❛ ❧❡tr❛ s✉❜s❡q✉❡♥t❡✳ ❖❜s❡r✈❡ ♥❛ ❚❛❜❡❧❛ ✷✳✶ q✉❡ ❛s ❧❡tr❛s ❞❛ ♣r✐♠❡✐r❛ ❡ t❡r❝❡✐r❛ ❧✐♥❤❛s r❡♣r❡s❡♥t❛♠ ♦ ❛❧❢❛❜❡t♦ ❝♦♠✉♠ ❡ ❛s ❧❡tr❛s ❞❛ s❡❣✉♥❞❛ ❡ q✉❛rt❛ ❧✐♥❤❛s r❡♣r❡s❡♥t❛♠ ♦ ❝ó❞✐❣♦✳ ❆ ❉ ◆ ◗ ❇ ❊ ❖ ❘ ❈ ❋ P ❙ ❉ ● ◗ ❚ ❊ ❋ ● ❍ ■ ❏ ❘ ❙ ❚ ❯ ❱ ❲ ❍ ■ ❏ ❑ ❑ ▲ ▼ ◆ ❯ ❱ ❲ ❳ ❳ ❨ ❩ ❆ ▲ ▼ ❖ P ❨ ❩ ❇ ❈ ❚❛❜❡❧❛ ✷✳✶✿ ❈♦❞✐✜❝❛çã♦ ❞❡ ❈és❛r ◆❡ss❡ t✐♣♦ ❞❡ ❝♦❞✐✜❝❛çã♦✱ ❛ ❢♦♥t❡ ❆ ❡s❝r❡✈❡ ✉♠❛ ♠❡♥s❛❣❡♠ ❡♠ ❝ó❞✐❣♦ ❡ ❡♥✈✐❛ ♣❛r❛ ❛ ❢♦♥t❡ ❇✳ ❆♣ós r❡❝❡❜❡r ❛ ♠❡♥s❛❣❡♠✱ ❛ ❢♦♥t❡ ❇ ✉t✐❧✐③❛ ❛ ❚❛❜❡❧❛ ✷✳✶ ❡ tr❛♥s❢♦r♠❛ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ❡♠ ❛❧❢❛❜❡t♦ ❝♦♠✉♠✱ ❞❡❝✐❢r❛♥❞♦ ❛ ♠❡♥s❛❣❡♠✳ P♦rt❛♥t♦✱ ❛ ❝❤❛✈❡ ❞❡ss❛ ❝♦♠✉♥✐❝❛çã♦ é ❛ ❚❛❜❡❧❛ ✷✳✶✳ ❊①❡♠♣❧♦ ✷✳✷✳ ❈♦♥s✐❞❡r❡ ❛ ♠❡♥s❛❣❡♠✿ ❉●❘❯❘ ❋❯▲❙❳❘❏❯❉■▲❉✳ ❙❡ ❛ ❢♦♥t❡ ❆ ❡♥✈✐❛ ❡st❛ ♠❡♥s❛❣❡♠ ♣❛r❛ ❛ ❢♦♥t❡ ❇ ❡♥tã♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✶✱ ❛ ❢♦♥t❡ ❇ ❞❡❝✐❢r❛ ❛ ♠❡♥s❛❣❡♠ ❡ ✈❡r ♦ q✉❡ ♦ t❡①t♦ ❞✐③✿ ❆❉❖❘❖ ❈❘■P❚❖●❘❆❋■❆✳ ❈♦♠♦ ✈❡r❡♠♦s ♥❛ ♣ró①✐♠❛ s❡çã♦✱ ♦ ❝ó❞✐❣♦ ❞❡ ❈és❛r é ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❝ó❞✐❣♦ ❆✜♠✳ P❛r❛ t♦r♥❛r ✐st♦ ❝❧❛r♦✱ r❡♣r❡s❡♥t❛r❡♠♦s ❝❛❞❛ ❧❡tr❛ ❞♦ ❛❧❢❛❜❡t♦ ♣♦r ✉♠ ♥ú♠❡r♦ ❞❡ ❞♦✐s ❞í❣✐t♦s✱ ♦✉ s❡❥❛✱ ❛ ❧❡tr❛ ❆ é r❡♣r❡s❡♥t❛❞❛ ♣♦r 00✱ ❛ ❧❡tr❛ ❇ ♣♦r 01✱ ❡ s✉❝❡ss✐✈❛♠❡♥t❡ ❛té ❛ ❧❡tr❛ ❩ ♣♦r 25✱ ❝♦♥❢♦r♠❡ ♠♦str❛❞♦ ♥❛ ❚❛❜❡❧❛ ✷✳✷✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦ q✉❡ ❝♦♥s✐st❡ ♦ ❝ó❞✐❣♦ ❞❡ ❈és❛r ❡ ❢❡✐t❛ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛s ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ ♣♦r ♥ú♠❡r♦s ❞❡ ❞♦✐s ❞í❣✐t♦s✱ ♦ ❝ó❞✐❣♦ ♣❛ss❛ ❛ s❡r ❝♦♥st✐t✉í❞♦ ♣❡❧❛ tr♦❝❛ ❞❡ ❝❛❞❛ ♥ú♠❡r♦ ♣❡❧♦ t❡r❝❡✐r♦ ♥ú♠❡r♦ s✉❜s❡q✉❡♥t❡✱ ✈❡❥❛ ❛ ❚❛❜❡❧❛ ✷✳✸✳ P♦❞❡♠♦s ♥♦t❛r q✉❡ ♦ ❝ó❞✐❣♦ ❞❡ ❏ú❧✐♦ ❈és❛r ❜❛s❡✐❛✲s❡ ♥❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ✭❱❡❥❛ ❚❛❜❡❧❛ ✷✳✸✮✿ α ≡ (β + 03) mod 26 ✺✷ ✭✷✳✶✮

(260) ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ ✵✵ ✵✶ ✵✷ ✵✸ ✵✹ ✵✺ ✵✻ ✵✼ ✵✽ ✵✾ ✶✵ ✶✶ ✶✷ ◆ ❖ P ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ❚❛❜❡❧❛ ✷✳✷✿ ❘❡♣r❡s❡♥t❛çã♦ ❞♦ ❆❧❢❛❜❡t♦ ♣♦r ❉í❣✐t♦s ✵✵ ✵✸ ✶✸ ✶✻ ✵✶ ✵✹ ✶✹ ✶✼ ✵✷ ✵✺ ✶✺ ✶✽ ✵✸ ✵✻ ✶✻ ✶✾ ✵✹ ✵✼ ✶✼ ✷✵ ✵✺ ✵✽ ✶✽ ✷✶ ✵✻ ✵✾ ✶✾ ✷✷ ✵✼ ✶✵ ✷✵ ✷✸ ✵✽ ✶✶ ✷✶ ✷✹ ✵✾ ✶✷ ✷✷ ✷✺ ✶✵ ✶✸ ✷✸ ✵✶ ✶✶ ✶✹ ✷✹ ✵✷ ✶✷ ✶✺ ✷✺ ✵✸ ❚❛❜❡❧❛ ✷✳✸✿ ❈♦❞✐✜❝❛çã♦ ❞❡ ❈és❛r ❡♠ ◆ú♠❡r♦s ❞❡ ❉♦✐s ❉í❣✐t♦s ♦♥❞❡ α r❡♣r❡s❡♥t❛ ♦s ♥ú♠❡r♦s ❝♦❞✐✜❝❛❞♦s ✭✈❡❥❛ ❚❛❜❡❧❛ ✷✳✸✮ ❡ β ♦s ♥ú♠❡r♦s q✉❡ r❡♣r❡s❡♥t❛♠ ❝❛❞❛ ❧❡tr❛ ❞♦ ❛❧❢❛❜❡t♦ ✭✈❡❥❛ ❚❛❜❡❧❛ ✷✳✷✮✳ P♦❞❡♠♦s t❛♠❜é♠ ❢❛③❡r ♦ ♣r♦❝❡ss♦ ✐♥✈❡rs♦✱ ✐st♦ é✱ ❞❡❝✐❢r❛r ❛ ♠❡♥s❛❣❡♠✳ P❛r❛ ✐ss♦ é ♥❡❝❡ssár✐♦ ❝❛❧❝✉❧❛r β ❡♠ t❡r♠♦s ❞❡ α✱ ♣♦❞❡♥❞♦ s❡r ❝❛❧❝✉❧❛❞♦ ❛tr❛✈és ❞❡✿ β ≡ (α − 03) mod 26. ✭✷✳✷✮ ❊①❡♠♣❧♦ ✷✳✸✳ ❖❜s❡r✈❡ ❛ ♠❡♥s❛❣❡♠✿ 1114172407011723 ❯s❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✸✱ ♦ t❡①t♦ ❞❡❝✐❢r❛❞♦ é ■ ▲❖❱❊ ❨❖❯✳ ❊①❡♠♣❧♦ ✷✳✹✳ ❈♦❞✐✜q✉❡ ❛ ♠❡♥s❛❣❡♠ ✏❖ ❈❖P❖ ◗❯❊❇❘❖❯✑✳ ❯s❛♥❞♦ ✭✷✳✶✮ ❡ ❛ ❚❛❜❡❧❛ ✷✳✷✱ t❡♠♦s✿ O −→ 14 + 03 C −→ 02 + 03 O −→ 14 + 03 P −→ 15 + 03 O −→ 14 + 03 Q −→ 16 + 03 U −→ 20 + 03 E −→ 04 + 03 B −→ 01 + 03 R −→ 17 + 03 O −→ 14 + 03 U −→ 20 + 03 ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ 17mod 05mod 17mod 18mod 17mod 19mod 23mod 07mod 04mod 20mod 17mod 23mod 26 −→ R 26 −→ F 26 −→ R 26 −→ S 26 −→ R 26 −→ T 26 −→ X 26 −→ H 26 −→ E 26 −→ U 26 −→ R 26 −→ X ✳ ▲♦❣♦✱ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ é ✏❘❋❘❙❘❚❳❍❊❯❘❳✑✳ ✷✳✷ ❈ó❞✐❣♦s ❆✜♥s ❈♦♠♦ ❥á ♠❡♥❝✐♦♥❛❞♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✱ ♦ ❝ó❞✐❣♦ ❛✜♠ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝ó❞✐❣♦ ❞❡ ❏ú❧✐♦ ❈és❛r ✲ ❏❈✳ ❘❡❧❡♠❜r❛♥❞♦ q✉❡ ♥♦ ❝ó❞✐❣♦ ❞❡ ❏❈ ❛ ♣♦s✐çã♦ ❞❛s ❧❡tr❛s ✺✸

(261) ❞♦ t❡①t♦ ❡ ♦s ❝ó❞✐❣♦s ♦❜❡❞❡❝❡♠ ❛ ❝♦♥❣r✉ê♥❝✐❛ α ≡ (β + 3)mod 26✳ ❆❣♦r❛✱ s❡❥❛ k ✭❝❤❛♠❛❞♦ ❝❤❛✈❡✮ ✉♠ ✐♥t❡✐r♦ s❛t✐s❢❛③❡♥❞♦ 0 ≤ k ≤ 25✳ ❊♠ ✈❡③ ❞❡ ✉t✐❧✐③❛r♠♦s ♦ ♥ú♠❡r♦ ✸ ♥♦ ❝ó❞✐❣♦ ❏❈✱ ✐r❡♠♦s ✉s❛r ❦ ❡ ❞❡✜♥✐r ✉♠ ♥♦✈♦ ❝ó❞✐❣♦ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ❝ó❞✐❣♦ ❏❈ ❣❡♥❡r❛❧✐③❛❞♦✳ P♦❞❡rí❛♠♦s ✐♠❛❣✐♥❛r ❡ss❡ ❝ó❞✐❣♦ ❝♦♠♦ ✉♠ ❝ó❞✐❣♦ ❜❛s❡❛❞♦ ♥❛ s✉❜st✐t✉✐çã♦ ❝í❝❧✐❝❛ ❞♦ ❛❧❢❛❜❡t♦✱ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ❞♦✐s ❞✐s❝♦s ❝♦♥❝ê♥tr✐❝♦s ❝♦♥t❡♥❞♦ t♦❞❛s ❛s ❧❡tr❛s✱ t♦r♥❛♥❞♦ ❞❡ss❛ ❢♦r♠❛ ❛ s✉❜st✐t✉✐çã♦ ♠❛✐s s✐♠♣❧❡s ✭✈❡❥❛ ❛ ❋✐❣✉r❛ ✷✳✷✱ r❡t✐r❛❞❛ ❞❡ ❬✷✺❪✮✳ ❋✐❣✉r❛ ✷✳✷✿ ❉✐s❝♦s ❈♦♥❝ê♥tr✐❝♦s ❈♦♥t❡♥❞♦ ❚♦❞❛s ❛s ▲❡tr❛s ❞♦ ❆❧❢❛❜❡t♦ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ♦ ❝ó❞✐❣♦ ❏❈ ❣❡♥❡r❛❧✐③❛❞♦ ❜❛s❡❛❞♦ ♥❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛✿ α ≡ (β + k) mod 26, ♦♥❞❡ k ✱ ✉♠ ✐♥t❡✐r♦ s❛t✐s❢❛③❡♥❞♦ 0 ≤ k ≤ 25✱ é ❛ ❝❤❛✈❡ ❞❡ ❝♦❞✐✜❝❛çã♦✱ β é ❛ ♣♦s✐çã♦ ❞❛ ❧❡tr❛ ✭✈❡❥❛ ❛ ❚❛❜❡❧❛ ✷✳✷✮ ❡ α r❡♣r❡s❡♥t❛ ❛ ♣♦s✐çã♦ ❞❛ ♥♦✈❛ ❧❡tr❛✳ ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✳ ❖❜s❡r✈❡ q✉❡ q✉❛♥❞♦ k = 0 ♦ ❝ó❞✐❣♦ ♦❜t✐❞♦ é ❡①❛t❛♠❡♥t♦ ♦ t❡①t♦ ❞❛ ♠❡♥s❛❣❡♠ s❡♠ ❛❧t❡r❛çõ❡s ❡ q✉❛♥❞♦ k = 3 t❡♠♦s ♦ ❝ó❞✐❣♦ ❏❈✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ❞❡❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠✱ é ♥❡❝❡ssár✐♦ ❡♥❝♦♥tr❛r ❛ ❝❤❛✈❡ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦✳ P❛r❛ t❛♥t♦✱ ❜❛st❛ ❡♥❝♦♥tr❛r j t❛❧ q✉❡ (j + k) ≡ 0 mod 26 ✭❱❡r ❉❡✜✐çã♦ ✶✳✹✼✱ s✐♠étr✐❝♦ ❛❞✐t✐✈♦✮✱ tr♦❝❛r k ♣♦r j ❡ ♣r♦❝❡❞❡r ❞❛ ♠❡s♠❛ ♠❛♥❡✐r❛✳ ❊①❡♠♣❧♦ ✷✳✺✳ ❖❜s❡r✈❡ ❝♦♠♦ ❝♦❞✐✜❝❛r ❛ ❢r❛s❡ ✏❊❙❚❖❯ ❈❖▼ ❋❘■❖✑✱ ✉s❛♥❞♦ ❝♦♠♦ ❝❤❛✈❡ k = 15✱ ♦✉ s❡❥❛✱ α ≡ (β + 15) mod 26✳ E −→ 04 + 15 ≡ 19mod 26 −→ T S −→ 18 + 15 ≡ 07mod 26 −→ H T −→ 19 + 15 ≡ 08mod 26 −→ I O −→ 14 + 15 ≡ 03mod 26 −→ D U −→ 20 + 15 ≡ 09mod 26 −→ J C −→ 02 + 15 ≡ 17mod 26 −→ R O −→ 14 + 15 ≡ 03mod 26 −→ D M −→ 12 + 15 ≡ 01mod 26 −→ B F −→ 05 + 15 ≡ 20mod 26 −→ U R −→ 17 + 15 ≡ 06mod 26 −→ G I −→ 08 + 15 ≡ 23mod 26 −→ X O −→ 14 + 15 ≡ 03mod 26 −→ D✳ P♦rt❛♥t♦✱ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ é ✏❚❍■❉❏❘❉❇❯●❳❉✑✳ ✺✹

(262) P❛r❛ ❞❡❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ 15 + 11 ≡ 0 mod 26 ✭❉❡✜♥✐çã♦ ✶✳✹✼✱ ✐♥✈❡rs♦ ❛❞✐t✐✈♦✮✱ t❡♠♦s q✉❡ ❛ ❝❤❛✈❡ ❞❡❝♦❞✐✜❝❛❞♦r❛ é 11✳ ▲♦❣♦✱ ♣r♦❝❡❞❡✲s❡ ❛ss✐♠✿ T −→ 19 + 11 ≡ 04mod 26 −→ E H −→ 07 + 11 ≡ 18mod 26 −→ S I −→ 08 + 11 ≡ 19mod 26 −→ T D −→ 03 + 11 ≡ 14mod 26 −→ O J −→ 09 + 11 ≡ 20mod 26 −→ U R −→ 17 + 11 ≡ 02mod 26 −→ C D −→ 03 + 11 ≡ 14mod 26 −→ O B −→ 01 + 11 ≡ 12mod 26 −→ M U −→ 20 + 11 ≡ 05mod 26 −→ F G −→ 06 + 11 ≡ 17mod 26 −→ R X −→ 23 + 11 ≡ 08mod 26 −→ I D −→ 03 + 11 ≡ 14mod 26 −→ O✳ ❉❡✜♥✐çã♦ ✷✳✻✳ ❈❤❛♠❛r❡♠♦s ❞❡ ❝ó❞✐❣♦ ❛✜♠ ❛ ❝♦❞✐✜❝❛çã♦ ❜❛s❡❛❞❛ ♥❛ tr♦❝❛ ❞❡ ❧❡tr❛s ❞♦ ❛❧❢❛❜❡t♦ ❛tr❛✈és ❞❛ r❡❣r❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ α ≡ (a · β + b) mod 26, ✭✷✳✸✮ ♦♥❞❡ a ❡ b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❝♦♠ 0 ≤ a ≤ 25 ❡ 0 ≤ b ≤ 25 ❡ ♦ mdc(a, 26) = 1✳ ❖s ♥ú♠❡r♦s a ❡ b sã♦ ❝❤❛♠❛❞♦s ❝❤❛✈❡s ❞♦ ❝ó❞✐❣♦ ❛✜♠✳ P❛r❛ ❞❡❝♦❞✐✜❝❛r ✉♠❛ ♠❡♥s❛❣❡♠ ♥♦ ❝ó❞✐❣♦ ❛✜♠✱ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❛ss✐♠✿ a.β ≡ (α − b)mod 26. ❈♦♠♦ ♦ mdc(a, 26) = 1✱ ✐ss♦ ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ i t❛❧ q✉❡ a.i ≡ 1 mod 26 ✭❉❡✜♥✐çã♦ ✶✳✹✽✱ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮✳ ❆ss✐♠✱ ♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r i t❡r❡♠♦s✿ ▲♦❣♦✱ i.a.β ≡ i.(α − b)mod 26. ✭✷✳✹✮ P♦rt❛♥t♦✱ ❝♦♠ ❡ss❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ♦ t❡①t♦ ❡ ♦ ❝♦♥t❡ú❞♦ ❞❡ss❛ ♠❡♥s❛❣❡♠✳ ❖❜s❡r✈❡ q✉❡✱ ❛ ♣❛rt✐r ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ✭✷✳✸✮✱ ♦❜t❡♠♦s ♦ ❝ó❞✐❣♦ ❏❈ ❣❡♥❡r❛❧✐③❛❞♦ ❢❛③❡♥❞♦ a = 1✳ ❈♦♠ ✐st♦ s✉r❣❡ ♦ s❡❣✉✐♥t❡ q✉❡st✐♦♥❛♠❡♥t♦✿ q✉❛♥t♦s ❝ó❞✐❣♦s ❞❡ ❏❈ ❣❡♥❡r❛❧✐③❛❞♦s ❡ q✉❛♥t♦s ❝ó❞✐❣♦s ❛✜♥s ❡①✐st❡♠❄ β ≡ i.(α − b)mod 26. ❚❡♦r❡♠❛ ✷✳✼✳ ✐✮ ❊①✐st❡♠ 26 ❝ó❞✐❣♦s ❞❡ ❏❈ ❣❡♥❡r❛❧✐③❛❞♦s❀ ✐✐✮ ❊①✐st❡♠ 312 ❝ó❞✐❣♦s ❛✜♥s✳ ❉❡♠♦♥str❛çã♦✳ ✐✮ ❖s ❝ó❞✐❣♦s ❞❡ ❏❈ ❞❡♣❡♥❞❡♠ ✉♥✐❝❛♠❡♥t❡ ❞♦ ✈❛❧♦r ❞❡ k ✱ ♦♥❞❡ 0 ≤ k ≤ 25✳ ▲♦❣♦ ❡①✐st❡♠ 26 ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ ❛ ❡s❝♦❧❤❛ ❞❡ k ✳ ✐✐✮ ◆♦s ❝ó❞✐❣♦s ❛✜♥s✱ ❛ ❝♦♥❞✐çã♦ mdc(a, 26) = 1 ♠♦str❛ q✉❡ ❤á ✶✷ ♣♦ss✐❜✐❧✐❞❛❞❡s (1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25) ♣❛r❛ ❡s❝♦❧❤❡r a✳ ❯♠❛ ✈❡③ q✉❡ ♦ ✈❛❧♦r ❞❡ a ❢♦✐ ❡s❝♦❧❤✐❞♦✱ t❡♠♦s 26 ♦♣çõ❡s ♣❛r❛ b ❡✱ ♣♦rt❛♥t♦✱ ♥♦ t♦t❛❧ 12 × 26 = 312 ♠❛♥❡✐r❛s ❞❡ ❡s❝♦❧❤❡r a ❡ b✳ ✺✺

(263) ❊①❡♠♣❧♦ ✷✳✽✳ ❯t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✷ ❡ s❛❜❡♥❞♦ q✉❡ a=5 ❡ b = 11 sã♦ ❛s ❝❤❛✈❡s ❞❛ ❝♦❞✐✜❝❛çã♦✱ ❝♦❞✐✜q✉❡ ❛ ♣❛❧❛✈r❛ ✏❯◆■❱❊❘❙❖✑✳ ❙♦❧✉çã♦✿ ❱❛♠♦s r❡♣r❡s❡♥t❛r ❡♠ ✉♠❛ t❛❜❡❧❛ ❛ ♠❡♥s❛❣❡♠ q✉❡ s❡rá ❝♦❞✐✜❝❛❞❛ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ❞❡ β β✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✹ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ❯ ◆ ■ ❱ ❊ ❘ ❙ ❖ ✷✵ ✶✸ ✵✽ ✷✶ ✵✹ ✶✼ ✶✽ ✶✹ ❚❛❜❡❧❛ ✷✳✹✿ ✭✷✳✸✮✱ t❡♠♦s✿ α ≡ (5 · 20 + 11)mod 26 ⇒ α = 7 = H; α ≡ (5 · 13 + 11)mod 26 ⇒ α = 24 = Y ; α ≡ (5 · 8 + 11)mod 26 ⇒ α = 25 = Z; α ≡ (5 · 21 + 11)mod 26 ⇒ α = 12 = M ; α ≡ (5 · 4 + 11)mod 26 ⇒ α = 5 = F ; α ≡ (5 · 17 + 11)mod 26 ⇒ α = 18 = S; α ≡ (5 · 18 + 11)mod 26 ⇒ α = 23 = X; α ≡ (5 · 14 + 11)mod 26 ⇒ α = 3 = D; P♦rt❛♥t♦✱ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ é ✏❍❨❩▼❋❙❳❉✑✳ ❊①❡♠♣❧♦ ✷✳✾✳ ❯t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✷ ❡ s❛❜❡♥❞♦ q✉❡ a=7 ❡ b = 12 sã♦ ❛s ❝❤❛✈❡s ❞❛ ❝♦❞✐✜❝❛çã♦✱ ❞❡❝♦❞✐✜q✉❡ ❛ ♣❛❧❛✈r❛ ✏❚❇▼■◗▲✑✳ ❙♦❧✉çã♦✿ ❱❛♠♦s r❡♣r❡s❡♥t❛r ❡♠ ✉♠❛ t❛❜❡❧❛ ❛ ♠❡♥s❛❣❡♠ q✉❡ s❡rá ❞❡❝♦❞✐✜❝❛❞❛ ❝♦♠ s❡✉s r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ❞❡ α✳ α ❈♦♠♦ ♦ mdc(7, 26) = 1✱ ❚ ❇ ▼ ■ ◗ ▲ ✶✾ ✵✶ ✶✷ ✵✽ ✶✻ ✶✶ ✐ss♦ ❣❛r❛♥t❡ q✉❡ ❡①✐st❡ i ❚❛❜❡❧❛ ✷✳✺✿ t❛❧ q✉❡ 7.i ≡ 1 mod 26 ✭❉❡✜♥✐çã♦ ✶✳✹✽✱ α ✭✈❡❥❛ ❚❛❜❡❧❛ ✷✳✺✮ ❡ ❜✱ ❈♦♥❤❡❝❡♥❞♦ ✐✱ ✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✮ ❡ ♣♦rt❛♥t♦ ❝♦♥s❡❣✉✐♠♦s ❞❡❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠✳ ❖❜s❡r✈❡ q✉❡✿ β β β β β β i = 15✳ ❡ ❛♣❧✐❝❛♥❞♦ ❡ss❛s ✐♥❢♦r♠❛çõ❡s ❡♠ ✭✷✳✹✮✱ ≡ 15 · (19 − 12)mod 26 ⇒ β = 1 = B; ≡ 15 · (1 − 12)mod 26 ⇒ β = 17 = R; ≡ 15 · (12 − 12)mod 26 ⇒ β = 0 = A; ≡ 15 · (8 − 12)mod 26 ⇒ β = 18 = S; ≡ 15 · (16 − 12)mod 26 ⇒ β = 8 = I; ≡ 15 · (11 − 12)mod 26 ⇒ β = 11 = L. P♦rt❛♥t♦✱ ❛ ♠❡♥s❛❣❡♠ ❞❡❝♦❞✐✜❝❛❞❛ é ✏❇❘❆❙■▲✑✳ ✺✻

(264) ✷✳✸ ❈ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡ ❖ ❈ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡✶ é ✉♠ ♠ét♦❞♦ ❞❡ ❝♦❞✐✜❝❛çã♦ q✉❡ ✉s❛ ✉♠❛ sér✐❡ ❞❡ ❞✐❢❡r❡♥t❡s ❝ó❞✐❣♦s ❞❡ ❏ú❧✐♦ ❈és❛r ❣❡♥❡r❛❧✐③❛❞♦ ❝♦♠ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❜❛s❡❛❞♦ ❡♠ ❧❡tr❛s ❞❡ ✉♠❛ ❝❤❛✈❡✳ ❆ ❝♦❞✐✜❝❛çã♦ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❛❧❣❡❜r✐❝❛♠❡♥t❡ ❝♦♠♦✿ Ci ≡ (Pi + ai )(mod m), ✭✷✳✺✮ Pi ≡ (Ci − ai )(mod m) ✭✷✳✻✮ ai ≡ (Ci − Pi )(mod m) ✭✷✳✼✮ ❡ ❛ ❞❡❝♦❞✐✜❝❛çã♦ ❝♦♠♦ ❡ ❛ ❝❤❛✈❡ ❝♦♠♦ ♦♥❞❡ Pi ❝♦rr❡s♣♦♥❞❡ ❛♦s ✈❛❧♦r❡s ❞❛s ❧❡tr❛s ❛ s❡r❡♠ ❝♦❞✐✜❝❛❞❛s✱ ai ❛♦s ✈❛❧♦r❡s ❞❛s ❧❡tr❛s ❞❛ ❝❤❛✈❡✱ Ci ❛♦s ✈❛❧♦r❡s ❞❛s ❧❡tr❛s ❝♦❞✐✜❝❛❞❛s ❡ m r❡♣r❡s❡♥t❛ ❛ ❝♦♥❣r✉ê♥❝✐❛ ❡♥✈♦❧✈✐❞❛ ✭Zm ✮✳ ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ P ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❄ ➪ ➹ ➱ [] ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✵ ❚❛❜❡❧❛ ✷✳✻✿ ❊①❡♠♣❧♦ ✷✳✶✵✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✻✱ ❝♦❞✐✜q✉❡ ❛ ♠❡♥s❛❣❡♠ ✏❋❘❖◆❚❊■❘❆❙ ❉❖ ❇❘❆❙■▲✑ ✉t✐❧✐③❛♥❞♦ ❛ ❝❤❛✈❡ ❚■●❘❊✳ ❙♦❧✉çã♦✿ ❱❛♠♦s r❡♣r❡s❡♥t❛r ❡♠ ✉♠❛ t❛❜❡❧❛ ❛ ♠❡♥s❛❣❡♠ q✉❡ s❡rá ❝♦❞✐✜❝❛❞❛ ❝♦♠ s✉❛ r❡s♣❡❝t✐✈❛ ❝❤❛✈❡✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✼ ❡ ❛♣❧✐❝❛♥❞♦ ❋ ❘ ❖ ◆ ❚ ❊ ■ ❘ ❆ ❙ 6 18 15 14 20 5 9 18 1 19 ❚ ■ ● ❘ ❊ ❚ ■ ● ❘ ❊ 20 9 7 18 5 20 9 7 18 5 ❉ ❖ 4 15 ❚ ■ 20 9 0 ✭✷✳✺✮✱ t❡♠♦s✿ ❇ ❘ ❆ ❙ ■ ▲ 0 2 18 1 19 9 12 ● ❘ ❊ ❚ ■ ● ❘ ❊ 7 18 5 20 9 7 18 5 ❚❛❜❡❧❛ ✷✳✼✿ C1 C2 C3 C4 C5 C6 C7 C8 C9 ✶ ❊st❡ ≡ (6 + 20)(mod 31) ⇒ C1 = 26 = Z; ≡ (18 + 9)(mod 31) ⇒ C2 = 27 =?; ≡ (15 + 7)(mod 31) ⇒ C3 = 22 = V ; ≡ (14 + 18)(mod 31) ⇒ C4 = 1 = A; ≡ (20 + 5)(mod 31) ⇒ C5 = 25 = Y ; ≡ (5 + 20)(mod 31) ⇒ C6 = 25 = Y ; ≡ (9 + 9)(mod 31) ⇒ C7 = 18 = R; ≡ (18 + 7)(mod 31) ⇒ C8 = 25 = Y ; ≡ (1 + 18)(mod 31) ⇒ C9 = 19 = S; ♥♦♠❡ é ✉♠❛ ❤♦♠❡♥❛❣❡♠ ❡rr❛❞❛♠❡♥t❡ ❛tr✐❜✉í❞❛ ❛ ❇❧❛✐s❡ ❞❡ ❱✐❣❡♥èr❡✱ ✉♠❛ ✈❡③ q✉❡ ♦ ❝ó❞✐❣♦ ❢♦✐ ✐♥✈❡♥t❛❞♦ ❡ ♦r✐❣✐♥❛❧♠❡♥t❡ ❞❡s❝r✐t♦ ♣♦r ●✐♦✈❛♥ ❇❛t✐st❛ ❇❡❧❛s♦ ❡♠ s❡✉ ❧✐✈r♦ ❞❛t❛❞♦ ❞❡ ✶✺✺✸ ❝♦♠ ♦ tít✉❧♦ ▲❛❝✐❢r❛ ❞❡❧✳ ❙✐❣✳ ●✐♦✈❛♥ ❇❛t✐st❛ ❇❡❧❛s♦✳ ✺✼

(265) C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 ≡ (19 + 5)(mod 31) ⇒ C10 = 24 = X; ≡ (0 + 20)(mod 31) ⇒ C11 = 20 = T ; ≡ (4 + 9)(mod 31) ⇒ C12 = 13 = M ; ≡ (15 + 7)(mod 31) ⇒ C13 = 22 = V ; ≡ (0 + 18)(mod 31) ⇒ C14 = 18 = R; ≡ (2 + 5)(mod 31) ⇒ C15 = 7 = G; ≡ (18 + 20)(mod 31) ⇒ C16 = 7 = G; ≡ (1 + 9)(mod 31) ⇒ C17 = 10 = J; ≡ (19 + 7)(mod 31) ⇒ C18 = 26 = Z; ≡ (9 + 18)(mod 31) ⇒ C19 = 27 =?; ≡ (12 + 5)(mod 31) ⇒ C20 = 17 = Q; P♦rt❛♥t♦✱ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ é ✏❩❄❱❆❨❨❘❨❙❳❚▼❱❘●●❏❩❄◗✑✳ ❊①❡♠♣❧♦ ✷✳✶✶✳ ❯t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✻ ❡ ✭✷✳✻✮✱ ❞❡❝♦❞✐✜q✉❡ ❛ ♠❡♥s❛❣❡♠ ❛❜❛✐①♦✱ ✉s❛♥❞♦ ❛ ♣❛❧❛✈r❛✲❝❤❛✈❡ ❈❆❙❆✿ F P DBCECAP V BER. ❙♦❧✉çã♦✿ ❱❛♠♦s r❡♣r❡s❡♥t❛r ❡♠ ✉♠❛ t❛❜❡❧❛ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ❝♦♠ s✉❛ r❡s♣❡❝t✐✈❛ ❝❤❛✈❡✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✽ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ✭✷✳✻✮✱ t❡♠♦s✿ ❋ P ❉ ❇ ❈ ❊ ❈ ❆ P ❱ ❇ ❊ ❘ 6 16 4 2 3 5 3 1 16 22 2 5 18 ❈ ❆ ❙ ❆ ❈ ❆ ❙ ❆ ❈ ❆ ❙ ❆ ❈ 3 1 19 1 3 1 19 1 3 1 19 1 3 ❚❛❜❡❧❛ ✷✳✽✿ P1 ≡ (6 − 3)(mod 31) ⇒ P1 = 3 = C; P2 ≡ (16 − 1)(mod 31) ⇒ P2 = 15 = O; P3 ≡ (4 − 19)(mod 31) ⇒ P3 = 16 = P ; P4 ≡ (2 − 1)(mod 31) ⇒ P4 = 1 = A; P5 ≡ (3 − 3)(mod 31) ⇒ P5 = 0 = [ ]; P6 ≡ (5 − 1)(mod 31) ⇒ P6 = 4 = D; P7 ≡ (3 − 19)(mod 31) ⇒ P7 = 15 = O; P8 ≡ (1 − 1)(mod 31) ⇒ P8 = 0 = [ ]; P9 ≡ (16 − 3)(mod 31) ⇒ P9 = 13 = M ; P10 ≡ (22 − 1)(mod 31) ⇒ P10 = 21 = U ; P11 ≡ (2 − 19)(mod 31) ⇒ P11 = 14 = N ; P12 ≡ (5 − 1)(mod 31) ⇒ P12 = 4 = D; P13 ≡ (18 − 3)(mod 31) ⇒ P13 = 15 = O; P♦rt❛♥t♦✱ ❛ ♠❡♥s❛❣❡♠ ❞❡❝♦❞✐✜❝❛❞❛ é ✏❈❖P❆ ❉❖ ▼❯◆❉❖✑✳ ❊①❡♠♣❧♦ ✷✳✶✷✳ ❉❛❞❛ ❛ ♣❛❧❛✈r❛ ❡♠ t❡①t♦ s✐♠♣❧❡s ✏❚❊❈◆❖▲❖●■❆✑✱ ❡ ♦ s❡✉ r❡s♣❡❝t✐✈♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ✏❊❚❯❈P➪➱❨➹❇✑✱ ❡♥❝♦♥tr❡ ❛ ❝❤❛✈❡ q✉❡ ❢♦✐ ✉t✐❧✐③❛❞❛ ♥❛ ❝♦❞✐✜❝❛çã♦✳ ❙♦❧✉çã♦✿ ❱❛♠♦s r❡♣r❡s❡♥t❛r ❡♠ ✉♠❛ t❛❜❡❧❛ ❛ ♠❡♥s❛❣❡♠ ❡ s✉❛ r❡s♣❡❝t✐✈❛ ❝♦❞✐✜❝❛çã♦✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✶✼ ❡ ❛♣❧✐❝❛♥❞♦ ❛ ❢ór♠✉❧❛ ✭✷✳✼✮✱ t❡♠♦s✿ ✺✽

(266) ❚ ❊ ❈ ◆ ❖ ▲ ❖ ● ■ ❆ 20 5 3 14 15 12 15 7 9 1 ❊ ❚ ❯ ❈ P ➪ ➱ ❨ ➹ ❇ 5 20 21 3 16 28 30 25 29 2 ❚❛❜❡❧❛ ✷✳✾✿ a1 ≡ (5 − 20)(mod 31) ⇒ a1 = 16 = P ; a2 ≡ (20 − 5)(mod 31) ⇒ a2 = 15 = O; a3 ≡ (21 − 3)(mod 31) ⇒ a3 = 18 = R; a4 ≡ (3 − 14)(mod 31) ⇒ a4 = 20 = T ; a5 ≡ (16 − 15)(mod 31) ⇒ a5 = 1 = A; a6 ≡ (28 − 12)(mod 31) ⇒ a6 = 16 = P ; a7 ≡ (30 − 15)(mod 31) ⇒ a7 = 15 = O; a8 ≡ (25 − 7)(mod 31) ⇒ a8 = 18 = R; a9 ≡ (29 − 9)(mod 31) ⇒ a9 = 20 = T ; a10 ≡ (2 − 1)(mod 31) ⇒ a10 = 1 = A; P♦rt❛♥t♦✱ ❛ ❝❤❛✈❡ ❞❡ ❝♦❞✐✜❝❛çã♦ é ✏P❖❘❚❆✑✳ ✷✳✹ ❈ó❞✐❣♦ ❞❡ ❍✐❧❧ ❖s ❝ó❞✐❣♦s ❞❡ s✉❜st✐t✉✐çã♦ ♣♦ss✉❡♠ ✉♠❛ ❞❡s✈❛♥t❛❣❡♠ ♣♦r s❡r❡♠ r❡❧❛t✐✈❛♠❡♥t❡ ❢á❝❡✐s ❞❡ s❡r❡♠ ❞❡❝♦❞✐✜❝❛❞♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❡①✐st❡♠ ❝ó❞✐❣♦s q✉❡✱ ❛♦ ✐♥✈és ❞❡ ❝r✐♣t♦❣r❛❢❛r ❧❡tr❛ ♣♦r ❧❡tr❛✱ ❞✐✈✐❞❡♠ ♦ t❡①t♦ ❡♠ ❣r✉♣♦s ❞❡ ❧❡tr❛s✳ ❊ss❡ t✐♣♦ ❞❡ ❝ó❞✐❣♦ ❢❛③ ♣❛rt❡ ❞❡ ✉♠ s✐st❡♠❛ ♣♦❧✐❣rá✜❝♦ ♥♦ q✉❛❧ ✉♠ t❡①t♦ ❝♦♠✉♠ é ❞✐✈✐❞✐❞♦ ❡♠ ❝♦♥❥✉♥t♦s ❞❡ n ❧❡tr❛s✱ ❝❛❞❛ ✉♠ ❞♦s q✉❛✐s é s✉❜st✉t✉í❞♦ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ ❞❡ n ❧❡tr❛s ❝♦❞✐✜❝❛❞❛s✳ ❖ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧✷ r❡♣r❡s❡♥t❛ ✉♠❛ ❝❧❛ss❡ ❞❡ s✐st❡♠❛s ♣♦❧✐❣rá✜❝♦s✳ ◆❡st❡ t✐♣♦ ❞❡ ❝ó❞✐❣♦✱ ✈❛♠♦s ❡st❛❜❡❧❡❝❡r ♣❛r❛ ❝❛❞❛ ❧❡tr❛ ❞♦ t❡①t♦ ❝♦♠✉♠ ❡ ❞♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ✉♠ ✈❛❧♦r ♥✉♠ér✐❝♦ q✉❡ ❡s♣❡❝✐✜❝❛ s✉❛ ♣♦s✐çã♦ ♥♦ ❛❧❢❛❜❡t♦ ♣❛❞rã♦ ✭✈❡r ❚❛❜❡❧❛ ✷✳✶✵✮✳ ❖s ❡s♣❛ç♦s ❡♠ ❜r❛♥❝♦ ❡♥tr❡ ❧❡tr❛s ♦✉ ♣❛❧❛✈r❛s s❡rã♦ r❡♣r❡s❡♥t❛❞♦s ♣❡❧♦ sí♠❜♦❧♦ [ ]✳ ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ◆ ❖ P ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ✷✸ ✷✹ ✷✺ ✵ ❚❛❜❡❧❛ ✷✳✶✵✿ ◆♦s ❝❛s♦s ♠❛✐s s✐♠♣❧❡s ❞♦ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧✱ tr❛♥s❢♦r♠❛♠♦s ♣❛r❡s s✉❝❡ss✐✈♦s ❞❡ t❡①t♦ ❝♦♠✉♠ ❡♠ t❡①t♦ ❝♦❞✐✜❝❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✶✵ ❡ ❛ ❡str✉t✉r❛ ❞♦ Z26 ✱ ✉s❛♥❞♦ ♦ s❡❣✉✐♥t❡ ♣r♦❝❡❞✐♠❡♥t♦✿ ✶✳ ❊s❝♦❧❤❛ ✉♠❛ ♠❛tr✐③ 2 × 2 ✷ ❊♠ ✶✾✷✾ ▲❡st❡r ❙✳ ❍✐❧❧ ♣✉❜❧✐❝♦✉ s❡✉ ❧✐✈r♦ ❈r②♣t♦❣r❛♣❤② ✐♥ ❛♥ ❆❧❣❡❜r❛✐❝ ❆❧♣❤❛❜❡t✱ ♥♦ q✉❛❧ ✉♠ ❜❧♦❝♦ ❞❡ t❡①t♦ ❝♦♠✉♠ é ❝♦❞✐✜❝❛❞♦ ❛tr❛✈és ❞❡ ♦♣❡r❛çõ❡s ❝♦♠ ♠❛tr✐③❡s ✺✾

(267) A=  a11 a12 a21 a22  ❝♦♠ ❡♥tr❛❞❛s ✐♥t❡✐r❛s ♣❛r❛ ❡❢❡t✉❛r ❛ ❝♦❞✐✜❝❛çã♦✳ ✷✳ ❆❣r✉♣❡ ❧❡tr❛s s✉❝❡ss✐✈❛s ❞❡ t❡①t♦ ❝♦♠✉♠ ❡♠ ♣❛r❡s ❡ s✉❜st✐t✉❛ ❝❛❞❛ ❧❡tr❛ ❞❡ t❡①t♦ ❝♦♠✉♠ ♣♦r s❡✉ ✈❛❧♦r ♥✉♠ér✐❝♦✳ ❙❡ ♦ t❡①t♦ ❝♦♠✉♠ t❡♠ ✉♠ ♥ú♠❡r♦ í♠♣❛r ❞❡ ❧❡tr❛s✱ ❛❞✐❝✐♦♥❡ ✉♠❛ ❧❡tr❛ ✜❝tí❝✐❛ ♣❛r❛ ❝♦♠♣❧❡t❛r ♦ ú❧t✐♠♦ ♣❛r ❡ ♣r♦❝❡❞❛ ❝♦♠♦ ❛♥t❡s✳ ✸✳ ❈♦♥✈❡rt❛ ❝❛❞❛ ♣❛r s✉❝❡ss✐✈♦ p1 , p2 ❞❡ ❧❡tr❛s ❞❡ t❡①t♦ ❝♦♠✉♠ ❡♠ ✉♠ ✈❡t♦r✲ ❝♦❧✉♥❛   p= p1 p2 ❡ ❢♦r♠❡ ♦ ♣r♦❞✉t♦ Ap✱ ♦♥❞❡ p é ♦ ✈❡t♦r ❝♦♠✉♠ ❡ Ap ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❡t♦r ❝♦❞✐✜❝❛❞♦✳ ✹✳ ❈♦♥✈❡rt❛ ❝❛❞❛ ✈❡t♦r ❝♦❞✐✜❝❛❞♦ ❡♠ s❡✉ ❡q✉✐✈❛❧❡♥t❡ ❛❧❢❛❜ét✐❝♦✳ ❊①❡♠♣❧♦ ✷✳✶✸✳ ❯s❡ ❛ ♠❛tr✐③ A=  1 2 0 3  ♣❛r❛ ♦❜t❡r ♦ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧ ❞❛ ♠❡♥s❛❣❡♠ ❞❡ t❡①t♦ ❝♦♠✉♠ ❈❆❉❊■❘❆✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ❞❡✈❡♠♦s ❛❣r✉♣❛r ♦ t❡①t♦ ❝♦♠✉♠ ❡♠ ♣❛r❡s ❞❡ ❧❡tr❛s ❡ ❛❞✐❝✐♦♥❛r ❛ ❧❡tr❛ ✜❝tí❝✐❛ ❆ ♣❛r❛ ❝♦♠♣❧❡t❛r ♦ ú❧t✐♠♦ ♣❛r✱ ❥á q✉❡ ♦ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❡tr❛s q✉❡ ❢♦r♠❛ ❛ ♣❛❧❛✈r❛ é í♠♣❛r✱ ❞❛í t❡♠♦s✿ ❙♦❧✉çã♦✿ ❈❆ ❉❊ ■❘ ❆❆ ♦✉ ❛✐♥❞❛✱ ✉s❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✶✵✱ 31 45 9 18 1 1. ❆❣♦r❛✱ ❞❡✈❡♠♦s ❡❢❡t✉❛r ♦ ♣r♦❞✉t♦ ♠❛tr✐❝✐❛❧ r❡❢❡r❡♥t❡ ❛ ❝❛❞❛ ♣❛r ❞❡ ❧❡tr❛s✳ P❛r❛ ✐ss♦✱ ♦❜s❡r✈❡ q✉❡ ❛ t❛❜❡❧❛ ✈❛✐ ❛té ♦ ♥ú♠❡r♦ 25✱ ❧♦❣♦✱ s❡♠♣r❡ q✉❡ ♦❝♦rr❡r ✉♠ ✐♥t❡✐r♦ ♠❛✐♦r ❞♦ q✉❡ 26✱ ❡❧❡ s❡rá s✉❜st✐t✉í❞♦ ♣❡❧♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡st❡ ✐♥t❡✐r♦ ♣♦r 26✱ ♦✉ s❡❥❛✱ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ✉♠ ♥♦✈♦ ♥ú♠❡r♦ b q✉❡ s❡❥❛ ❝♦♥❣r✉❡♥t❡ ❛ a ♠ó❞✉❧♦ 26✱ (a ≡ b mod 26)✳ ❈♦♠❡ç❛r❡♠♦s✱ ❡♥tã♦✱ ❛ ❝♦❞✐✜❝❛çã♦ ♣❡❧♦ ♣❛r ❈❆✱ ❛ss✐♠ t❡♠♦s✿  1 2 0 3  3 1  =  3+2 0+3  =  5 3  mod 26, q✉❡ ❢♦r♥❡❝❡ ♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ❊❈ ♣❡❧❛ ❚❛❜❡❧❛ ✷✳✶✵✳ ❉❛♥❞♦ s❡q✉ê♥❝✐❛✱ ✐r❡♠♦s ❝♦❞✐✜❝❛r ♦ ♣❛r ❉❊✱ ✻✵

(268)   1 2 0 3 4 5   =  14 15 mod 26, q✉❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✶✵✱ ❢♦r♥❡❝❡ ♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ◆❖✳ ❈♦♥t✐♥✉❛♥❞♦✱ t❡♠♦s ♦ ❝♦❞✐✜❝❛çã♦ ❞♦ ♣❛r ■❘✱  1 2 0 3   9 18 =  45 54   ≡ 19 2  mod 26. ❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡♠♦s ♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ❙❇ ♣❡❧❛ ❚❛❜❡❧❛ ✷✳✶✵ ♣❛r❛ ♦ ♣❛r ■❘✳ ❏á ♦ ♣❛r ❆❆✱ t❡♠♦s✿  1 2 0 3  1 1  =  3 3  mod 26, ❞❡ ♠♦❞♦ q✉❡ ♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ é ❈❈✳ ❏✉♥t❛♥❞♦ ♦s ♣❛r❡s ❝♦❞✐✜❝❛❞♦s✱ ♦❜t❡♠♦s ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ❝♦♠♣❧❡t❛ q✉❡✱ ♥♦r♠❛❧♠❡♥t❡✱ s❡r✐❛ tr❛♥s♠✐t✐❞❛ ❝♦♠♦ ✉♠ ú♥✐❝♦ t❡①t♦ s❡♠ ❡s♣❛ç♦s✿ ECN OSBCC. ❖❜s❡r✈❡ q✉❡ ♥♦ ❊①❡♠♣❧♦ ✷✳✶✸ ♦ t❡①t♦ ❝♦♠✉♠ ❢♦✐ ❛❣r✉♣❛❞♦ ❡♠ ♣❛r❡s ❡ ❝r✐♣t♦❣r❛❢❛❞♦ ♣♦r ✉♠❛ ♠❛tr✐③ 2 × 2✳ ◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧ ❞♦ ❡①❡♠♣❧♦ é ✉♠ ✷ ✲ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧✳ ◆♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ❛❣r✉♣❛r ♦ t❡①t♦ ❝♦♠✉♠ ❡♠ ❝♦♥❥✉♥t♦s ❞❡ ♥ ❧❡tr❛s ❡ ❝♦❞✐✜❝❛r♠♦s ❝♦♠ ✉♠❛ ♠❛tr✐③ ❝♦❞✐✜❝❛❞♦r❛ n × n ❞❡ ❡♥tr❛❞❛s ✐♥t❡✐r❛s✱ ♥❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧ é ✉♠ ♥✲ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧✳ ❊①❡♠♣❧♦ ✷✳✶✹✳ ❯t✐❧✐③❛♥❞♦ ❛ ♠❛tr✐③  A= ✼ ✽ ✶ ✶✷ ✷✸ ✶✹ ✷✷ ✹ ✷✶   ❝♦♠♦ ❝❤❛✈❡✱ ❝♦❞✐✜q✉❡ ❛ ♠❡♥s❛❣❡♠ ✏▼❆❚❊▼➪❚■❈❆ ➱ ▲❊●❆▲✑ ❡♠ Z32 t❡♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ❛ ❚❛❜❡❧❛ ✷✳✶✶✳ ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ 1 2 3 4 5 6 7 8 9 10 11 12 13 ◆ ❖ P ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ 14 15 16 17 18 19 20 21 22 23 24 25 26 ✃ ❄ ➪ ➹ ➱ ❬ ❪ 27 28 29 30 31 0 ❚❛❜❡❧❛ ✷✳✶✶✿ ❙♦❧✉çã♦✿ ❈♦♥✈❡rt❡♥❞♦ ❛ ♠❡♥s❛❣❡♠ ❝♦♥❢♦r♠❡ ❚❛❜❡❧❛ ✷✳✶✶✱ t❡♠♦s✿  13 M AT =  1  , 20  ❊▼➪    20 5 =  13  , T IC =  9  , A 3 29  ✻✶  1 ➱ =  0 , 31 

(269)  0 LE =  12  5   7 ❡ GAL =  1  . 12  P❛r❛ ❝♦❞✐✜❝❛r♠♦s ❛ ♠❡♥s❛❣❡♠✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ❝❛❞❛ ♠❛tr✐③ 3×1 ♦❜t✐❞❛ ♣❡❧❛ ♠❛tr✐③ ❝♦❞✐✜❝❛❞♦r❛ ❆✳ ✷✸ ✶✸ ✼ ✽ ✶  ✶✷ ✷✸ ✶✹   ✶  =  ✶✶  mod 32 ✻ ✷✵ ✷✷ ✹ ✷✶      ✽ ✺ ✼ ✽ ✶  ✶✷ ✷✸ ✶✹   ✶✸  =  ✷✾  mod 32 ✸ ✷✾ ✷✷ ✹ ✷✶      ✷✸ ✷✵ ✼ ✽ ✶  ✶✷ ✷✸ ✶✹   ✾  =  ✾  mod 32 ✷✼ ✸ ✷✷ ✹ ✷✶      ✶ ✼ ✽ ✶ ✻  ✶✷ ✷✸ ✶✹   ✵  =  ✸✵  mod 32 ✶ ✸✶ ✷✷ ✹ ✷✶      ✺ ✵ ✼ ✽ ✶  ✶✷ ✷✸ ✶✹   ✶✷  =  ✷✻  mod 32 ✷✺ ✺ ✷✷ ✹ ✷✶      ✼ ✼ ✽ ✶ ✺  ✶✷ ✷✸ ✶✹   ✶  =  ✶✾  mod 32 ✷✻ ✶✷ ✷✷ ✹ ✷✶      ❆ss✐♠✱ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ é ❲❑❋❍➪❈❲■✃❋➹❆❊❩❨❊❙❩✳ ✷✳✹✳✶ ❉❡❝♦❞✐✜❝❛♥❞♦ P❛r❛ ❞❡❝♦❞✐✜❝❛r ♦s ❝ó❞✐❣♦s ❞❡ ❍✐❧❧✱ ✉s❛♠♦s ❛ ✐♥✈❡rs❛ (mod m) ❞❛ ♠❛tr✐③ ❝♦❞✐✜❝❛❞♦r❛✳ ❙✉♣♦♥❤❛ q✉❡  a11 a12  a21 a22  A=  ... ... an1 an2 ✻✷  . . . a1n . . . a2n    ✳✳✳ ...  . . . ann

(270) é ✐♥✈❡rtí✈❡❧ ♠ó❞✉❧♦ m ❡ q✉❡ ❡st❛ ♠❛tr✐③ é ✉s❛❞❛ ♣❛r❛ ✉♠ n−❝ó❞✐❣♦ ❞❡ ❍✐❧❧✳ ❙❡  p1  p2    p =  ✳✳   ✳  pn ✭✷✳✽✮ c = Ap ✭✷✳✾✮  é ✉♠ ✈❡t♦r ❝♦♠✉♠✱ ❡♥tã♦ é ♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❡t♦r ❝♦❞✐✜❝❛❞♦ ❡ p = A−1 c. ❆ss✐♠✱ ❝❛❞❛ ✈❡t♦r ❝♦♠✉♠ ♣♦❞❡ s❡r r❡❝✉♣❛r❛❞♦ ❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ✈❡t♦r ❝♦❞✐✜❝❛❞♦ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ à ❡sq✉❡r❞❛ ♣♦r A−1 (mod m)✳ ❖ ✐♠♣♦rt❛♥t❡ ❛q✉✐✱ é s❛❜❡r q✉❛✐s ❛s ♠❛tr✐③❡s sã♦ ✐♥✈❡rtí✈❡✐s ♠ó❞✉❧♦ m ❡ ❝♦♠♦ ♦❜t❡r s✉❛s ✐♥✈❡rs❛s✱ ❡ ♣❛r❛ ✐ss♦ ❢❛③❡♠♦s ✉s♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✷✵✱ ❈♦r♦❧ár✐♦ ✶✳✶✷✶ ❡ ❈♦r♦❧ár✐♦ ✶✳✶✷✷✳ ❊①❡♠♣❧♦ ✷✳✶✺✳ ❉❡❝♦❞✐✜q✉❡ ❛ ♠❡♥s❛❣❡♠ SAKN OXAOJX s❛❜❡♥❞♦ q✉❡ é ✉♠ ❝ó❞✐❣♦ ❞❡ ❍✐❧❧ ❝♦♠ ♠❛tr✐③ ❝♦❞✐✜❝❛❞♦r❛ A=  4 1 3 2  . Pr✐♠❡✐r❛♠❡♥t❡ ❞❡✈❡♠♦s ❡♥❝♦♥tr❛r ❛ ♠❛tr✐③ ✐♥✈❡rs❛ ❞❛ ♠❛tr✐③ ❝♦❞✐✜❝❛❞♦r❛ ❙♦❧✉çã♦✿ A=  4 1 3 2  ♠ó❞✉❧♦ 26✳ ❆ss✐♠ ♣♦r ✭✶✳✾✮✱ t❡♠♦s q✉❡ det(A) = ad − bc = 4 · 2 − 1 · 3 = 8 − 3 = 5 ❞❡ ♠♦❞♦ q✉❡ (ad − bc)−1 = 5−1 = 21(mod 26) ✭✐♥✈❡rs♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦✱ ✈❡❥❛ ❉❡✜♥✐çã♦ ✶✳✹✽✮✳ ▲♦❣♦✱ ♣♦r ✭✶✳✾✮✱ t❡♠✲s❡ −1 A = 21  2 −1 −3 4  =  42 −21 −63 84  =  16 5 15 6  (mod 26). ❆❣♦r❛✱ ❡♥❝♦♥tr❛❞❛ ❛ ♠❛tr✐③ ✐♥✈❡rs❛✱ ❞❛r❡♠♦s ✐♥í❝✐♦ à ❞❡❝♦❞✐✜❝❛çã♦✳ P❡❧❛ ❚❛❜❡❧❛ ✷✳✶✵✱ ♦ ❡q✉✐✈❛❧❡♥t❡ ♥✉♠ér✐❝♦ ❞♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ é ❛ ❚❛❜❡❧❛ ✷✳✶✷✱ ❞❡ ♠♦❞♦ q✉❡ ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❡t♦r❡s ❝♦❞✐✜❝❛❞♦s sã♦✿ p1 =  19 1  , p2 =  11 14  , p3 =  15 24 ✻✸  , p4 =  1 15  e p5 =  10 24  .

(271) ❙ ❆ ❑ ◆ ❖ ❳ ❆ ❖ ❏ ❳ 19 1 11 14 15 24 1 15 10 24 ❚❛❜❡❧❛ ✷✳✶✷✿ ❊♥tã♦✱ ♣❛r❛ ♦❜t❡r ♦s ♣❛r❡s ❞❡ t❡①t♦ ❝♦♠✉♠✱ ♠✉❧t✐♣❧✐❝❛♠♦s ❝❛❞❛ ✈❡t♦r ❝♦❞✐✜❝❛❞♦ ♣♦r A−1 ❡ ♣❡❧❛ ❚❛❜❡❧❛ ✷✳✶✵ ❡♥❝♦♥tr❛♠♦s ♦s ❡q✉✐✈❛❧❡♥t❡s ❛❧❢❛❜ét✐❝♦s ❞❡st❡s ✈❡t♦r❡s✿   ✶✻ ✺ ✶✺ ✻ ✶✻ ✺ ✶✺ ✻  ✶✻ ✺ ✶✺ ✻  ✶✻ ✺ ✶✺ ✻  ✶✻ ✺ ✶✺ ✻   ✶✾ ✶ ✶✶ ✶✹  ✶✺  ✶  ✷✹ ✶✺ ✶✵ ✷✹  =  ✷✸ =  ✶✷ =  ✷✷ =  ✶✸ =  ✷✵     ✺ ✶✺ ✺ ✶ ✽  (mod 26) : W E  (mod 26) : L O  (mod 26) : V E  (mod 26) : M A  (mod 26) : T . H ▲♦❣♦✱ ❞❡❝♦❞✐✜❝❛♥❞♦ ♦ ❝ó❞✐❣♦ t❡♠♦s ❛ s❡❣✉✐♥t❡ ♠❡♥s❛❣❡♠ ✏❲❊ ▲❖❱❊ ▼❆❚❍✑✳ ✷✳✹✳✷ ◗✉❡❜r❛♥❞♦ ✉♠ ❈ó❞✐❣♦ ❞❡ ❍✐❧❧ ❖ ♦❜❥❡t✐✈♦ ❛♦ ❝r✐♣t♦❣r❛❢❛r ✉♠❛ ♠❡♥s❛❣❡♠ é ❢❛③❡r ❝♦♠ q✉❡ ❡ss❛ ❝❤❡❣✉❡ ❝♦♠ s❡❣✉r❛♥ç❛ ❛♦ s❡✉ ❞❡st✐♥♦✳ ◆♦ ❡♥t❛♥t♦✱ ❛❣♦r❛ s❡rá ❞✐s❝✉t✐❞♦ ✉♠❛ té❝♥✐❝❛ ❞❡ q✉❡❜r❛r ♦ ❈ó❞✐❣♦ ❞❡ ❍✐❧❧✳ P❛r❛ ✐ss♦✱ é ♥❡❝❡ssár✐♦ ♣r✐♠❡✐r♦ q✉❡ s❡❥❛ ❢❡✐t♦ ✉♠❛ ❛♥á❧✐s❡ ❞♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦✳ ❆ ♣❛rt✐r ❞❡ss❛ ❛♥á❧✐s❡ é ♣♦ssí✈❡❧ q✉❡ ✈♦❝ê ❞❡s❝✉❜r❛ ❛❧❣✉♠❛ ✐♥❢♦r♠❛çã♦ ❡ ❡♥tã♦ s❡❥❛ ♣♦ssí✈❡❧ ❞❡t❡r♠✐♥❛r ❛ ♠❛tr✐③ ❞❡❝♦❞✐✜❝❛❞♦r❛ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♦❜t❡♥❤❛ ❛❝❡ss♦ ❛♦ r❡st♦ ❞❛ ♠❡♥s❛❣❡♠✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ s✉♣♦♥❤❛ q✉❡ ✈♦❝ê✱ ❛♦ ❛♥❛❧✐s❛r ✉♠ t❡①t♦ ❝♦❞✐✜❝❛❞♦✱ ❞❡s❝✉❜r❛ q✉❡ ❡ss❡ t❡①t♦ r❡♣r❡s❡♥t❛ ✉♠❛ ❝❛rt❛ ❡ q✉❡ ❝♦♠❡ç❛ ♣♦r ❉❊❆❘ ❙■❘✳ ❊ss❡ ♣❡q✉❡♥♦ ❞❛❞♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡❝♦❞✐✜❝❛r ♦ r❡st♦ ❞♦ t❡①t♦✳ ❖ t❡♦r❡♠❛ ❛ s❡❣✉✐r ❢♦r♥❡❝❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ ❢❛③❡r ✐st♦✳ ❚❡♦r❡♠❛ ✷✳✶✻ ✭❉❡t❡r♠✐♥❛♥❞♦ ❛ ▼❛tr✐③ ❉❡❝♦❞✐✜❝❛❞♦r❛✮✳  p1i  p2i  pi =  ✳✳  ✳ pni ♦♥❞❡ p 1 , p2 , . . . , p n   c1i  c2i  ci =  ✳✳  ✳ cni   ,  sã♦ ✈❡t♦r❡s ❝♦♠✉♥s ❡ c 1 , c2 , . . . , c n ❙❡❥❛♠    ,  ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❡t♦r❡s ❝♦❞✐✜❝❛❞♦s ❞❡ ✉♠ ♥✲❝ó❞✐❣♦ ❞❡ ❍✐❧❧✱    p11 p21 . . . pn1 pt1  pt   p12 p22 . . . pn2  2   P =  ✳✳  =  ✳✳ ✳✳ ✳✳  ✳   ✳ ✳ ✳ t p1n p2n . . . pnn pn      ❡ ✻✹   c11 c21 . . . cn1 ct1  ct   c12 c22 . . . cn2  2   C =  ✳✳  =  ✳✳ ✳✳ ✳✳  ✳   ✳ ✳ ✳ t c1n c2n . . . cnn cn     , 

(272) ♦♥❞❡ pti ❡ cti r❡♣r❡s❡♥t❛♠ ❛ tr❛♥s♣♦st❛s ❞❡ pi ❡ ci r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❙❡ P é ✐♥✈❡rsí✈❡❧✱ ❡♥tã♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❧✐♥❤❛s q✉❡ r❡❞✉③ C ❛ I tr❛♥s❢♦r♠❛ P ❡♠ (A−1 )t ✳ ❉❡♠♦♥str❛çã♦✳ P♦r ✭✷✳✾✮ ❡ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ P ❡ C ✱ ❡❢❡t✉❛♥❞♦ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛s ♠❛tr✐③❡s✱ t❡♠♦s C t = AP t ✳ ❆ss✐♠✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r C = P · At ✳ ❙❡♥❞♦ A ❡ P ✐♥✈❡rsí✈❡✐s✱ r❡s✉❧t❛ ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❞❡t❡r♠✐♥❛♥t❡ ❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✵✹ q✉❡ C é ✉♠❛ ♠❛tr✐③ ✐♥✈❡rsí✈❡❧✳ ❙❡❥❛♠ E1 , . . . , Ek ❛s ♠❛tr✐③❡s ❡❧❡♠❡♥t❛r❡s q✉❡ ❝♦rr❡s♣♦♥❞❡♠ às ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s ❝♦♠ ❛s ❧✐♥❤❛s q✉❡ r❡❞✉③❡♠ C ❛ I ✱ ♦✉ s❡❥❛✱ Ek . . . E1 C = I ✳ ❙✉❜st✐t✉✐♥❞♦ ❡♠ C = P · AT ✱ ❡♥❝♦♥tr❛♠♦s Ek . . . E1 P AT = I ✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r (A−1 )t ✱ t❡♠♦s Ek . . . E1 P = (A−1 )t ✱ ♦✉ s❡❥❛✱ ❛ ♠❡s♠❛ s❡q✉ê♥❝✐❛ ❞❡ ♦♣❡r❛çõ❡s ❝♦♠ ❛s ❧✐♥❤❛s q✉❡ r❡❞✉③ C ❛ I ❝♦♥✈❡rt❡ P ❛ (A−1 )t ✳ ❆ ♣❛rt✐r ❞❡ss❡ t❡♦r❡♠❛✱ ❝❤❡❣❛♠♦s ❛ ❝♦♥❝❧✉sã♦ q✉❡ ♣❛r❛ ❡♥❝♦♥tr❛r ❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❞❡❝♦❞✐✜❝❛❞♦r❛ A−1 ❞❡✈❡♠♦s ❡❢❡t✉❛r ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❧✐♥❤❛s q✉❡ r❡❞✉③ C ❛ I ❡ ❡ss❛s ♠❡s♠❛s ♦♣❡r❛çõ❡s ❛♣❧✐❝❛r s♦❜r❡ ❧✐♥❤❛s ❞❡ P ✳ ❊①❡♠♣❧♦ ✷✳✶✼✳ ❉❡❝♦❞✐✜q✉❡ ♦ ✷✲❝ó❞✐❣♦ ❞❡ ❍✐❧❧ LN GIHGY BV REN JY QO s❛❜❡♥❞♦ q✉❡ ❛s q✉❛tr♦ ú❧t✐♠❛s ❧❡tr❛s ❞♦ t❡①t♦ ❝♦♠✉♠ sã♦ AT OM ✳ ❙♦❧✉çã♦✿ P❡❧❛ ❚❛❜❡❧❛ ✷✳✶✵✱ ♦ ❡q✉✐✈❛❧❡♥t❡ ♥✉♠ér✐❝♦ ❞♦ t❡①t♦ ❝♦♠✉♠ ❝♦♥❤❡❝✐❞♦ é✿ ❆ ❚ ❖ ▼ 1 20 15 13 ❀ ❚❛❜❡❧❛ ✷✳✶✸✿ ❡ ♦ ❡q✉✐✈❛❧❡♥t❡ ♥✉♠ér✐❝♦ ❞♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ❝♦rr❡s♣♦♥❞❡♥t❡ é✿ ❏ ❨ ◗ ❖ 10 25 17 15 ❚❛❜❡❧❛ ✷✳✶✹✿ ❞❡ ♠♦❞♦ q✉❡ ♦s ✈❡t♦r❡s ❝♦♠✉♥s ❡ ❝♦rr❡s♣♦♥❞❡♥t❡s ✈❡t♦r❡s ❝♦❞✐✜❝❛❞♦s sã♦✿ p1 =  1 20  p2 =  15 13  ❡ ◗✉❡r❡♠♦s r❡❞✉③✐r C=  ct1 ct2 ←→ c1 =  10 25  ←→ c2 =  17 15   = ✻✺  ✶✵ ✷✺ ✶✼ ✶✺  .

(273) ❛ I ✭♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡✮ ♣♦r ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s s♦❜r❡ ❧✐♥❤❛s ❡ s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❛♣❧✐❝❛r ❡st❛s ♦♣❡r❛çõ❡s ❛ P =  pt1 pt2  =  ✶ ✷✵ ✶✺ ✶✸  ♣❛r❛ ♦❜t❡r (A−1 )t ✭❛ tr❛♥s♣♦st❛ ❞❛ ♠❛tr✐③ ❞❡❝♦❞✐✜❝❛❞♦r❛✮✳ P❛r❛ ✐ss♦✱ ❝♦❧♦❝❛♠♦s P à ❞✐r❡✐t❛ ❞❡ C ✱ [C|P ]✱ ❡ ❛♣❧✐❝❛♠♦s ♥❡st❛ ❛s ♦♣❡r❛çõ❡s s♦❜r❡ ❧✐♥❤❛s ❛té q✉❡ ♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❡st❡❥❛ r❡❞✉③✐❞♦ ❛ I ✳ ❆ ♠❛tr✐③ ✜♥❛❧ ❡♥tã♦ t❡rá ♦ ❢♦r♠❛t♦ [I|(A−1 )T ]✳ ❆ss✐♠✱ t❡♠♦s✿   ✶✵ ✷✺ ✶ ✷✵ [C|P ] = . ✶✼ ✶✺ ✶✺ ✶✸ ❖❜s❡r✈❡ q✉❡ 7 · 15 ≡ 1 mod 26✳ ❆ss✐♠✱ ♠✉❧t✐♣❧✐❝❛♠♦s ❛ ❧✐♥❤❛ 2 ♣♦r 7 ❡ ♦❜t❡♠♦s ✶✵ ✷✺ ✶ ✷✵ ✶✶✾ ✶✵✺ ✶✵✺ ✾✶   . ❙✉❜st✐t✉✐♥❞♦ 119, 105 ❡ 91 ♣❡❧♦s s❡✉s r❡sí❞✉♦s ♠ó❞✉❧♦ 26✱ t❡♠✲s❡ ✶✵ ✷✺ ✶ ✷✵ ✶✺ ✶ ✶ ✶✸   . ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♣♦r −25 ❡ s♦♠❛♥❞♦ à ♣r✐♠❡✐r❛ ❧✐♥❤❛✱ ♦❜t❡♠♦s  ✲✸✻✺ ✵ ✲✷✹ ✲✸✵✺ ✶✸ ✶✺ ✶ ✶  . ❙✉❜st✐t✉í♠♦s −365✱ −24 ❡ −305 ♣❡❧♦s s❡✉s r❡sí❞✉♦s ♠ó❞✉❧♦ 26  ✷✺ ✵ ✷ ✼ ✶✺ ✶ ✶ ✶✸  . ▼✉❧t✐♣❧✐❝❛♠♦s ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ♣♦r 25−1 = 25✱ t❡♠♦s ✶ ✵ ✺✵ ✶✼✺ ✶✺ ✶ ✶ ✶✸   . ❙✉❜st✐t✉í♠♦s 50 ❡ 175 ♣❡❧♦s s❡✉s r❡sí❞✉♦s ♠ó❞✉❧♦ 26  ✶ ✵ ✷✹ ✶✾ ✶✺ ✶ ✶ ✶✸  . ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❧✐♥❤❛ ♣♦r −15 ❡ s♦♠❛♥❞♦ à s❡❣✉♥❞❛✱ ♦❜té♠✲s❡  ✶✾ ✶ ✵ ✷✹ ✵ ✶ ✲✸✺✾ ✲✷✼✷  . ❙✉❜st✐t✉í♠♦s −359 ❡ −272 ♣❡❧♦s s❡✉s r❡sí❞✉♦s ♠ó❞✉❧♦ 26✱ ❝♦♥❝❧✉í♠♦s q✉❡  ✶ ✵ ✷✹ ✶✾ ✵ ✶ ✺ ✶✹ ✻✻  .

(274) ▲♦❣♦✱  −1 T (A ) = ✷✹ ✶✾ ✺ ✶✹  ❡ ♣♦rt❛♥t♦ ❛ ♠❛tr✐③ ❞❡❝♦❞✐✜❝❛❞♦r❛ é A −1 =  ✷✹ ✺ ✶✾ ✶✹  . P❛r❛ ❞❡❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠✱ ♣r✐♠❡✐r♦ ❛❣r✉♣❛♠♦s ♦ t❡①t♦ ❝♦❞✐✜❝❛❞♦ ❡♠ ♣❛r❡s ❡ ❡♥❝♦♥tr❛♠♦s ♦s ❡q✉✐✈❛❧❡♥t❡s ♥✉♠ér✐❝♦s ❞❡ ❝❛❞❛ ❧❡tr❛ ✭✈❡❥❛ ❛ ❚❛❜❡❧❛ ✷✳✶✺✮✳ ▲ ◆ ● ■ ❍ ● ❨ ❇ ❱ ❘ ❊ ◆ ❏ ❨ ◗ ❖ 12 14 7 9 8 7 25 2 22 18 5 14 10 25 17 15 ❊♠ ❚❛❜❡❧❛ ✷✳✶✺✿ s❡❣✉✐❞❛✱ ♠✉❧t✐♣❧✐❝❛♠♦s ♦s ✈❡t♦r❡s ❝♦❞✐✜❝❛❞♦s s✉❝❡ss✐✈❛♠❡♥t❡ ♣❡❧❛ ❡sq✉❡r❞❛ ♣♦r A−1 ❡ ❡♥❝♦♥tr❛♠♦s ♦s ❡q✉✐✈❛❧❡♥t❡s ❛❧❢❛❜ét✐❝♦s ❞♦s ♣❛r❡s ❞❡ t❡①t♦ ❝♦♠✉♠ r❡s✉❧t❛♥t❡s✿         ✷✹ ✺ ✶✾ ✶✹  ✷✹ ✺ ✶✾ ✶✹ ✷✹ ✺ ✶✾ ✶✹ ✷✹ ✺ ✶✾ ✶✹ ✷✹ ✺ ✶✾ ✶✹  ✺ ✶✾ ✶✹ ✺ ✶✾ ✶✹ ✷✹ ✺ ✶✾ ✶✹   ✷✹ ✷✹      ✶✷ ✶✹ ✼ ✾ ✽ ✼   ✷✺ ✷ ✷✷ ✶✽ ✺ ✶✹ ✶✵ ✷✺ ✶✼ ✶✺ =  ✷✵ =  ✺ =   ✶✾ ✶✻ = ✶✷ =  ✷✵ =  ✽ =  ✶ =    ✷✺    ✽ ✾ ✷✵ ✺ ✷✵ ✶✺ ✶✸  (mod 26) =⇒ T H  (mod 26) =⇒ E Y  (mod 26) =⇒ S P  (mod 26) =⇒ L I  (mod 26) =⇒ T T  (mod 26) =⇒ H E  (mod 26) =⇒ A T  (mod 26) =⇒ O M ❋✐♥❛❧♠❡♥t❡✱ ❝♦♥str✉í♠♦s ❛ ♠❡♥s❛❣❡♠ ❛ ♣❛rt✐r ❞♦s ♣❛r❡s ❞❡ t❡①t♦ ❝♦♠✉♠✿ ✏❚❍❊❨ ❙P▲■❚ ❚❍❊ ❆❚❖▼✑✳ ✻✼

(275) ✷✳✺ ❙✐st❡♠❛ ❘❙❆ ❖ s✐st❡♠❛ ❘❙❆✸ é s✐♠♣❧❡s ❡ ❜❛s❡✐❛✲s❡ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ♣❛r❛ ❞❡s❝♦❜r✐r ♦s ❢❛t♦r❡s ♣r✐♠♦s ❡①✐st❡♥t❡s ❡♠ ♥ú♠❡r♦s ♠✉✐t♦ ❣r❛♥❞❡s✳ ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ ♠♦str❛ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ♦ s✐st❡♠❛ ❘❙❆✱ ❝♦♠♦ ❡stã♦ ❞❡✜♥✐❞♦s ♦s ❝ó❞✐❣♦s ❡ ❝♦♠♦ ♣♦❞❡♠♦s ❞❡❝♦❞✐✜❝á✲ ❧♦s✳ ❚❡♦r❡♠❛ ✷✳✶✽✳ ❙✉♣♦♥❤❛♠♦s q✉❡✿ ✶✳ n = pq ✱ ♦♥❞❡ p ❡ q sã♦ ♥ú♠❡r♦s ♣r✐♠♦s ❞✐st✐♥t♦s❀ ✷✳ e é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ✐♥✈❡rtí✈❡❧ ♠ó❞✉❧♦ φ(n)✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s mdc(e, φ(n)) = mdc(e, (p − 1)(q − 1)) = 1❀ ✸✳ b é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✈♦ t❛❧ q✉❡ b 6≡ 0(mod p) ❡ b 6≡ 0(mod q)✱ r❡♣r❡s❡♥t❛ ❝❛❞❛ ❜❧♦❝♦ ♥✉♠ér✐❝♦ ❡ b < n ❀ ✹✳ C(b) é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ r❡♣r❡s❡♥t❛ ❝❛❞❛ ❜❧♦❝♦ ❝♦❞✐✜❝❛❞♦ ❞❡✜♥✐❞♦ ♣♦r C(b) ≡ be (mod n), ♦♥❞❡ 0 ≤ C(b) < n❀ ✺✳ d é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡ é ♦ ✐♥✈❡rs♦ ❞❡ e ♠ó❞✉❧♦ φ(n)✱ ♦✉ s❡❥❛✱ ed ≡ 1(modφ(n))✱ 1 ≤ d < (p − 1)(q − 1). ✻✳ D(c) é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ r❡♣r❡s❡♥t❛ ♦ ❜❧♦❝♦ ❞❡❝♦❞✐✜❝❛❞♦ ❞❡✜♥✐❢♦ ♣♦r D(c) ≡ cd (mod n), ♦♥❞❡ 0 ≤ D(c) < n✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ D(C(b)) = b✱ ♦✉ s❡❥❛✱ ❞❡❝♦❞✐✜❝❛♥❞♦ ✉♠ ❜❧♦❝♦ ❞❡ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✱ ❡♥❝♦♥tr❛♠♦s ✉♠ ❜❧♦❝♦ ❞❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❡♠♦s ❡♥tã♦ n = pq ✳ ❱❛♠♦s ♣r♦✈❛r q✉❡ D(C(b)) ≡ b(mod n). ❖❜s❡r✈❡ q✉❡ D(C(b)) ❡ b sã♦ ♠❡♥♦r❡s q✉❡ n − 1✳ P♦r ✐ss♦ ❡s❝♦❧❤❡♠♦s b ♠❡♥♦r q✉❡ n ❡ ♠❛♥t✐✈❡♠♦s ♦s ❜❧♦❝♦s s❡♣❛r❛❞♦s ❞❡♣♦✐s ❞❛ ❝♦❞✐✜❝❛çã♦✳ P♦r ❞❡✜♥✐çã♦✱ t❡♠♦s q✉❡ D(C(b)) ≡ (be )d ≡ bed (mod n). ▼❛s d é ♦ ✐♥✈❡rs♦ ❞❡ e ♠ó❞✉❧♦ φ(n)✳ ▲♦❣♦ ❡①✐st❡ ✐♥t❡✐r♦ ❑ t❛❧ q✉❡ ed = 1 + kφ(n). ▲♦❣♦✱ bed ≡ b1+kφ(n) ≡ (bφ(n) )k b(mod n). ❙❡ mdc(b, n) = 1✱ ❡♥tã♦ ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ✶✳✻✺✿ ✸❖ bed ≡ (bφ(n) )k b ≡ b(mod n). s✐st❡♠❛ ❘❙❆ r❡❝❡❜❡ ❡ss❡ ♥♦♠❡ ❡♠ ❤♦♠❡♥❛❣❡♠ ❛ s❡✉s ✐♥✈❡♥t♦r❡s ❘♦♥❛❧❞ ❘✐✈❡st✱ ❆❞✐ ❙❤❛♠✐r ❡ ▲❡♦♥❛r❞ ❆❞❧❡♠❛♥✱ ❢♦✐ ♦ ♣r✐♠❡✐r♦ ❝r✐♣t♦ss✐st❡♠❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ✻✽

(276) ❙❡ b ❡ n ♥ã♦ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦❜s❡r✈❡ q✉❡ n = pq ✱ p ❡ q ♣r✐♠♦s ❞✐st✐♥t♦s✳ ▲♦❣♦✱ bed ≡ b1+kφ(n) ≡ (b(p−1) )k(q−1) b(mod p). mdc(b, p) = 1✱ ❡♥tã♦ ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ✶✳✺✼ (bp−1 ≡ 1(mod p))✳ t❡♠♦s q✉❡ p|b ❡ ♣♦rt❛♥t♦✱ bed ≡ b ≡ o(mod p). ❙❡ ❙❡ ♥ã♦✱ ▲♦❣♦✱ bed ≡ b(mod p) q✉❛❧q✉❡r q✉❡ s❡❥❛ b✳ ❋❛③❡♥❞♦ ♦ ♠❡s♠♦ ♣❛r❛ ♦ ♣r✐♠♦ q✱ ♦❜t❡♥❞♦✿ bed ≡ b(mod q). P♦rt❛♥t♦✱ bed ≡ b(mod pq) ❝♦♠♦ q✉❡rí❛♠♦s✳ ❉❡✜♥✐çã♦ ✷✳✶✾✳ ❈❤❛♠❛r❡♠♦s ♦ ♥ú♠❡r♦ n = pq ❞❡ ♠ó❞✉❧♦✱ ♦ ♥ú♠❡r♦ e ❞❡ ♣♦tê♥❝✐❛ ❞❡ ❝♦❞✐✜❝❛çã♦✱ d ❞❡ ♣♦tê♥❝✐❛ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦ ❡ ❛ tr✐♣❧❛ (n, e, d) ❞❡ ❛ ❝❤❛✈❡ ❞♦ s✐st❡♠❛ ❘❙❆✳ ❖s ♥ú♠❡r♦s n, d ❡ e sã♦ t♦❞♦s ❡s❝♦❧❤✐❞♦s ♣♦r ✉s✉ár✐♦s ❞♦ s✐st❡♠❛ ❘❙❆ ❞❡s❞❡ q✉❡ s❡❥❛♠ s❛t✐s❢❡✐t❛ ❛s ❝♦♥❞✐çõ❡s ❞❡ ✶ ❛ ✺ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✽✳ ❉❡✜♥✐çã♦ ✷✳✷✵✳ ❖ ♣❛r (n, e) é ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞♦ s✐st❡♠❛ ❘❙❆ ❡ ♦ ♣❛r (n, d) é ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ❞♦ s✐st❡♠❛ ❘❙❆✳ P❛r❛ q✉❡ ❤❛❥❛ ❝♦♠✉♥✐❝❛çã♦ ❡♥tr❡ ❞✉❛s ❢♦♥t❡s é ♥❡❝❡ssár✐♦ ♦ ✉s♦ ❞❛s ❝❤❛✈❡s ♣ú❜❧✐❝❛ ❡ ♣r✐✈❛❞❛✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞❛ ❢♦♥t❡ ✈❡rs❛✳ ❉❡ss❛ ❢♦r♠❛ B ❡ A A ❞❡✈❡ s❡r ❝♦♥❤❡❝✐❞❛ ♣❡❧❛ ❢♦♥t❡ B ❡ ✈✐❝❡✲ ♣♦❞❡♠ tr♦❝❛r ♠❡♥s❛❣❡♥s s❡❝r❡t❛s✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ ✐ss♦ ❛❝♦♥t❡❝❡r ❛❧❣✉♠❛s ❡t❛♣❛s sã♦ ✐♠♣♦rt❛♥t❡s✿ ✶✳ B ❞❡✈❡ s❛❜❡r ❞❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ (n, e) ✷✳ B ❝♦♥✈❡rt❡ ❛ ♠❡♥s❛❣❡♠ ♣❛r❛ ♥ú♠❡r♦s ❛tr❛✈és ❞❡ ✉♠❛ t❛❜❡❧❛ ♦♥❞❡ ❝❛❞❛ ❧❡tr❛ ❞❡ A✳ ❞♦ ❛❧❢❛❜❡t♦ ❡stá r❡♣r❡s❡♥t❛❞♦ ♣♦r ✉♠ ♥ú♠❡r♦ ❢♦r♠❛❞♦ ♣♦r ✷ ❛❧❣❛r✐s♠♦s ♣❛r❛ ❡✈✐t❛r ❛♠❜✐❣✉✐❞❛❞❡ ✭❡ss❛ t❛❜❡❧❛ ❞❡✈❡ s❡r ❝♦♥❤❡❝✐❞♦ ♣♦r ❛♠❜❛s ❛s ❢♦♥t❡s✮✳ ✸✳ B ❡s❝r❡✈❡ ✉❧tr❛♣❛ss❛r b ❡♠ ❜❧♦❝♦s n = pq ✳ ♥✉♠ér✐❝♦s b1 , b2 , · · · , br ✱ ❡ss❡s ❜❧♦❝♦s ♥ã♦ ❞❡✈❡♠ ❖❜s❡r✈❛çã♦ ✷✳✷✶✳ ❆ ♠❛♥❡✐r❛ ❞❡ ❡s❝♦❧❤❡r ♦s ❜❧♦❝♦s ♥ã♦ é ú♥✐❝❛✱ ♠❛s é ✐♠♣♦rt❛♥t❡ ❡✈✐t❛r q✉❡ ❛❧❣✉♠ ❜❧♦❝♦ ❝♦♠❡❝❡ ❝♦♠ ♦ ♥ú♠❡r♦ 0 ✭♣♦r ♣r♦❜❧❡♠❛s ♥❛ ❞❡❝♦❞✐✜❝❛çã♦✮✳ ✹✳ B ❡♥❝r✐♣t❛ ♦s ❜❧♦❝♦s b1 , b2 , · · · , br ✉s❛♥❞♦ ❛ ❝♦♥❞✐çã♦ 4 ❞♦ ❚❡♦r❡♠❛ ✷✳✶✽ ❡ ❛ss✐♠ C(b1 ), C(b2 ), · · · , C(br )✳ ❡st❛❜❡❧❡❝❡ ♦s ❝ó❞✐❣♦s ✺✳ B tr❛♥s♠✐t❡ ♦s ❝ó❞✐❣♦s C(b1 ), C(b2 ), · · · , C(br ) ✻✾ ♣❛r❛ A✳

(277) ✻✳ ❆♦ r❡❝❡❜❡r ♦ ❝ó❞✐❣♦✱ ❛ ❢♦♥t❡ A ❞❡❝♦❞✐✜❝❛ ♦s ❝ó❞✐❣♦s C(b1 ), C(b2 ), · · · , C(br ) ✉s❛♥❞♦ ♦ ✐t❡♠ 6 ❞♦ ❚❡♦r❡♠❛ ✷✳✶✽✳ ✼✳ ❯♠❛ ✈❡③ q✉❡ D(c1 ), D(c2 ), · · · , D(cr ) sã♦ ❝♦♥❤❡❝✐❞♦s ♣♦r A✱ ❜❛st❛ ✉s❛r ❛ t❛❜❡❧❛ ❡ tr❛♥s❢♦r♠❛r ❡ss❡s ❜❧♦❝♦s ♥✉♠ér✐❝♦s ♥❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✳ ❉❡ss❛ ❢♦r♠❛ ♦ ♣r♦❝❡ss♦ ✜❝❛ ❝♦♥❝❧✉í❞♦✳ ❯s❛♥❞♦ ❛ ❝❤❛✈❡s ♣ú❜❧✐❝❛ (n, e) = (1037, 7) ❡ ❛ ❚❛❜❡❧❛ ✷✳✶✻✱ ❝♦❞✐✜q✉❡ ❡ ❞❡❝♦❞✐✜q✉❡ ❛ s❡❣✉✐♥t❡ ♣❛❧❛✈r❛✿ ❙❖◆❍❖✳ ❊①❡♠♣❧♦ ✷✳✷✷✳ ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ❏ ❑ ▲ ▼ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ✶✾ ✷✵ ✷✶ ✷✷ ◆ ❖ P ◗ ❘ ❙ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✸✷ ✸✸ ✸✹ ✸✺ ❬ ❪ ✾✾ ❚❛❜❡❧❛ ✷✳✶✻✿ ■♥✐❝✐❧♠❡♥t❡✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✷✳✶✻✱ ❞❡✈❡♠♦s tr♦❝❛r ❝❛❞❛ ❧❡tr❛ ♣❡❧♦ s❡✉ r❡s♣❡❝t✐✈♦ ✈❛❧♦r ♥✉♠ér✐❝♦✿ ❙♦❧✉çã♦✿ ❙❖◆❍❖ ✷✽ ✷✹ ✷✸ ✶✼ ✷✹ ❊♠ s❡❣✉✐❞❛✱ ❞❡✈❡♠♦s tr♦❝❛r ❛ ♠❡♥s❛❣❡♠ ❛❝✐♠❛ ❡♠ ❜❧♦❝♦s✱ q✉❡ ❞❡✈❡♠ t❡r ✈❛❧♦r ♠❡♥♦r q✉❡ ✶✵✸✼ ❡ ♠❛✐♦r q✉❡ ③❡r♦✿ ✷✽✷✱ ✹✷✸✱ ✶✼✷✱ ✹ ❆ ♣❛rt✐r ❞♦ ❚❡♦r❡♠❛ ✷✳✶✽ ✐t❡♠ 4✱ ❞❡s❝♦❜r✐r❡♠♦s s❡✉ r❡s♣❡❝t✐✈♦ ✈❛❧♦r ❝♦❞✐✜❝❛❞♦✳ C1 (282) = 2827 mod 1037 C1 (282) = 2822 .2824 .282 mod 1037 C1 (282) = 712.2824 282 mod 1037 C1 (282) = 712.7122 .282 mod 1037 C1 (282) = 712.888.282 mod 1037 C1 (282) = 723.282 mod 1037 C1 (282) = 634 mod 1037 Pr♦❝❡❞❡♥❞♦ ❞❡ss❛ ♠❛♥❡✐r❛ ❝♦♠ t♦❞♦s ♦s ❜❧♦❝♦s✱ t❡r❡♠♦s✿ C2 (423) = 4237 mod 1037 = 934, C3 (172) = 1727 mod 1037 = 621 ❡ C4 (4) = 47 mod 1037 = 829. ✼✵

(278) ▲♦❣♦✱ ♦❜t❡♠♦s ❛ s❡❣✉✐♥t❡ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✿ 634 − 934 − 621 − 829 P❛r❛ ❢❛③❡r ❛ ❞❡❝♦❞✐✜❝❛çã♦✱ ❞❡✈❡♠♦s ♣r✐♠❡✐r♦ ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❡ ♦ ♣❛r (n, d) d q✉❡ ❞❡t❡r♠✐♥❛ q✉❡ é ❛ ❝❤❛✈❡ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦✳ ed = l(p − 1)(q − 1) + 1 ❚❡♠♦s q✉❡✱ n = pq = 1037 = 17.61✱ ❧♦❣♦✱ p = 17 ❡ q = 61✳ ❉❛í✱ t❡♠♦s✿ 7d = l.(61 − 1).(17 − 1) + 1 7d = l.60.16 + 1 + 17 d = 960l 7 + 7l + 17 d = 959l 7 959l d = 7 + l+1 7 d = 137l + l+1 7 ❋❛③❡♥❞♦ l = 6✱ t❡♠♦s✿ d = 823. ❙❛❜❡♥❞♦ ♦ ✈❛❧♦r ❞❡ d✱ ❡ ✉s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✷✳✶✽ tó♣✐❝♦ ✺✱ t❡♠♦s✿ T1 (634) = 634823 mod 1037 T1 (634) = (6342 )411 .634 mod 1037 T1 (634) = (637)411 .634 mod 1037 T1 (634) = (6372 )205 .637.634 mod 1037 T1 (634) = (3022 )102 .302.637.634 mod 1037 T1 (634) = (9852 )51 .302.637.634 mod 1037 T1 (634) = (6302 )25 .630.302.637.634 mod 1037 T1 (634) = (7662 )12 .766.630.302.637.634 mod 1037 T1 (634) = (8512 )6 .766.630.302.637.634 mod 1037 T1 (634) = (3752 )3 .766.630.302.637.634 mod 1037 T1 (634) = 6302 .6302 .766.302.637.634 mod 1037 T1 (634) = 7662 .766.302.637.634 mod 1037 T1 (634) = 851.766.302.637.634 mod 1037 T1 (634) = 630.302.637.634 mod 1037 T1 (634) = 489.637.634 mod 1037 T1 (634) = 393.634 mod 1037 T1 (634) = 282 mod 1037 Pr♦❝❡❞❡♥❞♦ ❞❡ss❛ ♠❛♥❡✐r❛ ❝♦♠ t♦❞♦s ♦s ❜❧♦❝♦s✱ t❡r❡♠♦s✿ T2 (934) = 934823 mod 1037 = 423, T3 (621) = 621823 mod 1037 = 172 ❡ T4 (829) = 829823 mod 1037 = 4. ❉❡ss❛ ❢♦r♠❛✱ ✉t✐❧✐③❛♥❞♦ ❛ ❚❛❜❡❧❛ ✷✳✶✻✱ t❡r❡♠♦s ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ ✏❙❖◆❍❖✑✳ ✼✶

(279) ✷✳✻ ❈ó❞✐❣♦ ❞❡ ❘❛❜✐♥ ❖ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✱ ❛ss✐♠ ❝♦♠♦ ♦ s✐st❡♠❛ ❘❙❆✱ é ❜❛s❡❛❞♦ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❢❛t♦r❛r ✐♥t❡✐r♦s✱ ♠❛s ❡♠ ❝♦♥tr❛st❡ ❝♦♠ ♦ ❘❙❆✱ ♣♦❞❡ s❡r ♠♦str❛❞♦ q✉❡ ❛❧❣✉é♠ q✉❡ q✉❡❜r❡ ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✱ ♣♦❞❡ t❛♠❜é♠ ❢❛t♦r❛r ✐♥t❡✐r♦s ❞❡ ♠❛♥❡✐r❛ ❡✜❝✐❡♥t❡ ❡ ♣♦rt❛♥t♦✱ ♣♦❞❡ t❛♠❜é♠ q✉❡❜r❛r ♦ ❘❙❆✳ ◆♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✱ t❛❧ ❝♦♠♦ ♥♦ ❘❙❆✱ ♣r❡❝✐s❛♠♦s ❞❡ ❞♦✐s ♣r✐♠♦s ❣r❛♥❞❡s p ❡ q ✱ só q✉❡ ♥❡st❡ ❝ó❞✐❣♦ ❝♦st✉♠❛✲s❡ ✐♠♣♦r ❛ ❝♦♥❞✐çã♦ ❛❞✐❝✐♦♥❛❧ p, q ≡ 3 mod 4, ♣❛r❛ s✐♠♣❧✐✜❝❛r ♦s ❝á❧❝✉❧♦s✳ ◆♦t❛✲s❡ q✉❡ ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥ ❢✉♥❝✐♦♥❛ ♠❡s♠♦ q✉❡ ♦s ♣r✐♠♦s ♥ã♦ ✈❡r✐✜q✉❡♠ ❡st❛ ❝♦♥❞✐çã♦✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ é n = pq ✱ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ é ♦ ♣❛r (p, q)✳ ❖ ❡s♣❛ç♦ ❞❛s ♠❡♥s❛❣❡♥s ♦r✐❣✐♥❛✐s é {0, 1, . . . , n − 1}✳ P❛r❛ ❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠ m ∈ {0, 1, . . . , n − 1}✱ ❞❡✈❡♠♦s ❞❡t❡r♠✐♥❛r✿ c ≡ m2 mod n. P❛r❛ r❡❝✉♣❡r❛r ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ m ❞❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ c✱ ❞❡✈❡♠♦s ❞❡t❡r♠✐♥❛r✿ mp ≡ c p+1 4 mod p ❡ mq ≡ c q+1 4 mod p ❆ss✐♠✱ ±mp sã♦ ❛s ❞✉❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ c mod p ❡ ±mq sã♦ ❛s ❞✉❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ c mod q ✳ ❉❛í✱ t❡♠♦s ♦s s❡❣✉✐♥t❡s ♣❛r❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛✳ ✶✳ m ≡ mp mod p ❡ m ≡ mq mod q ✷✳ m ≡ mp mod p ❡ m ≡ −mq mod q ✸✳ m ≡ −mp mod p ❡ m ≡ mq mod q ✹✳ m ≡ −mp mod p ❡ m ≡ −mq mod q ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✶✳✼✺ ❡♠ ❝❛❞❛ ✉♠ ❞♦s ♣❛r❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ♦❜té♠✲s❡ q✉❛tr♦ ✐♥t❡✐r♦s x1 , x2 , x3 ❡ x4 ❝✉❥♦ q✉❛❞r❛❞♦ é ❝♦♥❣r✉❡♥t❡ ❛ c mod n ❡ ✉♠ ❞❡❧❡s é ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ m✳ ❍á ✈ár✐♦s ♠ét♦❞♦s ♣❛r❛ ❡s❝♦❧❤❡r ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ ❞❛s q✉❛tr♦ r❛í③❡s q✉❛❞r❛❞❛s ❞❡ c mod n✳ P♦r ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛q✉❡❧❛ q✉❡ ❢❛③ s❡♥t✐❞♦ ❛♣ós t❡r s✐❞♦ ❞❡❝♦❞✐✜❝❛❞❛✳ ◆♦ ❡♥t❛♥t♦✱ ❡st❡ ♠ét♦❞♦ ♥❡♠ s❡♠♣r❡ ❢✉♥❝✐♦♥❛✳ ❊①❡♠♣❧♦ ✷✳✷✸✳ ❖❜s❡r✈❡ ❛ s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ ❇♦❜ ❝♦❞✐✜❝❛ ❛ ♠❡♥s❛❣❡♠ m = 158 ❝❛❧❝✉❧❛♥❞♦ c ≡ m2 mod n ❡ ♦❜té♠ c = 170✳ ❆❧✐❝❡ ❛♦ r❡❝❡❜❡r ❛ ♠❡♥s❛❣❡♠✱ ♣❛r❛ ❞❡❝♦❞✐✜❝á✲❧❛✱ ❡s❝♦❧❤❡ ❞♦✐s ♥ú♠❡r♦s ♣r✐♠♦s p = 11 ❡ q = 23 ❡ ❝❛❧❝✉❧❛ mp ❡ mq ✳ ❱❡❥❛✿ mp mp mp mp mp mp ≡ c 4 mod p ❡ mq ≡ c 4 mod q 11+1 23+1 ≡ 170 4 mod 11 ❡ mq ≡ 170 4 mod 23 ≡ 1703 mod 11 ❡ mq ≡ 1706 mod 23 ≡ 53 mod 11 ❡ mq ≡ 96 mod 23 ≡ 125 mod 11 ❡ mq ≡ 531441 mod 23 ≡ 4 mod 11 ❡ mq ≡ 3 mod 23 p+1 q+1 ✼✷

(280) ▲♦❣♦✱ mp = 4 ❡ mq = 3✳ ❆ss✐♠ ±4 sã♦ ❛s ❞✉❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ 170 mod 11 ❡ ±3 sã♦ ❛s ❞✉❛s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ 170 mod 23✳ ❉❛í✱ t❡♠♦s ♦s s❡❣✉✐♥t❡s ♣❛r❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛✿ ✶✳ m ≡ 4 mod 11 ✷✳ m ≡ 4 mod 11 ❡ m ≡ 3 mod 23 ❡ m ≡ −3 mod 23 ⇒ m ≡ 20 mod 23 ✸✳ m ≡ −4 mod 11 ⇒ m ≡ 7mod 11 ❡ ✹✳ m ≡ −4 mod 11 ⇒ m ≡ 7mod 11 m ≡ 3 mod 23 ❡ m ≡ −3 mod 23 ⇒ m ≡ 20 mod 23 ❯s❛♥❞♦ ♦ ❚❡♦r❡♠❛ ✶✳✼✺✱ ❡♠ ❝❛❞❛ ✉♠ ❞♦s ♣❛r❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛✱ ❆❧✐❝❡ ♦❜té♠ q✉❛tr♦ ✐♥t❡✐r♦s q✉❡ sã♦ 26✱ 95✱ 158 ❡ 227 ❝✉❥♦ q✉❛❞r❛❞♦ é ❝♦♥❣r✉❡♥t❡ ❝♦♠ c mod n ❡ ✉♠ ❞❡❧❡s é ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ m✱ q✉❡ ♥❡st❡ ❝❛s♦✱ m = 158✳ ◗✉❡❜r❛r ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥ é tã♦ ❞✐❢í❝✐❧ ❝♦♠♦ ❢❛t♦r❛r ✐♥t❡✐r♦s✳ P♦r ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ ❛❧❣✉é♠ ❞❡s❝♦❜r✐r ✉♠ ❛❧❣♦r✐t♠♦ q✉❡ q✉❡❜r❡ ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✱ ❡❧❡ ♣♦❞❡ ✉s❛r ❡st❡ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❢❛t♦r❛r ✐♥t❡✐r♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❡✜❝✐❡♥t❡✳ ❚❡♦r❡♠❛ ✷✳✷✹✳ ❉❡♠♦♥str❛çã♦✳ ❈❧❛r❛♠❡♥t❡✱ q✉❛❧q✉❡r ♣❡ss♦❛ q✉❡ ❝♦♥s✐❣❛ ❢❛t♦r❛r n✱ ❝♦♥s❡❣✉❡ t❛♠❜é♠ q✉❡❜r❛r ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✳ ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ ✉♠❛ ♣❡ss♦❛ ❞❡s❝♦❜r✐✉ R✱ ♣❛r❛ q✉❡❜r❛r ♦ ❝ó❞✐❣♦ ❞❡ ❘❛❜✐♥✳ ❙❡❥❛ n✱ ♦ ♠ó❞✉❧♦ ♣ú❜❧✐❝♦✱ ❡ p ❡ q ✱ ♦s ❢❛t♦r❡s ♣r✐♠♦s✳ ❉❛❞❛ ✉♠❛ ♠❡♥s❛❣❡♠ ❝✐❢r❛❞❛ c mod n✱ ❛ ♣❡ss♦❛ ♦❜té♠ m = R(c)✳ P♦rt❛♥t♦✱ ❞❛❞♦ ✉♠ q✉❛❞r❛❞♦ c mod n✱ ♦ ❛❧❣♦r✐t♠♦ R✱ ♣❡r♠✐t❡ ❞❡t❡r♠✐♥❛r♠♦s ✉♠❛ r❛✐③ q✉❛❞r❛❞❛ ❞❡ c mod n✳ ❱❡❥❛♠♦s ❝♦♠♦ ♣♦❞❡♠♦s ✉s❛r ❡st❡ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❢❛t♦r❛r n✿ ✉♠❛ ♣❡ss♦❛ ❡s❝♦❧❤❡✱ ❛❧❡❛t♦r✐❛♠❡♥t❡✱ ✉♠ ✐♥t❡✐r♦ 1 ≤ x ≤ n − 1✳ ❙❡ (n, x) = d 6= 1 ✉♠ ❛❧❣♦r✐t♠♦✱ s❡❥❛♠ ❡♥tã♦ d é ✉♠ ❢❛t♦r ❞❡ n ❡ ❛ ❢❛t♦r❛çã♦ ❞❡ n ❡stá ❡♥❝♦♥tr❛❞❛ (n = d × n ). d ❈❛s♦ ❝♦♥trár✐♦✱ ❛ ♣❡ss♦❛ ❞❡t❡r♠✐♥❛ c = x2 mod n ❙❛❜❡♠♦s q✉❡ m m = R(c)✳ ❡ é ✉♠❛ ❞❛s r❛í③❡s q✉❛❞r❛❞❛s✱ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✐❣✉❛❧ ❛ x✳ ◆♦ ❡♥t❛♥t♦✱ m mod n✱ ❞❡ c✱ t❛❧ ❝♦♠♦ x✱ ♠❛s ♥ã♦ s❛t✐s❢❛③ ✉♠ ❞♦s s❡❣✉✐♥t❡s ♣❛r❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s✿ m ≡ x mod p ❡ m ≡ x mod q m ≡ x mod p ❡ m ≡ −x mod q m ≡ −x mod p ❡ m ≡ x mod q m ≡ −x mod p ❡ m ≡ −x mod q m = x ❡ (m − x, n) = n✱ ♥♦ s❡❣✉♥❞♦ ❝❛s♦✱ (m − x, n) = p✱ ♥♦ (m−x, n) = q ❡ ♥♦ ú❧t✐♠♦ ❝❛s♦✱ m = n−x ❡✱ ❝♦♠♦ (n, x) = 1✱ ♦❜t❡♠♦s (m − x, n) = 1✳ ❉❡♣♦✐s ❞❡ ❛♣❧✐❝❛r♠♦s ❡st❡ ♣r♦❝❡❞✐♠❡♥t♦ k ✈❡③❡s✱ n é ❢❛t♦r✐③❛❞♦ ❝♦♠ ◆♦ ♣r✐♠❡✐r♦ ❝❛s♦✱ t❡r❝❡✐r♦ ❝❛s♦✱ ♣r♦❜❛❜✐❧✐❞❛❞❡ 1 1 − ( )k 2 ✳ ✼✸

(281) n = 253✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❖❧❣❛ ❝♦♥s❡❣✉❡ ❞❡t❡r♠✐♥❛r r❛í③❡s mod 253 ❝♦♠ ♦ s❡✉ ❛❧❣♦r✐t♠♦ R✳ ❊❧❛ s❡❧❡❝✐♦♥❛✱ x = 17 ❡ ♦❜té♠ (17, 253) = 1✳ ❉❡♣♦✐s ❝❛❧❝✉❧❛ ❊①❡♠♣❧♦ ✷✳✷✺✳ ❙❡❥❛ q✉❛❞r❛❞❛s c ≡ 172 ≡ 36 mod 253. ❆s r❛í③❡s q✉❛❞r❛❞❛s ❞❡ 36 mod253 sã♦✱ (6 − 17, 253) = 11 ♣♦rt❛♥t♦✱ s❡ ♦ ❛❧❣♦r✐t♠♦ 253✱ R ♦❜t❡✈❡ 6 6✱ 17✱ 236 ❡ ♦✉ ❡ 247✳ ❚❡♠♦s (247 − 17, 253) = 23✱ 247 ❡♥tã♦ ❖❧❣❛ ❡♥❝♦♥tr♦✉ ❛ ❢❛t♦r❛çã♦ ❞❡ ❝❛s♦ ❝♦♥trár✐♦✱ ❖❧❣❛ ❡s❝♦❧❤❡ ♦✉tr♦ ✐♥t❡✐r♦ x ❡ ❛♣❧✐❝❛ ♦ ♣r♦❝❡❞✐♠❡♥t♦ ♦✉tr❛ ✈❡③✳ ❉❡♣♦✐s ❞❡ ♣♦✉❝❛s ❛♣❧✐❝❛çõ❡s é ♠✉✐t♦ ♣r♦✈á✈❡❧ q✉❡ ❖❧❣❛ t❡♥❤❛ ❡♥❝♦♥tr❛❞♦ ❛ ❢❛t♦r❛çã♦ ❞❡ n s❡♠ ❞❡♠♦r❛r ❞❡♠❛s✐❛❞♦ t❡♠♣♦✳ ✷✳✼ ❖ ▼ét♦❞♦ ▼❍ ✭▼❡r❦❧❡ ❡ ❍❡❧❧♠❛♥✮ ❊st❡ ♠ét♦❞♦ ❢♦✐ ❝r✐❛❞♦ ♣♦r ▼❡r❦❧❡ ❡ ❍❡❧❧♠❛♥ ❡♠ 1978 ❜❛s❡❛♥❞♦✲s❡ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞♦ ❝❤❛♠❛❞♦ Pr♦❜❧❡♠❛ ❞❛ ▼♦❝❤✐❧❛✳ ➱ ✉♠ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ♠♦♥♦❛❧❢❛❜ét✐❝♦ ❡ ❛ss✐♠étr✐❝♦ ♣♦✐s ♦ ❛❧❣♦r✐t♠♦ ❞❡ ❝♦❞✐✜❝❛çã♦ é ❞✐❢❡r❡♥t❡ ❞♦ ❛❧❣♦r✐t♠♦ ❞❡ ❞❡❝♦❞✐✜❝❛çã♦✳ ✷✳✼✳✶ ❖ Pr♦❜❧❡♠❛ ❞❛ ▼♦❝❤✐❧❛ ❉❛❞♦ ♦ ✈❡t♦r a = (a1 , a2 , . . . , an ) ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥❛t✉r❛✐s ❡ b t❛♠❜é♠ ♥❛t✉r❛❧✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ❝♦♥s✐st❡ ❡♠ s❛❜❡r s❡ ❡①✐st❡ X = (x1 , x2 , . . . , xn ) ♦♥❞❡ ❝❛❞❛ xi é 0 ♦✉ 1✱ t❛❧ q✉❡✿ n X ai xi = b. i=1 ❉❡✜♥✐♠♦s ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❞❡ ❝❛❞❛ ❞❡st✐♥❛tár✐♦ ♥♦ ▼ét♦❞♦ ▼❍ ♣❡❧♦ ✈❡t♦r P = (c1 , c2 , . . . , cn ) ❞❡ ♥❛t✉r❛✐s✱ ♦♥❞❡ n ≈ 100✳ P❛r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ s❡❥❛ ❞❡ ❢á❝✐❧ r❡s♦❧✉çã♦✱ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♥ã♦ ♣♦❞❡ s❡r q✉❛❧q✉❡r✳ P❛r❛ ✐ss♦✱ ♦ ❞❡st✐♥❛tár✐♦ ❞❡✈❡ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❛♥t❡s ❞❡ ❞✐✈✉❧❣❛r ❛ s✉❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❝r✐❛r ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s s = (s1 , s2 , . . . , sn ) ❡ t❛♠❜é♠ t ❡ k t❛✐s q✉❡ r X ✭✷✳✶✵✮ si < sr+1 < t i=1 ♣❛r❛ 1 ≤ r < n − 1 ❡ mdc(k, t) = 1✳ ❆ss✐♠✱ ❛ s❡q✉ê♥❝✐❛ s = (s1 , s2 , . . . , sn ) é ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛✳ ❖ ❞❡st✐♥❛tár✐♦ ♠❛♥té♠ ♦ ✈❡t♦r s ❡ ♦s ✈❛❧♦r❡s ❞❡ t ❡ k s❡❝r❡t♦s ❡ ♣✉❜❧✐❝❛ ♦ ✈❡t♦r c✱ ❞❛❞♦ ♣♦r ci = ksi (mod t), ❝♦♠ 1 ≤ i ≤ n✳ ❆❧é♠ ❞✐ss♦✱ ♦ ❞❡st✐♥❛tár✐♦ ❡s❝♦❧❤❡ ❡ ♠❛♥té♠ s❡❝r❡t♦ ♦ ♥ú♠❡r♦ l q✉❡ ❞❡✈❡ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦✿ lk (mod t) = 1. ✼✹

(282) ✷✳✼✳✷ ❈♦❞✐✜❝❛♥❞♦ P❛r❛ ❝♦❞✐✜❝❛r ✉♠❛ ♠❡♥s❛❣❡♠ ❡ ❡♥✈✐❛r ❛♦ ❞❡st✐♥❛tár✐♦✱ ♦ ❡♠✐ss♦r ❞❡✈❡ ❝♦♥s✉❧t❛r ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ P = (c1 , c2 , . . . , cn ) ❞♦ ❞❡st✐♥❛tár✐♦✱ ❝♦♥✈❡rt❡r ❝❛❞❛ sí♠❜♦❧♦ ❞❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ ❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s m ♠❡♥♦r❡s ❞♦ q✉❡ 2n ❡ ❡s❝r❡✈ê✲❧♦ ♥❛ ❜❛s❡ ❜✐♥ár✐❛✱ ✐st♦ é✱ m = [m1 m2 . . . mn ]2 , s❡♥❞♦ mi = 0 ♦✉ 1✳ ❊♥tã♦✱ ❝❛❧❝✉❧❛✲s❡ P (m) = n X mi ci . i=1 ❆ss✐♠✱ ♦ tr❛❜❛❧❤♦ ❞♦ ❞❡st✐♥❛tár✐♦ ❡♠ ❞❡❝♦❞✐✜❝❛r P (m) é ❞❡t❡r♠✐♥❛r ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ s❛❜❡♥❞♦✲s❡ P = (c1 , c2 , . . . , cn ) ❡ P (m)✳ ✷✳✼✳✸ ❆❧❣♦r✐t♠♦ ♣❛r❛ ❛ ❘❡s♦❧✉çã♦ ❞♦ Pr♦❜❧❡♠❛ ❞❛ ▼♦❝❤✐❧❛ ✲ ❉❡❝♦❞✐✜❝❛♥❞♦ ❆❧❣♦r✐t♠♦ ❞❛ ▼♦❝❤✐❧❛ ❆ ❞❡❝♦❞✐✜❝❛çã♦ ❞❛ ♠❡♥s❛❣❡♠ s❡ ❞á ❧❡tr❡ ♣♦r ❧❡tr❛ ✉t✐❧✐③❛♥❞♦ ♣❛r❛ ✐ss♦ ♦ ❆❧❣❛r✐t♠♦ ❞❛ ▼♦❝❤✐❧❛✳ ❖ ❞❡st✐♥❛tár✐♦ ❞❡✈❡ ♣r✐♠❡✐r♦ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ d = l.P (m)(mod t). ❊♠ s❡❣✉✐❞❛✱ ❢❛ç❛ ♦s s❡❣✉✐♥t❡s ♣r♦❝❡❞✐♠❡♥t♦s✿ ❊♥tr❛❞❛✿ (n, (s1 , s2 , . . . , sn ), d)✱ ♦♥❞❡ s = (s1 , s2 , . . . , sn ) é ❛ s❡qüê♥❝✐❛ ✷✳✶✵ ❡ d ≡ l · P (m) (mod t). ❙❛í❞❛✿ m✳ ❊t❛♣❛ 1✿ ❋❛ç❛ y = d✳ ❊t❛♣❛ 2✿ P❛r❛ ❝❛❞❛ i = n, n − 1, n − 2, . . . , 1✱ ♦✉ s❡❥❛✱ ♣❛r❛ ♦s ✈❛❧♦r❡s ❞❡ i s❡rã♦ ❛tr✐❜✉í❞♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡❝r❡s❝❡♥t❡ ❞❡ n ❛té 1✱ ❢❛ç❛✿ ✶✳ ❙❡ y < si ✱ ❡♥tã♦✱ mi = 0✳ ✷✳ ❙❡ y ≥ si ✱ ❡♥tã♦ ❢❛ç❛ y = y − si ❡ t♦♠❡ mi = 1✳ ❊t❛♣❛ 3✿ ✶✳ ❙❡ y = 0✱ ❡♥tã♦ r❡t♦r♥❡ ♦ ✈❡t♦r✿ m = (m1 , m2 , . . . , mn )✳ ✷✳ ❙❡ y 6= 0✱ ❡♥tã♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠♦❝❤✐❧❛ ♥ã♦ t❡♠ s♦❧✉çã♦✳ ❙❡❥❛ ❛ ♠❡♥s❛❣❡♠ BRASIL 2014✳ ❆ss♦❝✐❛♥❞♦ ❛ ♠❡♥s❛❣❡♠ ❛♦s ♥ú♠❡r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❛ ❚❛❜❡❧❛ ✷✳✶✼ ❡ ✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ▼❍✱ ❢❛ç❛ ❛ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦✳ ❊①❡♠♣❧♦ ✷✳✷✻✳ ✼✺

(283) ❆ ❇ ❈ ❉ ❊ ❋ ● ❍ ■ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ◆ ❖ P ◗ ❘ ❙ ❚ ❯ ❱ ✷✸ ✷✹ ✷✺ ✷✻ ✷✼ ✷✽ ✷✾ ✸✵ ✸✶ ✵ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✸✻ ✸✼ ✸✽ ✸✾ ✹✵ ✹✶ ✹✷ ✹✸ ✹✹ ❏ ❑ ▲ ▼ ✶✾ ✷✵ ✷✶ ✷✷ ❲ ❳ ❨ ❩ ✸✷ ✸✸ ✸✹ ✸✺ ✽ ✾ ✹✺ ✹✻ ❚❛❜❡❧❛ ✷✳✶✼✿ ❙♦❧✉çã♦✿ Pr✐♠❡✐r❛♠❡♥t❡✱ é ♣r❡❝✐s♦ q✉❡ ♦ ❞❡st✐♥❛tár✐♦ ❞❡t❡r♠✐♥❡ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ q✉❡ s❡rá ♦ ✈❡t♦r P = (c1 , c2 , . . . , cn ✳ P❛r❛ ✐ss♦✱P❡❧❡ ❞❡✈❡rá ❡s❝♦❧❤❡r ✉♠❛ s❡q✉ê♥❝✐❛ s ❝♦♠♦ ✷✳✶✵✳ ❆❧é♠ ❞✐ss♦✱ k ❡ t✱ ❞❡ ♠♦❞♦ q✉❡ ni=1 si < t ❡ mdc(k, t) = 1✳ P❛r❛ ♦ ❡①❡♠♣❧♦ ❡s❝♦❧❤❡♠♦s ❛ s❡q✉ê♥❝✐❛✿ s = (3, 5, 11, 22, 50, 107), k = 27 ❡ t = 199✱ ♣♦✐s ♦ mdc(27, 199) = 1 ❡ t > 3 + 5 + 11 + 22 + 50 + 107 = 198✳ ❚❡♠♦s ❡♥tã♦ ❛ ❡①♣r❡ssã♦✿ 27l(mod 199) = 1 =⇒ 199x + 27l = 1. ❈❛❧❝✉❧❡♠♦s ♦ ✈❛❧♦r ❞❡ l ❛ ♣❛rt✐r ❞♦ ❆❧❣♦r✐t♠♦ ❊✉❝❧✐❞❡s ❊st❡♥❞✐❞♦ ✷✳✶✹✳ ❈♦❧♦❝❛♥❞♦ ♦s ✈❛❧♦r❡s ❡♠ ✉♠❛ t❛❜❡❧❛✿ ❘❡st♦s ◗✉♦❝✐❡♥t❡s xi yi ✶✾✾ ✶ ✵ ✷✼ ✵ ✶ ✶✵ ✼ ✶ ✲✼ ✼ ✷ ✲✷ ✶✺ ✸ ✶ ✲✷✹ ✶✼✼ ✶ ✷ ✲✽ ✺✾ ❚❛❜❡❧❛ ✷✳✶✽✿ ❚❡♠♦s✱ ❡♥tã♦✿ l = yi = 59. ❉❡st❡ ♠♦❞♦✱ ♦ ❞❡st✐♥❛tár✐♦ ♣ú❜❧✐❝❛ ♦ ✈❡t♦r c = (c1 , c2 , . . . , cn )✱ ♦♥❞❡ n = 6 ❡ ❝✉❥♦ ci = ksi (mod t)✳ c1 c2 c3 c4 c5 c6 = 27.s1 (mod = 27.s2 (mod = 27.s3 (mod = 27.s4 (mod = 27.s5 (mod = 27.s6 (mod 199) =⇒ c1 199) =⇒ c2 199) =⇒ c3 199) =⇒ c4 199) =⇒ c5 199) =⇒ c6 = 27.3(mod 199) =⇒ c1 = 81(mod 199) = 27.5(mod 199) =⇒ c2 = 135(mod 199) = 27.11(mod 199) =⇒ c3 = 98(mod 199) = 27.22(mod 199) =⇒ c4 = 196(mod 199) = 27.50(mod 199) =⇒ c5 = 156(mod 199) = 27.107(mod 199) =⇒ c6 = 103(mod 199) ❆ss✐♠✱ t❡♠♦s q✉❡ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ é P = (81, 135, 98, 196, 156, 103)✳ ❈♦❞✐✜❝❛♥❞♦ ❆❣♦r❛ q✉❡ ❝♦♥❤❡❝❡♠♦s ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ♣♦❞❡♠♦s ❝♦❞✐✜❝❛r ❛ ♠❡♥s❛❣❡♠✳ P❛r❛ ✐ss♦✱ ✼✻

(284) ❛ss♦❝✐❛♠♦s ❛ ♠❡♥s❛❣❡♠ ❛♦s ♥ú♠❡r♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ♥❛ ❚❛❜❡❧❛ ✷✳✶✼✱ ❞❛í t❡♠♦s ❛ s❡❣✉✐♥t❡ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s✿ 11, 27, 10, 28, 18, 21, 36, 39, 37, 38, 41 P❛ss❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ❛❝✐♠❛ ♣❛r❛ ❛ ❜❛s❡ ❜✐♥ár✐❛ ✷✳✶✹✱t❡♠♦s✿ 11 = [001011]2 , 27 = [011011]2 , 10 = [001010]2 , 28 = [011100]2 18 = [010010]2 , 21 = [010101]2 , 36 = [100100]2 , 39 = [100111]2 37 = [100101]2 , 38 = [100110]2 , 41 = [101001]2 ▲♦❣♦✱ ❛ ♣r✐♠❡✐r❛ ❧❡tr❛ ❞❛ ♠❡♥s❛❣❡♠✱ q✉❡ é B ✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ 11 = [001011]2 é ❝♦❞✐✜❝❛❞❛ ❡♠✿ P (11) = 6 X mi ci = 0.81 + 0.135 + 1.98 + 0.196 + 1.156 + 1.103 = 357 i=1 Pr♦❝❡❞❡♥❞♦ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ ❝♦♠ ♦s ❞❡♠❛✐s sí♠❜♦❧♦s ❞❛ ♠❡♥s❛❣❡♠✱ t❡♠♦s✿ t❡♠♦s ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s✿ 11 27 10 28 18 21 36 39 37 38 41. P❛ss❛♥❞♦ ❛ s❡q✉ê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ❛❝✐♠❛ ♣❛r❛ ❛ ❜❛s❡ ❜✐♥ár✐❛✱ t❡♠♦s✿ 357 492 254 429 291 434 277 536 380 433 282 ❉❡❝♦❞✐✜❝❛♥❞♦ P❛r❛ ❞❡❝✐❢r❛r ❛ ♠❡♥s❛❣❡♠ ♦ ❞❡st✐♥❛tár✐♦ ❞❡✈❡ ♣r✐♠❡✐r♦ ❞❡t❡r♠✐♥❛r ♦s ✈❛❧♦r❡s ❞❡ d = l.P (m)(mod t). ▲♦❣♦✱ t❡♠♦s✿ P❛r❛ P (11) P❛r❛ P (27) P❛r❛ P (10) P❛r❛ P (28) P❛r❛ P (18) P❛r❛ P (21) P❛r❛ P (36) P❛r❛ P (39) P❛r❛ P (37) P❛r❛ P (38) P❛r❛ P (41) ❡♥tã♦ d = 168 ❡♥tã♦ d = 173 ❡♥tã♦ d = 61 ❡♥tã♦ d = 38 ❡♥tã♦ d = 55 ❡♥tã♦ d = 134 ❡♥tã♦ d = 25 ❡♥tã♦ d = 182 ❡♥tã♦ d = 132 ❡♥tã♦ d = 75 ❡♥tã♦ d = 121 ❈♦♥t✐♥✉❛♥❞♦ ❛ ❞❡❝♦❞✐✜❝❛çã♦ ❞♦ ▼ét♦❞♦ M H ✱ ✈❛♠♦s ❝♦♠❡ç❛r ❞❡❝♦❞✐✜❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❧❡tr❛ ❞❛ ♥♦ss❛ ♠❡♥s❛❣❡♠ ✉t✐❧✐③❛♥❞♦ ♣❛r❛ ✐ss♦ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ▼♦❝❤✐❧❛✳ ❚❡♠♦s✿ (n, (s1 , s2 , . . . , sn ), d)✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ (6, (3, 5, 11, 22, 50, 107), 168)✳ ❊t❛♣❛ 1✿ ❋❛ç❛ y = 168✳ ✼✼

(285) ❊t❛♣❛ 2 P❛r❛ i = 6 ❈♦♠♦ y ≥ s6 ✱ ♦✉ s❡❥❛✱ y ≥ 107 ❡♥tã♦ ❢❛ç❛ y = 168 − 107 = 61 ❡ t♦♠❡ m6 = 1✳ P❛r❛ i = 5 ❈♦♠♦ y ≥ s5 ✱ ♦✉ s❡❥❛✱ y ≥ 50 ❡♥tã♦ ❢❛ç❛ y = 61 − 50 = 11 ❡ t♦♠❡ m5 = 1✳ P❛r❛ i = 4 ❈♦♠♦ y < s4 ✱ ♦✉ s❡❥❛✱ y < 22 ❡♥tã♦ t♦♠❡ m4 = 0✳ P❛r❛ i = 3 ❈♦♠♦ y ≥ s3 ✱ ♦✉ s❡❥❛✱ y ≥ 11 ❡♥tã♦ ❢❛ç❛ y = 11 − 11 = 0 ❡ t♦♠❡ m3 = 1✳ P❛r❛ i = 2 ❈♦♠♦ y < s2 ✱ ♦✉ s❡❥❛✱ y < 5 ❡♥tã♦ t♦♠❡ m2 = 0✳ P❛r❛ i = 1 ❈♦♠♦ y < s1 ✱ ♦✉ s❡❥❛✱ y < 3 ❡♥tã♦ t♦♠❡ m1 = 0✳ ❊t❛♣❛ 3✿ ❈♦♠♦ y = 0✱ ❡♥tã♦ m = [001011]2 = 11 q✉❡ ❝♦rr❡s♣♦♥❞❡ à ❧❡tr❛ B ✳ ❉❡ ♠♦❞♦ ❛♥á❧♦❣♦✱ ✉t✐❧✐③❛♥❞♦ ♦ ❆❧❣♦r✐t♠♦ ❞❛ ▼♦❝❤✐❧❛ ♣❛r❛ ♦s ❞❡♠❛✐s sí♠❜♦❧♦s ❞❛ ♠❡♥s❛❣❡♠✱ ❡♥❝♦♥tr❛♠♦s ♦s r❡s♣❡❝t✐✈♦s r❡s✉❧t❛❞♦s✿ [001011]2 , [011011]2 , [001010]2 , [011100]2 [010010]2 , [010101]2 , [100100]2 , [100111]2 [100101]2 , [100110]2 , [101001]2 q✉❡ ❝♦rr❡s♣♦♥❞❡♠ ❛ m = 11, m = 27, m = 10, m = 28, m = 18, m = 21, m = 36, m = 39, m = 37, m = 38, m = 41. ❋♦r♠❛♥❞♦ ❛ ♠❡♥s❛❣❡♠ ✐♥✐❝✐❛❧ BRASIL 2014✳ ✷✳✽ ❈ó❞✐❣♦ ❊❧●❛♠❛❧ ❈ó❞✐❣♦ ❊❧●❛♠❛❧ é ✉♠ s✐st❡♠❛ ❝♦♠ ♦ ✉s♦ ❞❡ ❝❤❛✈❡s ❛ss✐♠étr✐❝❛s ✹ ❝r✐❛❞♦ ♣❡❧♦ ❡st✉❞✐♦s♦ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡❣í♣❝✐♦ ❚❛❤❡r ❊❧❣❛♠❛❧ ❡♠ 1984✳ ❙✉❛ s❡❣✉r❛♥ç❛ s❡ ❜❛s❡✐❛ ♥❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ s♦❧✉çã♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦ ♣♦❞❡ ❛♣r❡s❡♥t❛r✳ ◆❛ ❣❡r❛çã♦ ❞❛s ❝❤❛✈❡s ❞❛ ❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧✱ t❡♠♦s q✉❡✿ ✶✳ ❊s❝♦❧❤❡r ✉♠ ♥ú♠❡r♦ ♣r✐♠♦ ❣r❛♥❞❡ p ❡ ✉♠ ❣❡r❛❞♦r α ❞♦ ❣r✉♣♦ ♠✉❧t✐♣❧✐❝❛t✐✈♦ Z∗p ✳ ✷✳ ❙❡❧❡❝✐♦♥❛r ❛♦ ❛❝❛s♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ a < p − 1 ❡ ❝❛❧❝✉❧❛r f = αa (mod p)✳ ✸✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ é (p, α, f ) ❡ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ é a✳ ✹ ❈❤❛✈❡s ❛ss✐♠étr✐❝❛s é ✉♠ ♣❛r ❞❡ ❝❤❛✈❡s ❢♦r♠❛❞♦ ♣♦r ✉♠❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ❡ ✉♠❛ ❝❤❛✈❡ ♣r✐✈❛❞❛✳ ❆ ❝❤❛✈❡ ♣ú❜❧✐❝❛ é ❞✐str✐❜✉í❞❛ ❧✐✈r❡♠❡♥t❡ ♣❛r❛ t♦❞♦s ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s✱ ❡♥q✉❛♥t♦ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ❞❡✈❡ s❡r ❝♦♥❤❡❝✐❞❛ ❛♣❡♥❛s ♣❡❧♦ s❡✉ ❞♦♥♦✳ ◆❡ss❡ t✐♣♦ ❞❡ s✐st❡♠❛ q✉❡ ✉t✐❧✐③❛ ❡ss❡ ♣❛r ❞❡ ❝❤❛✈❡s ❛ss✐♠étr✐❝❛✱ ✉♠❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ ❝♦♠ ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ ♣♦❞❡ s♦♠❡♥t❡ s❡r ❞❡❝♦❞✐✜❝❛❞❛ ♣❡❧❛ s✉❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ✼✽

(286) ✷✳✽✳✶ ❊t❛♣❛ ❞❡ ❈♦❞✐✜❝❛çã♦ ◆❡st❛ ❡t❛♣❛ ♦ ❡♠✐ss♦r A ❞❡✈❡rá✿ ✶✳ ❖❜t❡r ❛ ❝❤❛✈❡ ♣ú❜❧✐❝❛ (p, α, f ) ❞❡ B ✳ ✷✳ ❈♦♥✈❡rt❡r ❛s ❧❡tr❛s✱ ♥ú♠❡r♦s ❡ sí♠❜♦❧♦s ❞❛ ♠❡♥s❛❣❡♠ ❡♠ ♥ú♠❡r♦s (m) ❡♥tr❡ 0 ❡ p − 1✳ ✸✳ ❊s❝♦❧❤❡r ❛♦ ❛❝❛s♦ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ b✱ t❛❧ q✉❡ b < p − 1✳ ✹✳ P❛r❛ ❝❛❞❛ m ♦❜t✐❞♦ ❛❝✐♠❛✱ ❝❛❧❝✉❧❛r✿ β ≡ αb (mod p) ❡ γ ≡ m(αa )b (mod p)✳ ✺✳ ❊♥✈✐❛r ❛ ❝♦❞✐✜❝❛çã♦ c = (β, γ) ❞❡ m ♣❛r❛ B ✳ ✷✳✽✳✷ ❊t❛♣❛ ❞❡ ❉❡❝♦❞✐✜❝❛çã♦ ❯♠❛ ✈❡③ q✉❡ ♦ r❡❝❡♣t♦r B r❡❝❡❜❡ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛ c✱ ❡♥tã♦ ❞❡✈❡rá✿ ✶✳ ❯s❛r ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ ♣❛r❛ ❝❛❧❝✉❧❛r β p−1−a (mod p)✳ ✷✳ ❉❡❝♦❞✐✜❝❛r m ❝❛❧❝✉❧❛♥❞♦ β −a γ(mod p)✳ ✸✳ ❚❡♠♦s β −a γ ≡ α−ab mαab ≡ m(mod p) ❞❡✈✐❞♦ ❛♦ ❚❡♦r❡♠❛ ❞❡ ❋❡r♠❛t ✶✳✺✼✳ ❙❡❥❛ ❛ ❢r❛s❡ P ROF M AT 2014✳ ❚♦♠❡ p = 1999 ❡ ✉♠ ❣❡r❛❞♦r α = 7 ❞❡ ✳ ❖ ❞❡st✐♥❛tár✐♦ B ❡s❝♦❧❤❡ ❛ ❝❤❛✈❡ ♣r✐✈❛❞❛ a = 117✳ ❯s❛♥❞♦ ❛ ❈r✐♣t♦❣r❛✜❛ ❊❧●❛♠❛❧ ❢❛ç❛ ❛ ❝♦❞✐✜❝❛çã♦ ❡ ❞❡❝♦❞✐✜❝❛çã♦ ❞❛ ❧❡tr❛ M ❞❛ ♠❡♥s❛❣❡♠✱ q✉❡ ❝♦rr❡s♣♦♥❞❡ ❛ m = 22 ♥❛ ❚❛❜❡❧❛ ✷✳✶✼✳ ❙♦❧✉çã♦✿ ❙✉♣♦♥❤❛ q✉❡ ♦ ❡♠✐ss♦r A ❡s❝♦❧❤❛ b = 503✳ P❛r❛ ❝♦❞✐✜❝❛r ♦ ❡♠✐ss♦r A✱ ❞❡✈❡ ❝❛❧❝✉❧❛r ❊①❡♠♣❧♦ ✷✳✷✼✳ Z∗p f = αa (mod p) = 7117 (mod 1999). ❯s❛♥❞♦ ♦ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ✶✳✾✱ ❡♥❝♦♥tr❛♠♦s f = 54✳ ❉❡♣♦✐s ❝❛❧❝✉❧❛♠♦s β ≡ αb (mod p) = 7503 (mod 1999). ❯s❛♥❞♦ ♦ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ✶✳✾✱ ❡♥❝♦♥tr❛♠♦s β = 300✳ ❊♠ s❡❣✉✐❞❛ ❝❛❧❝✉❧❛♠♦s γ ≡ m(αa )b (mod p) = 22(54)503 (mod 1999). ❯s❛♥❞♦ t❛♠❜é♠ ♦ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ✶✳✾✱ ❡♥❝♦♥tr❛♠♦s γ = 77✳ ▲♦❣♦✱ A ❡♥✈✐❛ (β, γ) = (300, 77) ♣❛r❛ B ✳ P❛r❛ ❞❡❝♦❞✐✜❝❛r✱ B ❞❡✈❡ ❝❛❧❝✉❧❛r✿ β p−1−a = 3001999−1−117 (mod 1999) = 3001881 (mod 1999). ✼✾

(287) ❯s❛♥❞♦ ♦ ▼ét♦❞♦ ❞♦s ◗✉❛❞r❛❞♦s ❘❡♣❡t✐❞♦s ✶✳✾✱ ❡♥❝♦♥tr❛♠♦s ❋✐♥❛❧♠❡♥t❡✱ B ❝❛❧❝✉❧❛ m✱ β p−1−a = 857✳ ❞❡ ♠♦❞♦ q✉❡✿ m = β −a γ ≡ 857 × 77 (mod 1999). ❆♦ r❡s♦❧✈❡r ❛ ❝♦♥❣r✉ê♥❝✐❛ ❛❝✐♠❛✱ ❡♥❝♦♥tr❛♠♦s M ❞❛ ♠❡♥s❛❣❡♠ ✐♥✐❝✐❛❧ ❡♥✈✐❛❞❛✳ ✽✵ m = 22✱ ♦ q✉❡ ❝♦rr❡s♣♦♥❞❡ à ❧❡tr❛

(288) ❈❛♣ít✉❧♦ ✸ ❖ ❡st✉❞♦ ❞❡ ❛❧❣✉♥s ❝ó❞✐❣♦s ❝♦♠ ê♥❢❛s❡ ♥❛ ♠❛t❡♠át✐❝❛ ♠♦❞✉❧❛r ❊st✉❞❛r❡♠♦s ❛❣♦r❛ ❝ó❞✐❣♦s ✈♦❧t❛❞♦s ❛ ❚❡♦r✐❛ ❞♦s ❈ó❞✐❣♦s ♦♥❞❡ ♦ q✉❡ ✐♠♣♦rt❛ ❛q✉✐ é ❛ s❡❣✉r❛♥ç❛ ❝♦♥tr❛ ❞❛♥✐✜❝❛çã♦ ❞❛ ✐♥❢♦r♠❛çã♦✱ ♦✉ s❡❥❛✱ é ✐♠♣♦rt❛♥t❡ ♣r♦t❡❣❡r ♦ ❝♦♥t❡ú❞♦ ❞❛ ♠❡♥s❛❣❡♠ ❝♦♥tr❛ ❞❡str✉✐çã♦ ❡♥tr❡ ♦✉tr♦s ❛s♣❡❝t♦s ♥❛t✉r❛✐s q✉❡ ♣♦❞❡♠ ❝❛✉s❛r ❡rr♦s ❞✉r❛♥t❡ ❛ tr❛♥s♠✐ssã♦✱ ❛ss✐♠ ❛ ❢♦♥t❡ ❇ r❡❝❡❜❡ ❛ ♠❡♥s❛❣❡♠ ❝♦rr❡t❛♠❡♥t❡✳ ✸✳✶ ✸✳✶✳✶ ❈ó❞✐❣♦s ❞❡ ❜❛rr❛s ❍✐stór✐❛ ❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❜❛s❡❛❞❛ ♥♦s ❛rt✐❣♦s ❬✶✷❪✱ ❬✷✸❪ ❡ ❬✶✸❪✳ ❖ ♣r✐♠❡✐r♦ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s ✏♥❛s❝❡✉✑ ❡♠ 1949✱ ❢♦r♠❛❞♦ ♣♦r q✉❛tr♦ ❧✐♥❤❛s ❜r❛♥❝❛s s♦❜r❡ ✉♠ ❢✉♥❞♦ ♣r❡t♦✱ ♣♦ré♠ ❝♦♠♦ ♥ã♦ ❤❛✈✐❛ ✉♠ s✐st❡♠❛ ❞❡ ❧❡✐t✉r❛ ❞❡ ❜❛✐①♦ ❝✉st♦✱ ❡st❛ ✐❞é✐❛ ✜❝♦✉ ♥♦s ❛rq✉✐✈♦s✱ s❡♠ s❡r ✐♠♣❧❡♠❡♥t❛❞❛ ♥♦ ♠❡r❝❛❞♦✳ ❊♠ 1952✱ ❢♦✐ ❛tr✐❜✉í❞❛ ❛ ❏♦s❡♣❤ ❲♦♦❞❧❛♥❞ ❡ ❇❡r♥❛r❞ ❙✐❧✈❡r ❛ ♣r✐♠❡✐r❛ ♣❛t❡♥t❡ ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s✳ ❙❡✉ ❝ó❞✐❣♦ ❝♦♥s✐st✐❛ ♥✉♠ ♣❛❞rã♦ ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛s ❝♦♥❝ê♥tr✐❝❛s ❞❡ ❡s♣❡ss✉r❛ ✈❛r✐á✈❡❧✳ ❊♠ t♦r♥♦ ❞❡ 1970✱ ✉♠❛ ✜r♠❛ ❞❡ ❛ss❡ss♦r✐❛✱ ❛ ▼❝❑✐♥s❡② ✫ ❈♦✳✱ ❥✉♥t♦ ❝♦♠ ❛ ❯♥✐❢♦r♠ ●r♦❝❡r② Pr♦❞✉❝t ❈♦❞❡ ❈♦✉♥❝✐❧ 1 ❞❡✜♥✐✉ ✉♠ ❢♦r♠❛t♦ ♥✉♠ér✐❝♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♣r♦❞✉t♦s ❡ ♣❡❞✐✉ ❛ ❞✐✈❡rs❛s ❝♦♠♣❛♥❤✐❛s q✉❡ ❡❧❛❜♦r❛ss❡♠ ✉♠ ❝ó❞✐❣♦ ❛❞❡q✉❛❞♦✳ ❆ ❝♦♠♣❛♥❤✐❛ ✈❡♥❝❡❞♦r❛ ❢♦✐ ❛ ■❇▼ ❡ ♦ ❝ó❞✐❣♦ ❢♦✐ ❝r✐❛❞♦ ♣♦r ●❡♦r❣❡ ❏✳ ▲❛✉r❡r✳ ❖ ❝ó❞✐❣♦ ♣r♦♣♦st♦✱ ❢♦r♠❛❧♠❡♥t❡ ❛❝❡✐t♦ ❡♠ ♠❛✐♦ ❞❡ 1973✱ ♣❛ss♦✉ ❛ s❡r ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝ó❞✐❣♦ ❯P❈ ✭❯♥✐✈❡rs❛❧ Pr♦❞✉❝t ❈♦❞❡✮ ❡ ❢♦✐ ❛❞♦t❛❞♦ ♥♦s ❊st❛❞♦s ❯♥✐❞♦s ❡ ❈❛♥❛❞á✳ ❊❧❡ ❝♦♥s✐st✐❛ ❞❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞❡ ❊♠ ❞❡③❡♠❜r♦ ❞❡ 1976✱ ◆✉♠❜❡r✐♥❣ s②st❡♠✮ ❝♦♠ 12 ❞í❣✐t♦s✱ tr❛❞✉③✐❞♦s ♣❛r❛ ❜❛rr❛s✳ ▲❛✉r❡r ❝r✐♦✉ ✉♠ ♥♦✈♦ ❝ó❞✐❣♦ ♦ ❊❆◆ ✭❊✉r♦♣❡❛♥ ❆rt✐❝❧❡ 13 ❞í❣✐t♦s✱ ❜❛s❡❛❞♦ ♥♦ ❯P❈−❆✳ ❊ss❡ ❝ó❞✐❣♦ ♣❡r♠✐t✐✉ ✐❞❡♥t✐✜❝❛r ♦ ♣❛ís ❞❡ ♦r✐❣❡♠ ❞❡ ❝❛❞❛ ♣r♦❞✉t♦ ❝❧❛ss✐✜❝❛❞♦✳ ❖✉tr♦s ♣❛ís❡s t❛♠❜é♠ ❛❞♦t❛♠ ❡st❡ ♠❡s♠♦ s✐st❡♠❛✱ ❡♥tr❡t❛♥t♦ ✉t✐❧✐③❛♥❞♦ ♦✉tr♦ ♥♦♠❡✳ P♦r ❡①❡♠♣❧♦✱ ♥♦ ❏❛♣ã♦ ♦ s✐st❡♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❏❆◆ ✭❏❛♣❛♥❡s❡ ❆rt✐❝❧❡ ◆✉♠❜❡r✐♥❣ s②st❡♠✮✳ EAN − 13 ◆♦ ❡♥t❛♥t♦✱ ♥❡st❡ tr❛❜❛❧❤♦ tr❛t❛r❡♠♦s ❛♣❡♥❛s ❞♦ ❝ó❞✐❣♦ q✉❡ é ♦ ♣❛❞rã♦ ✉t✐❧✐③❛❞♦ ♥♦ ❇r❛s✐❧✳ ✽✶

(289) ✸✳✶✳✷ ❖ s✐❣♥✐✜❝❛❞♦ ❞♦s 13 ❞í❣✐t♦s ❆ ♠❛✐♦r✐❛ ❞♦s ♣r♦❞✉t♦s ✈❡♥❞✐❞♦s ♥♦s s✉♣❡r♠❡r❝❛❞♦s ❡ ❧♦❥❛s sã♦ ✐❞❡♥t✐✜❝❛❞♦s ❛ ♣❛rt✐r ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛✳ ◆❡ss❡ s✐st❡♠❛✱ ❛ ❧❛r❣✉r❛ ❞❛s ❜❛rr❛s ❡ ♦s ❡s♣❛ç♦s ❡♠ ❜r❛♥❝♦ ❡♥tr❡ ❡❧❛s ❝♦❞✐✜❝❛♠ ♥ú♠❡r♦s✱ q✉❡ ♣♦r s✉❛ ✈❡③ r❡♣r❡s❡♥t❛♠ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ♣r♦❞✉t♦✳ ❖ ❝ó❞✐❣♦ s❡❣✉❡ ♣❛❞rõ❡s ✐♥t❡r♥❛❝✐♦♥❛✐s✳ ❖ ♣❛❞rã♦ EAN − 13✱ é ♦ ♣❛❞rã♦ ✉t✐❧✐③❛❞♦ ♥♦ ❇r❛s✐❧ ❡ ❡♠ ♦✉tr♦s ♣❛ís❡s✳ ❊ss❡ ♣❛❞rã♦ é ❝♦♠♣♦st♦ ♣♦r 13 ❞í❣✐t♦s ❡ ❡①✐st❡ ✉♠ s✐❣♥✐✜❝❛❞♦ ♣r❛ t♦❞♦s ❡❧❡s✳ ✶✳ ❖s 3 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ✐❞❡♥t✐✜❝❛♠ ❛ ❡♥t✐❞❛❞❡ q✉❡ ❣❡r❡♥❝✐❛ ❡ ❝♦♥tr♦❧❛ ♦s ❝ó❞✐❣♦s ✉t✐❧✐③❛❞♦s ♣♦r ❡♠♣r❡s❛s ❡ s❡✉s ♣r♦❞✉t♦s✳ ◆♦ ❇r❛s✐❧✱ ❛ ❡♥t✐❞❛❞❡ r❡s♣♦♥sá✈❡❧ é ❛ GS1 ❇r❛s✐❧✳ ❆ GS1 tr❛❜❛❧❤❛ ❝♦♠ ❞✐✈❡rs♦s ❝ó❞✐❣♦s✱ ❡①❡♠♣❧♦s✿ • ❊❆◆/❯P❈ ❈ó❞✐❣♦ ❞❡s❡♥✈♦❧✈✐❞♦ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♣❛r❛ ❧❡✐t✉r❛ ♥♦ P❉❱ ❋✐❣✉r❛ ✸✳✶✿ ✭♣♦♥t♦ ❞❡ ✈❡♥❞❛✮✱ ❞❡✈✐❞♦ à ❛❣✐❧✐❞❛❞❡ ♣r♦♣✐❝✐❛❞❛ ♥❛ ❝❛♣t✉r❛ ❞❛ ✐♥❢♦r♠❛çã♦✳ P❡r♠✐t❡ ❝♦❞✐✜❝❛r ♦s ●❚■◆✲✽✱ ●❚■◆✲✶✷ ❡ ●❚■◆✲✶✸ ✶ ✳ • GS1 ❉❛t❛❇❛r ❋✐❣✉r❛ ✸✳✷✿ ❈♦♠♣r❡❡♥❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡ ❝ó❞✐❣♦s q✉❡ ♣♦❞❡♠ s❡r ❡s❝❛♥❡❞♦s ♥♦ ♣♦♥t♦ ❞❡ ✈❡♥❞❛✱ ♣♦❞❡♠ s❡r ♠✉✐t♦ ♠❡♥♦r❡s ❞♦ q✉❡ ♦ ♦s ❝ó❞✐❣♦s ❊❆◆✴❯P❈ ❡ ♣♦❞❡♠ ❛✐♥❞❛ ❝♦❞✐✜❝❛r ✐♥❢♦r♠❛çõ❡s ❛❞✐❝✐♦♥❛✐s ❝♦♠♦ ♥ú♠❡r♦ s❡r✐❛❧✱ ♥ú♠❡r♦ ❞❡ ❧♦t❡ ❡✴♦✉ ❞❛t❛ ❞❡ ✈❛❧✐❞❛❞❡✳ • GS1 − 128 ✶ ●❚■◆ ✭●❧♦❜❛❧ ❚r❛❞❡ ■t❡♠ ◆✉♠❜❡r ✲ ◆ú♠❡r♦ ●❧♦❜❛❧ ❞❡ ■t❡♠ ❈♦♠❡r❝✐❛❧✮ é ✉♠ ✐❞❡♥t✐✜❝❛❞♦r ♣❛r❛ ✐t❡♥s ❝♦♠❡r❝✐❛✐s ❞❡s❡♥✈♦❧✈✐❞♦ ❡ ❝♦♥tr♦❧❛❞♦ ♣❡❧❛ ●❙✶✳ ❖s ●❚■◆s ♣♦❞❡♠ t❡r ♦ t❛♠❛♥❤♦ ❞❡ ✽ ✭●❚■◆✲ ✽✮✱ ✶✷ ✭●❚■◆✲✶✷✮✱ ✶✸ ✭●❚■◆✲✶✸✮ ♦✉ ✶✹ ✭●❚■◆✲✶✹✮ ❞í❣✐t♦s ❡ ♣♦❞❡♠ s❡r ❝♦♥str✉í❞♦s ✉t✐❧✐③❛♥❞♦ q✉❛❧q✉❡r ✉♠❛ ❞❛s q✉❛tr♦ ❡str✉t✉r❛s ❞❡ ♥✉♠❡r❛çã♦ ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❛♣❧✐❝❛çã♦✳ ✽✷

(290) ❋✐❣✉r❛ ✸✳✸✿ ❈ó❞✐❣♦ ❞❡ ❜❛rr❛s q✉❡ ♣❡r♠✐t❡ ❝♦❞✐✜❝❛r t♦❞❛s ❛s ❈❤❛✈❡s GS1✳ ❯t✐❧✐③❛❞♦ ♥❛ ❣❡stã♦ ❧♦❣íst✐❝❛ ❡ ❞❡ r❛str❡❛❜✐❧✐❞❛❞❡ ♣♦r ♠❡✐♦ ❞❛ ❝♦❞✐✜❝❛çã♦ ❞❡ ✐♥❢♦r♠❛çõ❡s ❛❞✐❝✐♦♥❛✐s ❝♦♠♦ ♥ú♠❡r♦ s❡r✐❛❧✱ ♥ú♠❡r♦ ❞❡ ❧♦t❡✱ ❞❛t❛ ❞❡ ✈❛❧✐❞❛❞❡✱ q✉❛♥t✐❞❛❞❡s✱ ♥ú♠❡r♦ ❞♦ ♣❡❞✐❞♦ ❞♦ ❝❧✐❡♥t❡ ❡t❝✳ ◆ã♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ✐t❡♥s q✉❡ ♣❛ss❛rã♦ ♣❡❧♦ ♣♦♥t♦ ❞❡ ✈❡♥❞❛ ✭P❉❱✮ • IT F − 14 ❋✐❣✉r❛ ✸✳✹✿ ❈ó❞✐❣♦ ❞❡ ❜❛rr❛s ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ❝♦❞✐✜❝❛r ❛♣❡♥❛s ●❚■◆s✱ ♣♦❞❡ s❡r ✐♠♣r❡ss♦ ❞✐r❡t❛♠❡♥t❡ ❡♠ s✉❜str❛t♦ ❝♦rr✉❣❛❞♦ ✭❝❛✐①❛ ❞❡ ♣❛♣❡❧ã♦✮ ♦❢❡r❡❝❡♥❞♦ ✉♠ ❜♦♠ ❞❡s❡♠♣❡♥❤♦ ❞❡ ❧❡✐t✉r❛✳ ◆ã♦ ♣♦❞❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ✐t❡♥s ❝♦♠❡r❝✐❛✐s q✉❡ ♣❛ss❛rã♦ ♣❡❧♦ ♣♦♥t♦ ❞❡ ✈❡♥❞❛✳ • GS1 ❉❛t❛▼❛tr✐① ❋✐❣✉r❛ ✸✳✺✿ ❙í♠❜♦❧♦ ❜✐❞✐♠❡♥s✐♦♥❛❧ ♣❛r❛ ❛♣❧✐❝❛çõ❡s ❡s♣❡❝✐❛✐s✱ q✉❡ ♣❡r♠✐t❡ ❝♦❞✐✜❝❛r ✐♥❢♦r♠❛çõ❡s ❡♠ ❡s♣❛ç♦s ♠✉✐t♦ ♠❡♥♦r❡s q✉❡ ♦s ❝ó❞✐❣♦s ❧✐♥❡❛r❡s ❡ ❛❣r❡❣❛r ✐♥❢♦r♠❛çõ❡s ❛❞✐❝✐♦♥❛✐s ❝♦♠♦ ❝ó❞✐❣♦ ❞♦ ♣r♦❞✉t♦✱ ❧♦t❡ ❡ ✈❛❧✐❞❛❞❡✳ ❖ GS1 ❉❛t❛▼❛tr✐① ❡①✐❣❡ ✉♠ ❧❡✐t♦r ❞❡ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s ❜✐❞✐♠❡♥s✐♦♥❛❧ ♣♦r ✐ss♦ ♥ã♦ ❞❡✈❡ s❡r ✉t✐❧✐③❛❞♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ✐t❡♥s q✉❡ ♣r❡❝✐s❛♠ ♣❛ss❛r ♣❡❧♦ ♣♦♥t♦ ❞❡ ✈❡♥❞❛ q✉❡ ♣♦ss✉✐ ❛♣❡♥❛s ❧❡✐t♦r❡s ❧✐♥❡❛r❡s✳ ✷✳ ❆ ✐❞❡♥t✐✜❝❛çã♦ ❞❛ ❡♠♣r❡s❛ r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ♣r♦❞✉t♦ ♦❝✉♣❛ ❞❡ q✉❛tr♦ ❛ s❡✐s ❞í❣✐t♦s✱ ♣♦❞❡♥❞♦ s❡r ❞♦ 4♦ ❛♦ 7♦ ♦✉ ❞♦ 4♦ ❛♦ 9♦ ❞í❣✐t♦✳ ✽✸

(291) ✸✳ ❆ ✐❞❡♥t✐✜❝❛çã♦ ❞♦ ♣r♦❞✉t♦ ♦❝✉♣❛ ❞❡ três ❛ ❝✐♥❝♦ ❞í❣✐t♦s✱ ♣♦❞❡♥❞♦ s❡r ❞♦ 8♦ ❛♦ 12♦ ♦✉ ❞♦ 10♦ ❛♦ 12♦ ❞í❣✐t♦✳ ✹✳ ❖ 13♦ ❞í❣✐t♦ é ❞❡ s❡❣✉r❛♥ç❛✳ ❆ss✐♠✱ ❛s ✐♥❢♦r♠❛çõ❡s ❡stã♦ ♥♦s ♥ú♠❡r♦s❀ ♦ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s ❝♦❞✐✜❝❛ ❡ss❡s ♥ú♠❡r♦s ♣❛r❛ s❡r❡♠ ❧✐❞♦s ♣❡❧❛s ♠áq✉✐♥❛s ❝♦♠ ❧❡✐t♦r❡s ó♣t✐❝♦s✳ ❊ss❡s ❧❡✐t♦r❡s ♠❡❞❡♠ ❛ ❧❛r❣✉r❛ ❞❛s ❜❛rr❛s ♣r❡t❛s ❡ ♦s ❡s♣❛ç♦s ❡♠ ❜r❛♥❝♦s ❡ ❞❡❝♦❞✐✜❝❛♠ ♦s ❞❛❞♦s ♣❛r❛ ♦❜t❡r ♦s ♥ú♠❡r♦s ❡ ❡♠ s❡❣✉✐❞❛ ❡❢❡t✉❛♠ ❛ ✈❡r✐✜❝❛çã♦ ❞❡ s❡❣✉r❛♥ç❛✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s r❡❛❧✐③❛❞♦s ❝♦♠ ♦s 12 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ✭❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮✱ ♦❜t❡♥❞♦ ❝♦♠♦ r❡s✉❧t❛❞♦ 13◦ ❞í❣✐t♦✱ ❝❤❛♠❛❞♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛✳ ❊ss❡ ❞í❣ít♦ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ r❡s✉❧t❛❞♦ ❞❛s ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦s 12 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❞❡✈❡ s❡r ≡ 0 mod 10✳ ❙❡ ♦ r❡s✉❧t❛❞♦ ❢♦r ❞✐❢❡r❡♥t❡ ❞❡ss❡ ❞í❣✐t♦ s✐❣♥✐✜❝❛ q✉❡ ❤♦✉✈❡ ❡rr♦ ♥❛ ❧❡✐t✉r❛✳ ✸✳✶✳✸ ❈♦♠♦ sã♦ ❣❡r❛❞♦s ♦s ❝ó❞✐❣♦s ❞❡ ❜❛rr❛s❄ ❙❡❥❛♠ a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 ❡ a12 ✱ ♦s ♣r✐♠❡✐r♦s 12 ❞í❣✐t♦s ✭❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮ ❞♦ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s✳ P❛r❛ ❞❡t❡r♠✐♥❛r ♦ 13♦ ❞í❣✐t♦✱ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛✱ ❞❡✈❡♠♦s✿ ✶✳ ❆❞✐❝✐♦♥❛r ♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s í♠♣❛r❡s✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ♣♦r✿ S1 = a1 + a3 + a5 + a7 + a9 + a11 . ✷✳ ❆❞✐❝✐♦♥❛r ♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s ♣❛r❡s✳ ❘❡♣r❡s❡♥t❛r❡♠♦s ♣♦r✿ S2 = a2 + a4 + a6 + a8 + a10 + a12 . ✸✳ ▼✉❧t✐♣❧✐❝❛r ❛ s♦♠❛ ❞♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s ♣❛r❡s ♣♦r 3✱ ♦✉ s❡❥❛✱ S 3 = 3 × S2 ✳ ✹✳ ❆❞✐❝✐♦♥❛r ♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s✱ ♦✉ s❡❥❛✱ S4 = S1 + S3 ✳ ✺✳ ❊♥❝♦♥tr❛r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞♦ r❡s✉❧t❛❞♦ ♣♦r 10✿ S4 ÷ 10✳ ✽✹

(292) ✻✳ ❙❡ ♦ r❡st♦ ❢♦r ③❡r♦✱ ♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛ a13 é ♦ ♣ró♣r✐♦ ③❡r♦❀ ❝❛s♦ ❝♦♥trár✐♦✱ ♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛ a13 é ♦ r❡s✉❧t❛❞♦ ❞❡ (10− r❡st♦✮✳ ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❞❡✈❡ ❣❡r❛r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 10✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ ♦❜t✐❞❛ é S4 ✱ ♦ ♥ú♠❡r♦ S4 + a13 ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 10✱ ♦✉ s❡❥❛✱ S4 + a13 ≡ 0 mod 10✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ✈❛♠♦s ❡❢❡t✉❛r ❛ ✈❡r✐✜❝❛çã♦ ❞❡ s❡❣✉r❛♥ç❛ ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛ ❞❡ ✉♠❛ ❜❛rr❛ ❞❡ ❝❡r❛❧✱ q✉❡ ❝♦♥s✐st❡ ❡♠ r❡❛❧✐③❛r ❛s ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s ❛❜❛✐①♦ ♣❛r❛ ❝♦♥❢❡r✐r ♦ r❡s✉❧t❛❞♦ ❞❡❧❛s ❝♦♠♦ ♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛✳ ❋✐❣✉r❛ ✸✳✻✿ ❙❡❥❛♠ 789432161362✱ ♦s ♣r✐♠❡✐r♦s 12 ❞í❣✐t♦s ✭❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮ ❞♦ ❝ó❞✐❣♦ ❞❡ ❜❛rr❛s✳ P❛r❛ ❞❡t❡r♠✐♥❛r ♦ 13◦ ❞í❣✐t♦✱ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛✱ ❞❡✈❡♠♦s✿ ✶✳ ❆❞✐❝✐♦♥❛r ♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s í♠♣❛r❡s✿ S1 = 7 + 9 + 3 + 1 + 1 + 6 =⇒ S1 = 27. ✷✳ ❆❞✐❝✐♦♥❛r ♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s ♣❛r❡s✿ S2 = 8 + 4 + 2 + 6 + 3 + 2 =⇒ S2 = 25. ✸✳ ▼✉❧t✐♣❧✐❝❛r ❛ s♦♠❛ ❞♦s ❞í❣✐t♦s ❧♦❝❛❧✐③❛❞♦s ♥❛s ♣♦s✐çõ❡s ♣❛r❡s ♣♦r 3✱ ♦✉ s❡❥❛✱ S3 = 3 × 25 =⇒ S3 = 75. ✹✳ ❆❞✐❝✐♦♥❛r ♦s r❡s✉❧t❛❞♦s ❛♥t❡r✐♦r❡s✱ ♦✉ s❡❥❛✱ S4 = 27 + 75 =⇒ S4 = 102. ✺✳ ❊♥❝♦♥tr❛r ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ S4 ♣♦r 10✿ 102 ÷ 10 = 10 ❡ ❞❡✐①❛ r❡st♦ 2✳ ✻✳ ▲♦❣♦✱ ♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛ é ♦ r❡s✉❧t❛❞♦ ❞❡ (10 − 2 = 8) ❡ 102 + 8 ≡ 0 m 10✳ ✸✳✷ ❈P❋ ✲ ❈❛❞❛str♦ ❞❡ P❡ss♦❛s ❋ís✐❝❛s ❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❜❛s❡❛❞❛ ♥♦s ❛rt✐❣♦s ❬✶✸❪✱ ❬✶✼❪✱ ❬✶✽❪✱ ❬✶✾❪✱ ❡ ❬✷✷❪✳ ✽✺

(293) ✸✳✷✳✶ ❍✐stór✐❛ ❉❡s❞❡ ❛ é♣♦❝❛ q✉❡ ♦ ❇r❛s✐❧ ❡r❛ ❝♦❧ô♥✐❛ q✉❡ sã♦ ❝♦❜r❛❞♦s ✐♠♣♦st♦s ❞❛ ♣♦♣✉❧❛çã♦✳ ❉❡ 1534 ❛ 1700✱ ♥♦ ❇r❛s✐❧✱ ❡r❛♠ ❝♦❜r❛❞❛s ❛ ♣♦♣✉❧❛çã♦ 10% ❞♦s s❡✉s ❣❛♥❤♦s ❡ ✐♥t❡r❡ss❡s ❡ ♣❛rt❡ ❞❡ss❡s ✐♠♣♦st♦s ❡r❛♠ r❡♣❛ss❛❞♦s ❛ ❈♦r♦❛ P♦rt✉❣✉❡s❛✳ ❊♠ 1808✱ ❝♦♠ ❛ ❝❤❡❣❛❞❛ ❞❛ ❢❛♠í❧✐❛ r❡❛❧ ♥♦ ❇r❛s✐❧✱ ❢♦✐ ❝r✐❛❞♦ ♦ ❈♦♥s❡❧❤♦ ❞❛ ❋❛③❡♥❞❛✱ q✉❡ t✐♥❤❛ ❝♦♠♦ ♦❜❥❡t✐✈♦ ❛❞♠✐♥✐str❛r ❡ ✜s❝❛❧✐③❛r ❛ ❛rr❡❝❛❞❛çã♦ ❞♦s ✐♠♣♦st♦s✳ ❆ ♣r✐♠❡✐r❛ t❡♥t❛t✐✈❛ ❞❡ ✐♠♣❧❛♥t❛r ♦ ✐♠♣♦st♦ ❞❡ r❡♥❞❛ ♥♦ ❇r❛s✐❧ ❢♦✐ ❡♠ 1843✳ ▼❛s só ❡♠ 1922 é q✉❡ ❡❢❡t✐✈❛♠❡♥t❡ ❢♦✐ ✐♠♣❧❛♥t❛❞♦ s❡❣✉✐♥❞♦ ♦ ♠♦❞❡❧♦ q✉❡ ❡r❛ ❛♣❧✐❝❛❞♦ ♥❛ ❋r❛♥ç❛✳ ❈♦♠ ❛ ✐♠♣❧❛♥t❛çã♦ ❞♦ ❊st❛❞♦ ◆♦✈♦ ❞❡ ●❡tú❧✐♦ ❱❛r❣❛s✱ ❡♠ 1934✱ ♦ ✐♠♣♦st♦ ❞❡ r❡♥❞❛ ❣❛♥❤♦✉ st❛t✉s ❝♦♥st✐t✉❝✐♦♥❛❧ ❡ ♣❛ss♦✉ ❛ s❡r ❞❡ ❝♦♠♣❡tê♥❝✐❛ ❞❛ ❯♥✐ã♦✳ ❊♠ 1968✱ ❢♦✐ ❝r✐❛❞❛ ❛ ❙❡❝r❡t❛r✐❛ ❞❛ ❘❡❝❡✐t❛ ❋❡❞❡r❛❧✱ ♥❛ q✉❛❧ s❡ ✉♥✐✉ ♦s ♣❛♣❡✐s ❞❡ ❛rr❡❝❛❞❛çã♦✱ tr✐❜✉t❛çã♦ ❡ ✜s❝❛❧✐③❛çã♦ ❞❡ ✐♠♣♦st♦s✳ ◆❡st❡ ♠❡s♠♦ ❛♥♦✱ q✉❡♠ ❞❡❝❧❛r❛ss❡ ♦ ✐♠♣♦st♦ ❞❡ r❡♥❞❛✱ r❡❝❡❜❡r✐❛ ✉♠ ❞♦❝✉♠❡♥t♦ ❝❤❛♠❛❞♦ ❞❡ ■❞❡♥t✐✜❝❛çã♦ ❞♦ ❈♦♥tr✐❜✉✐♥t❡ ✭❈■❈✮ ❡♠✐t✐❞♦ ❡❧❡tr♦♥✐❝❛♠❡♥t❡ ❡ q✉❡ t✐♥❤❛ ♣r❛③♦ ❞❡ ✈❛❧✐❞❛❞❡ ❡ ❞❡✈❡r✐❛ s❡r s❡♠♣r❡ r❡♥♦✈❛❞♦✳ ❊ss❡ ❞♦❝✉♠❡♥t♦ ❡r❛ ❞❡ ♣❛♣❡❧ ❡ ❡r❛ ✉s❛❞♦ ❜❛s✐❝❛♠❡♥t❡ ♣❛r❛ ❛s ❞❡❝❧❛r❛çõ❡s ❞❡ ✐♠♣♦st♦ ❞❡ r❡♥❞❛✳ ❊♠ ♠❡❛❞♦s ❞♦s ❛♥♦s 80✱ ♦ ❈■❈ ❢♦✐ s✉❜st✐t✉í❞♦ ♣❡❧♦ ❈P❋ ✭❈❛❞❛str♦ ❞❡ P❡ss♦❛s ❋ís✐❝❛s✮✱ t❛♠❜é♠ ❞❡ ♣❛♣❡❧ ❡ ❝♦♠ ❛ ♠❡s♠❛ ✜♥❛❧✐❞❛❞❡ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✸✳✼✳ ❈♦♠ ♦ ♣❛ss❛r ❞♦s ❛♥♦s✱ ♦ ❈P❋ ❢♦✐ t♦r♥❛♥❞♦✲s❡ ❝❛❞❛ ✈❡③ ♠❛✐s ✐♠♣♦rt❛♥t❡ ♣❛ss❛♥❞♦ ❛ s❡r ♦❜r✐❣❛tór✐♦ s✉❛ ❛♣r❡s❡♥t❛çã♦ ❛té ❡♠ ❛❜❡rt✉r❛s ❞❡ ❝♦♥t❛s ❡♠ ❜❛♥❝♦✱ ♣♦r ❡①❡♠♣❧♦✳ ❈♦♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞♦ ❈P❋ ❛✉♠❡♥t❛♥❞♦✱ ❡❧❡ ❞❡✐①♦✉ ❞❡ s❡r ❞❡ ♣❛♣❡❧ ❡ ♣❛ss♦✉ ❛ s❡r ❞❡ ♣❧ást✐❝♦ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✸✳✽✳ ▼❛s ✐ss♦ ♠✉❞♦✉ ❡ ❡♠ s❡t❡♠❜r♦ ❞❡ 2010✱ ❛ ❘❡❝❡✐t❛ ❡❢❡t✉♦✉ ✉♠❛ ♠✉❞❛♥ç❛ ♥♦ ❈P❋✱ q✉❡ ❛❣♦r❛ ♣❛ss❛ ❛ s❡r ♦♥❧✐♥❡✳ ■st♦ é✱ ♥ã♦ s❡rá ♠❛✐s ❡♠✐t✐❞♦ ♦ ❝❛rtã♦ ❈P❋ q✉❡ ❛ ♠❛✐♦r✐❛ t❡♠ ❤♦❥❡✳ ❖ ❞♦❝✉♠❡♥t♦ ♣❛ss❛ ❛ s❡r ✐♠♣r❡ss♦ ♣❡❧❛ ✐♥t❡r♥❡t✳ ❋✐❣✉r❛ ✸✳✽✿ ♣❧ást✐❝♦ ❋✐❣✉r❛ ✸✳✼✿ ❈P❋ ❞❡ ♣❛♣❡❧ ✸✳✷✳✷ ❈P❋ ❞❡ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ❈P❋❄ ❖ ❈P❋ é ♦ r❡❣✐str♦ ❞❡ ✉♠ ❝✐❞❛❞ã♦ ♥❛ ❘❡❝❡✐t❛ ❋❡❞❡r❛❧ ❜r❛s✐❧❡✐r❛ ♥♦ q✉❛❧ ❞❡✈❡♠ ❡st❛r t♦❞♦s ♦s ❝♦♥tr✐❜✉✐♥t❡s ✭♣❡ss♦❛s ❢ís✐❝❛s ♥❛❝✐♦♥❛✐s ❡ ❡str❛♥❣❡✐r❛s ❝♦♠ ♥❡❣ó❝✐♦s ♥♦ ❇r❛s✐❧✮✳ ❊❧❡ ❛r♠❛③❡♥❛ ✐♥❢♦r♠❛çõ❡s ❢♦r♥❡❝✐❞❛s ♣❡❧♦ ♣ró♣r✐♦ ❝♦♥tr✐❜✉✐♥t❡ ❡ ♣♦r ♦✉tr♦s s✐st❡♠❛s ❞❛ ❘❡❝❡✐t❛ ❋❡❞❡r❛❧✳ ❖ ❈P❋ é ✉♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❞♦❝✉♠❡♥t♦ ♣❛r❛ ❝✐❞❛❞ã♦s ❜r❛s✐❧❡✐r♦s✱ ❛❧é♠ ❞❡ s❡r ú♥✐❝♦ ❡ ✐♥tr❛♥s❢❡rí✈❡❧✳ ❊❧❡ é ❝♦♠♣♦st♦ ♣♦r 11 ❞í❣✐t♦s✱ ♦♥❞❡ ♦ ♥♦♥♦ ❞í❣✐t♦ ✭❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✮ r❡✈❡❧❛ ❛ ✉♥✐❞❛❞❡ ❢❡❞❡r❛t✐✈❛ ❡♠ q✉❡ ❛ ♣❡ss♦❛ r❡❣✐str♦✉✲s❡ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✱ ❞❛❞♦ q✉❡ é ♣r♦✐❜✐❞♦ ✭❡♠ ❝♦♥❞✐çõ❡s ♥♦r♠❛✐s✮ tr♦❝❛r ❞❡ ♥ú♠❡r♦✳ ▲♦❣♦✱ ♦❜s❡r✈❛♥❞♦ ❡ss❡ ❡①❡♠♣❧♦✱ ♦ ❈P❋ XXX.XXX.XX ✺ − Y Y ✱ ♦ ✽✻

(294) ♥ú♠❡r♦ ✏ 5✑ ❡♠ ❞❡st❛q✉❡ ✐♥❞✐❝❛ q✉❡ ❛ ♦r✐❣❡♠ ❞❡st❡ ❈P❋ é ❇❛❤✐❛ ♦✉ ❙❡r❣✐♣❡✳ ❆❜❛✐①♦ s❡❣✉❡ ❛ ❚❛❜❡❧❛ ✸✳✶ ❝♦♠ t♦❞♦s ♦s ❡st❛❞♦s ❜r❛s✐❧❡✐r♦s✿ ◆♦♥♦ ❞í❣✐t♦ ❆ ✉♥✐❞❛❞❡ ❢❡❞❡r❛t✐✈❛ ♦♥❞❡ ♦r✐❣✐♥♦✉ ♦ ❈P❋ ✵ ❘✐♦ ●r❛♥❞❡ ❞♦ ❙✉❧ ✶ ❉✐str✐t♦ ❋❡❞❡r❛❧✱ ●♦✐ás✱ ▼❛t♦ ●r♦ss♦✱ ▼❛t♦ ●r♦ss♦ ❞♦ ❙✉❧ ❡ ❚♦❝❛♥t✐♥s ✷ ❆♠❛③♦♥❛s✱ P❛rá✱ ❘♦r❛✐♠❛✱ ❆♠❛♣á✱ ❆❝r❡ ❡ ❘♦♥❞ô♥✐❛ ✸ ❈❡❛rá✱ ▼❛r❛♥❤ã♦ ❡ P✐❛✉í ✹ P❛r❛í❜❛✱ P❡r♥❛♠❜✉❝♦✱ ❆❧❛❣♦❛s ❡ ❘✐♦ ●r❛♥❞❡ ❞♦ ◆♦rt❡ ✺ ❇❛❤✐❛ ❡ ❙❡r❣✐♣❡ ✻ ▼✐♥❛s ●❡r❛✐s ✼ ❘✐♦ ❞❡ ❏❛♥❡✐r♦ ❡ ❊s♣ír✐t♦ ❙❛♥t♦ ✽ ❙ã♦ P❛✉❧♦ ✾ P❛r❛♥á ❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ❚❛❜❡❧❛ ✸✳✶✿ ◆♦♥♦ ❞í❣✐t♦ ❯♥✐❞❛❞❡ ❋❡❞❡r❛t✐✈❛ ◆♦ ❡①❡♠♣❧♦ ♠❡♥❝✐♦♥❛❞♦✱ ♦s ❞♦✐s ú❧t✐♠♦s ❞í❣✐t♦s ✏ Y Y ✑✱ ❝❤❛♠❛❞♦s ❞❡ ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s✱ sã♦ ❣❡r❛❞♦s ❛ ♣❛rt✐r ❞♦s ♦✉tr♦s ❞í❣✐t♦s ❛tr❛✈és ❞❡ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ 11✳ ▲♦❣♦✱ ♣❛r❛ ❣❡r❛r ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❞❡✈❡♠♦s ❡♥tã♦ ❢❛③❡r ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦s ♥♦✈❡s ♣r✐♠❡✐r♦s ❞í❣✐t♦s✱ s❡♥❞♦ ❛ss✐♠✱ ✈❛♠♦s s✉♣♦r q✉❡ ♦s ♥♦✈❡ ♣r✐♠❡✐r♦ ❞í❣✐t♦s✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ s❡❥❛♠ a9 ✳ a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ❡ ❉✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {10, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✷✳ a1 a2 a3 a4 a5 a6 a7 a8 a9 ✶✵ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 10a1 9a2 8a3 7a4 6a5 5a6 4a7 3a8 2a9 ❚❛❜❡❧❛ ✸✳✷✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 9 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 . ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ✏ S ✑ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 2✱ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ s❡ ❢♦r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 11✱ s❡rá s✉❜tr❛í❞♦ ❞❡ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ♠❡♥♦r q✉❡ 2✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá 0✳ ♦ q✉♦❝✐❡♥t❡ ❞❡✈❡rá s❡r ✐♥t❡✐r♦ ❡ ♦ a10 ✳ 11✱ ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ♦❜t✐❞❛ é S✱ ♦ ♥ú♠❡r♦ S + a10 ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ r❡st♦ ❢♦r ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 11✱ 11✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ ♦✉ s❡❥❛✱ S + a10 ≡ 0 mod 11. P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛r❡♠♦s ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ♠❛s ❛❣♦r❛ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛rá ♣❛rt❡ ❞♦ ❝á❧❝✉❧♦✳ ❊♥tã♦ ❛❣♦r❛ t❡r❡♠♦s a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ❡ a10 ✳ 10 ❞í❣✐t♦s✱ ❉✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {11, 10, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✸✳ ✽✼

(295) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 ✶✶ ✶✵ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 11a1 10a2 9a3 8a4 7a5 6a6 5a7 4a8 3a9 2a10 ❚❛❜❡❧❛ ✸✳✸✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 10 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 11a1 + 10a2 + 9a3 + 8a4 + 7a5 + 6a6 + 5a7 + 4a8 + 3a9 + 2a10 . ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ✏ S ✑ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 2✱ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ s❡ ❢♦r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 11✱ s❡rá s✉❜tr❛í❞♦ ❞❡ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ♠❡♥♦r q✉❡ 2✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá 0✳ ♦ q✉♦❝✐❡♥t❡ ❞❡✈❡rá s❡r ✐♥t❡✐r♦ ❡ ♦ a11 ✳ 11✱ ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ♦❜t✐❞❛ é S✱ ♦ ♥ú♠❡r♦ S + a11 ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ r❡st♦ ❢♦r ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 11✱ 11✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ ♦✉ s❡❥❛✱ S + a11 ≡ 0 mod 11. ❉❡ss❛ ♠❛♥❡✐r❛✱ ✜❝❛ ❞❡♠♦♥str❛❞♦ ❝♦♠♦ ♦ ❈P❋ é ❣❡r❛❞♦✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ✈❛♠♦s ❣❡r❛r ✉♠ ❈P❋ ✈á❧✐❞♦ ❝❛❧❝✉❧❛♥❞♦ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❞❡ ✉♠ ❈P❋ ❤✐♣♦tét✐t♦✱ 543.736.128 − Y Y ✳ ❏á s❛❜❡♠♦s q✉❡ ❛ ♦r✐❣❡♠ ❞❡ss❡ ❈P❋✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❛❜❡❧❛ ✸✳✶✱ é ❙ã♦ P❛✉❧♦✳ ❆❣♦r❛ ✈❛♠♦s ❣❡r❛r ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ✶✳ ❉✐str✐❜✉❛ ♦s 9 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡♠ ✉♠ q✉❛❞r♦ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s 10, 9, 8, 7, 6, 5, 4, 3, 2 ❛❜❛✐①♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❝♦♥❢♦r♠❡ ❛ ❚❛❜❡❧❛ ✸✳✹✳ ✺ ✹ ✸ ✼ ✸ ✻ ✶ ✷ ✽ ✶✵ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ ✺✵ ✸✻ ✷✹ ✹✾ ✶✽ ✸✵ ✹ ✻ ✶✻ ❚❛❜❡❧❛ ✸✳✹✿ ✷✳ ❈❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞♦s r❡s✉❧t❛❞♦s ✸✳ ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ (50+36+24+49+18+30+4+6+16) = 233 (233) s❡rá ❞✐✈✐❞♦ ♣♦r 11✳ ❈♦♥s✐❞❡r❡ ❝♦♠♦ q✉♦❝✐❡♥t❡ ❛♣❡♥❛s ♦ ✈❛❧♦r ✐♥t❡✐r♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡rá r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❝á❧❝✉❧♦ ❞♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r✳ 2 ▲♦❣♦✱ 233 ❞✐✈✐❞✐❞♦ ♣♦r 11 ❝♦♠♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✳ ❙✉❜tr❛✐✲s❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ❞❡ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r é 11 − 2✱ ♦✉ s❡❥❛✱ 9✳ 21 ❝♦♠♦ 11✱ s❡♥❞♦ ♦❜t❡♠♦s ❖❜s❡r✈❡ t❛♠❜é♠ q✉❡ 233 + 9 ≡ 0 mod11. ❏á t❡♠♦s ♣♦rt❛♥t♦ ♣❛rt❡ ❞♦ ❈P❋✱ ❝♦♥✜r❛✿ ✽✽ 543.736.128 − 9Y ✳ q✉♦❝✐❡♥t❡ ❡ ❛ss✐♠ ♥♦ss♦

(296) ❈❛❧❝✉❧❛♥❞♦ ♦ ❙❡❣✉♥❞♦ ❉í❣✐t♦ ❱❡r✐✜❝❛❞♦r P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ s❡❣✉♥❞♦ ❞í❣✐t♦ s❡rá ✉s❛❞♦ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❥á ❝❛❧❝✉❧❛❞♦✳ ▼♦♥t❛r❡♠♦s ✉♠❛ t❛❜❡❧❛ s❡♠❡❧❤❛♥t❡ ❛ ❛♥t❡r✐♦r só q✉❡ ❞❡st❛ ✈❡③ ✉s❛r❡♠♦s ♥❛ s❡❣✉♥❞❛ ❧✐♥❤❛ ♦s ✈❛❧♦r❡s 11, 10, 9, 8, 7, 6, 5, 4, 3 ❡ 2✳ ✶✳ ❉✐str✐❜✉❛ ♦s 10 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡♠ ✉♠ q✉❛❞r♦ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s 11, 10, 9, 8, 7, 6, 5, 4, 3, 2 ❛❜❛✐①♦ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❝♦♥❢♦r♠❡ ❛ ❚❛❜❡❧❛ ✸✳✺✳ ✺ ✹ ✸ ✼ ✸ ✻ ✶ ✷ ✽ ✾ ✶✶ ✶✵ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ ✺✺ ✹✵ ✷✼ ✺✻ ✷✶ ✸✻ ✺ ✽ ✷✹ ✶✽ ❚❛❜❡❧❛ ✸✳✺✿ ✷✳ ❈❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞♦s r❡s✉❧t❛❞♦s (55+40+27+56+21+36+5+8+24+18) = 290 ✸✳ ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ (290) s❡rá ❞✐✈✐❞♦ ♣♦r 11✳ ❈♦♥s✐❞❡r❡ ❝♦♠♦ q✉♦❝✐❡♥t❡ ❛♣❡♥❛s ♦ ✈❛❧♦r ✐♥t❡✐r♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡rá r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❝á❧❝✉❧♦ ❞♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r✳ ▲♦❣♦✱ 290 ❞✐✈✐❞✐❞♦ ♣♦r 11 ♦❜t❡♠♦s 26 ❝♦♠♦ q✉♦❝✐❡♥t❡ ❡ 4 ❝♦♠♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✳ ❙✉❜tr❛✐✲s❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ❞❡ 11✱ s❡♥❞♦ ❛ss✐♠ ♥♦ss♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r é 11 − 4✱ ♦✉ s❡❥❛✱ 7✳ ❯♠❛ ✈❡③ q✉❡ 290 + 7 ≡ 0 mod 11. ◆❡st❡ ❝❛s♦ ❝❤❡❣❛♠♦s ❛♦ ✜♥❛❧ ❞♦s ❝á❧❝✉❧♦s ❡ ❞❡s❝♦❜r✐♠♦s q✉❡ ♦s ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s ❞♦ ♥♦ss♦ ❈P❋ ❤✐♣♦tét✐❝♦ sã♦ ♦s ♥ú♠❡r♦s 9 ❡ 7✱ ♣♦rt❛♥t♦ ♦ ❈P❋ ✜❝❛r✐❛ ❛ss✐♠✿ 543.736.128 − 97✳ ✸✳✸ ❈◆P❏ ✲ ❈❛❞❛str♦ ◆❛❝✐♦♥❛❧ ❞❛ P❡ss♦❛ ❏✉rí❞✐❝❛ ❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❜❛s❡❛❞❛ ♥♦s ❛rt✐❣♦s ❬✶✸❪✱ ❬✶✻❪❡ ❬✷✹❪✳ ✸✳✸✳✶ ❍✐stór✐❛ ❖ ❈❛❞❛str♦ ◆❛❝✐♦♥❛❧ ❞❛ P❡ss♦❛ ❏✉rí❞✐❝❛ ✭❈◆P❏✮ ❢♦✐ ✐♥st✐t✉í❞♦ ❡♠ 1998 ❡♠ s✉❜st✐t✉✐çã♦ ❛♦ ❛♥t✐❣♦ ❈❛❞❛str♦ ●❡r❛❧ ❞❡ ❈♦♥tr✐❜✉✐♥t❡s ✭❈●❈✮ ❝r✐❛❞♦ ❡♠ 1964 ❡ ❡①t✐♥t♦ ❡♠ 1999✳ ❖ ❈◆P❏ ❛❞♠✐♥✐str❛❞♦ ♣❡❧❛ ❘❡❝❡✐t❛ ❋❡❞❡r❛❧ q✉❡ r❡❣✐str❛ ❛s ✐♥❢♦r♠❛çõ❡s ❝❛❞❛str❛✐s ❞❛s ♣❡ss♦❛s ❥✉rí❞✐❝❛s ❡ ❞❡ ❛❧❣✉♠❛s ❡♥t✐❞❛❞❡s ♥ã♦ ❝❛r❛❝t❡r✐③❛❞❛s ❝♦♠♦ t❛✐s✳ ◆♦ ❡♥t❛♥t♦✱ ❛ ♣❛rt✐r ❞❡ 01/11/2002✱ ♦s ❝❛rtõ❡s ❈◆P❏ ♣❡r❞❡r❛♠ s✉❛ ✈❛❧✐❞❛❞❡ ❡✱ ♣♦rt❛♥t♦✱ ♥ã♦ ❡stã♦ s❡♥❞♦ ♠❛✐s ❡♠✐t✐❞♦s✳ ❈♦♠ ❛ ❡①t✐♥çã♦ ❞♦ ❈❛rtã♦ ❈◆P❏✱ ❛ ❝♦♠♣r♦✈❛çã♦ ❞❛ ❝♦♥❞✐çã♦ ❞❡ ✐♥s❝r✐t♦ ♣❛ss♦✉ ❛ s❡r ❢❡✐t❛ ♠❡❞✐❛♥t❡ ❝♦♥s✉❧t❛ ♥♦ s✐t❡ ❞❛ ❘❡❝❡✐t❛ ❋❡❞❡r❛❧✳ ◆❡st❡ s✐t❡ é ♣♦ssí✈❡❧ ❢❛③❡r ❛ ❡♠✐ssã♦ ❞♦ ❝♦♠♣r♦✈❛♥t❡ ❞❡ ✐♥s❝r✐çã♦ ❡ ❞❛ s✐t✉❛çã♦ ❝❛❞❛str❛❞❛ ✈✐❛ ♦♥✲❧✐♥❡ s❡♠ ♥❡❝❡ss✐❞❛❞❡ ❞♦ ❛♥t✐❣♦ ❝❛rtã♦✳ ✽✾

(297) ✸✳✸✳✷ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ❈◆P❏❄ ❖ ❈◆P❏ é ✉♠ ♥ú♠❡r♦ ú♥✐❝♦ q✉❡ ✐❞❡♥t✐✜❝❛ ✉♠❛ ♣❡ss♦❛ ❥✉rí❞✐❝❛ ❥✉♥t♦ à ❘❡❝❡✐t❛ ❋❡❞❡r❛❧ ❇r❛s✐❧❡✐r❛ ✭ór❣ã♦ ❞♦ ▼✐♥✐stér✐♦ ❞❛ ❋❛③❡♥❞❛✮✱ ♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ❛ ♣❡ss♦❛ ❥✉rí❞✐❝❛ t❡♥❤❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❢❛③❡r ❝♦♥tr❛t♦s ❡ ♣r♦❝❡ss❛r ♦✉ s❡r ♣r♦❝❡ss❛❞❛✳ ❊❧❡ é ❝♦♠♣♦st♦ ♣♦r • ❖s 8 • ❖s • ❡ ♦s 14 ❛❧❣❛r✐s♠♦s✱ ❞✐✈✐❞✐❞♦s ❡♠ 3 ♣❛rt❡s✿ ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❧♦❝❛❧✐③❛❞♦s ❛♥t❡s ❞❛ ❜❛rr❛ r❡♣r❡s❡♥t❛♠ ♦ ♥ú♠❡r♦ ❞❡ ✐♥s❝r✐çã♦ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦❀ 4 ♣r✐♠❡✐r♦s ♥ú♠❡r♦s ❧♦❝❛❧✐③❛❞♦s ❛♣ós ❛ ❜❛rr❛ r❡♣r❡s❡♥t❛♠ ✉♠ ❝ó❞✐❣♦ ú♥✐❝♦ ♣❛r❛ ❛ ♠❛tr✐① ♦✉ ✜❧✐❛❧❀ 2 ú❧t✐♠♦s ♥ú♠❡r♦s sã♦ ❝❤❛♠❛❞♦s ❞❡ ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s ✭❉❱✮❀ ▲♦❣♦✱ ♦❜s❡r✈❛♥❞♦ ❡ss❡ ❡①❡♠♣❧♦✱ ♦ ❈◆P❏ ❳❳✳❳❳❳✳❳❳❳✴❨❨❨❨✲ZZ ✱ ♦s ❞♦✐s ú❧t✐♠♦s ❞í❣✐t♦s ✏ Z ✑ ❝❤❛♠❛❞♦s ❞❡ ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s✱ sã♦ ❣❡r❛❞♦s ❛ ♣❛rt✐r ❞♦s ♦✉tr♦s ❞í❣✐t♦s ❛tr❛✈és ❞❡ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ ✶✶✳ ▲♦❣♦✱ ♣❛r❛ ❣❡r❛r ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❞❡✈❡♠♦s ❡♥tã♦ ❢❛③❡r ♦♣❡r❛çõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦s ❞♦③❡s ♣r✐♠❡✐r♦s ❞í❣✐t♦s✱ s❡♥❞♦ ❛ss✐♠✱ ✈❛♠♦s s✉♣♦r q✉❡ ♦s ❞♦③❡s ♣r✐♠❡✐r♦ ❞í❣✐t♦s✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ s❡❥❛♠ a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 , a10 , a11 ❡ a12 ✳ ❉✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {5, 4, 3, 2, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✻✳ a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 ✺ ✹ ✸ ✷ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 5a1 4a2 3a3 2a4 9a5 8a6 7a7 6a8 5a9 4a10 3a11 2a12 ❚❛❜❡❧❛ ✸✳✻✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 12 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 5a1 + 4a2 + 3a3 + 2a4 + 9a5 + 8a6 + 7a7 + 6a8 + 5a9 + 4a10 + 3a11 + 2a12 . ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ S é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 2✱ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ s❡ ❢♦r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 11✱ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ♠❡♥♦r q✉❡ 2✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá 0✳ ♦ q✉♦❝✐❡♥t❡ ❞❡✈❡rá s❡r ✐♥t❡✐r♦ ❡ ♦ s❡rá s✉❜tr❛í❞♦ ❞❡ a13 ✳ 11✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ r❡st♦ ❢♦r ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ 11✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ s❡❥❛✱ S + a13 ≡ 0 ♠♦❞ 11✳ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ♦❜t✐❞❛ é S✱ ♦ ♥ú♠❡r♦ S + a13 ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 11✱ ♦✉ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛r❡♠♦s ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ♠❛s ❛❣♦r❛ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛rá ♣❛rt❡ ❞♦ ❝á❧❝✉❧♦✳ ❊♥tã♦ ❛❣♦r❛ t❡r❡♠♦s ❞í❣✐t♦s✱ a1 ✱ a2 ✱ a3 ✱ a4 ✱ a5 ✱ a6 ✱ a7 ✱ a8 ✱ a9 ✱ a10 ✱ a11 ✱ a12 ❡ a13 ✳ 13 ❉✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {6, 5, 4, 3, 2, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✼✳ ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 6a1 + 5a2 + 4a3 + 3a4 + 2a5 + 9a6 + 8a7 + 7a8 + 6a9 + 5a10 + 4a11 + 3a12 + 2a13 . ✾✵

(298) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 ✻ ✺ ✹ ✸ ✷ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 6a1 5a2 4a3 3a4 2a5 9a6 8a7 7a8 6a9 5a10 4a11 3a12 2a13 ❚❛❜❡❧❛ ✸✳✼✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 13 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ ✏ S ✑ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 2✱ r❡st♦ ❞❛ ❞✐✈✐sã♦✱ s❡ ❢♦r ♠❛✐♦r ♦✉ ✐❣✉❛❧ ❛ 11✱ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ♠❡♥♦r q✉❡ 2✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá 0✳ ♦ q✉♦❝✐❡♥t❡ ❞❡✈❡rá s❡r ✐♥t❡✐r♦ ❡ ♦ s❡rá s✉❜tr❛í❞♦ ❞❡ a14 ✳ S✱ ♦ ♥ú♠❡r♦ S + a14 ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ ♦ r❡st♦ ❢♦r ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ♦❜t✐❞❛ é 11✱ ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 11✱ ♦✉ s❡❥❛✱ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✜❝❛ ❞❡♠♦♥str❛❞♦ ❝♦♠♦ ♦ ❈◆P❏ é ❣❡r❛❞♦✳ 11✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ S + a14 ≡ 0 mod 11✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r ♦ ♣r♦❝❡ss♦ ❡ t♦r♥❛r ♠❛✐s ❢á❝✐❧ ❛ ❡①♣❧✐❝❛çã♦ ✈❛♠♦s ❝❛❧❝✉❧❛r ♦s ❞í❣✐t♦s ✈❡r✐✜❝❛❞♦r❡s ❞❡ ✉♠ ❈◆P❏ ❤✐♣♦tét✐❝♦✱ ♣♦r ❡①❡♠♣❧♦✱ ZZ ✳ P❛r❛ ❣❡r❛r ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r✱ 11.444.777/0001 − ❞✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {5, 4, 3, 2, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✽✳ ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞♦s r❡s✉❧t❛❞♦s✿ 1 1 4 4 4 7 7 7 0 0 0 1 ✺ ✹ ✸ ✷ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 5 4 12 8 36 56 49 42 0 0 0 2 ❚❛❜❡❧❛ ✸✳✽✿ S = 5 + 4 + 12 + 8 + 36 + 56 + 49 + 42 + 0 + 0 + 0 + 2 = 214. ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ (214) s❡rá ❞✐✈✐❞♦ ♣♦r ✶✶✳ ❈♦♥s✐❞❡r❡ ❝♦♠♦ q✉♦❝✐❡♥t❡ ❛♣❡♥❛s ♦ ✈❛❧♦r ✐♥t❡✐r♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡rá r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❝á❧❝✉❧♦ ❞♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r✳ ❱❛♠♦s ❛❝♦♠♣❛♥❤❛r✿ 214 ❞✐✈✐❞✐❞♦ ♣♦r 11 ♦❜t❡♠♦s 19 ❝♦♠♦ q✉♦❝✐❡♥t❡ ❡ r❡st♦ ❞❛ ❞✐✈✐sã♦✳ ❈❛s♦ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡❥❛ ♠❡♥♦r q✉❡ ✈❡r✐✜❝❛❞♦r s❡ t♦r♥❛ 0 2✱ 11 − 5✱ ❝♦♠♦ ♦ ♥♦ss♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✭③❡r♦✮✱ ❝❛s♦ ❝♦♥trár✐♦ s✉❜tr❛✐✲s❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ❞❡ ♥♦ss♦ ❝❛s♦✳ ❙❡♥❞♦ ❛ss✐♠ ♥♦ss♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r é 5 ♦✉ s❡❥❛✱ 6✳ 11✱ q✉❡ é ❯♠❛ ✈❡③ q✉❡ 214 + 6 ≡ 0 mod 11. ❏á t❡♠♦s ♣♦rt❛♥t♦✱ ♣❛rt❡ ❞♦ ❈◆P❏✱ ❝♦♥✜r❛✿ 11.444.777/0001 − 6Z ✳ P❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦ s❡❣✉♥❞♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛r❡♠♦s ❞❡ ♠♦❞♦ s❡♠❡❧❤❛♥t❡✱ ♠❛s ❛❣♦r❛ ♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❢❛rá ♣❛rt❡ ❞♦ ❝á❧❝✉❧♦✳ ❊♥tã♦ ❛❣♦r❛ t❡r❡♠♦s 13 ❞í❣✐t♦s✳ ❉✐str✐❜✉❛ ❡ss❡s ♥ú♠❡r♦s ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {6, 5, 4, 3, 2, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✾✳ ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞♦s r❡s✉❧t❛❞♦s✿ S = 6 + 5 + 16 + 12 + 8 + 63 + 56 + 49 + 0 + 0 + 0 + 3 + 12 = 230. ✾✶

(299) 1 1 4 4 4 7 7 7 0 0 0 1 6 ✻ ✺ ✹ ✸ ✷ ✾ ✽ ✼ ✻ ✺ ✹ ✸ ✷ 6 5 16 12 8 63 56 49 0 0 0 3 12 ❚❛❜❡❧❛ ✸✳✾✿ ❖ r❡s✉❧t❛❞♦ ♦❜t✐❞♦ (230) s❡rá ❞✐✈✐❞♦ ♣♦r ✶✶✳ ❈♦♥s✐❞❡r❡ ❝♦♠♦ q✉♦❝✐❡♥t❡ ❛♣❡♥❛s ♦ ✈❛❧♦r ✐♥t❡✐r♦✱ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡rá r❡s♣♦♥sá✈❡❧ ♣❡❧♦ ❝á❧❝✉❧♦ ❞♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r✳ ❱❛♠♦s ❛❝♦♠♣❛♥❤❛r✿ 230 ❞✐✈✐❞✐❞♦ ♣♦r 11 ♦❜t❡♠♦s 20 ❝♦♠♦ q✉♦❝✐❡♥t❡ ❡ 10 ❝♦♠♦ r❡st♦ ❞❛ ❞✐✈✐sã♦✳ ❈❛s♦ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡❥❛ ♠❡♥♦r q✉❡ 2✱ ♦ ♥♦ss♦ ♣r✐♠❡✐r♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡ t♦r♥❛ 0 ✭③❡r♦✮✱ ❝❛s♦ ❝♦♥trár✐♦ s✉❜tr❛✐✲s❡ ♦ ✈❛❧♦r ♦❜t✐❞♦ ❞❡ 11✱ q✉❡ é ♥♦ss♦ ❝❛s♦✳ ❙❡♥❞♦ ❛ss✐♠ ♥♦ss♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r é 11 − 10✱ ♦✉ s❡❥❛✱ 1✳ ❯♠❛ ✈❡③ q✉❡ 230 + 1 ≡ 0 mod 11. ❏á t❡♠♦s ♣♦rt❛♥t♦✱ ♦ ❈◆P❏✱ ❝♦♥✜r❛✿ 11.444.777/0001 − 61✳ ✸✳✹ ■❙❇◆ ✲ ■♥t❡r♥❛t✐♦♥❛❧ ❙t❛♥❞❛r❞ ❇♦♦❦ ◆✉♠❜❡r ❡♠ ♣♦rt✉❣✉ês ◆ú♠❡r♦ P❛❞rã♦ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ▲✐✈r♦ ❊st❛ s❡çã♦ ❢♦✐ ❡❧❛❜♦r❛❞❛ ❜❛s❡❛❞❛ ♥♦ ❛rt✐❣♦ ❬✶✸❪✳ ✸✳✹✳✶ ❍✐stór✐❛ ❊♠ 1966 r❡❛❧✐③♦✉✲s❡ ❡♠ ❇❡r❧✐♠ ❛ t❡r❝❡✐r❛ ❈♦♥❢❡rê♥❝✐❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ■♥✈❡st✐❣❛çã♦ ❡ ❘❛❝✐♦♥❛❧✐③❛çã♦ ❞♦ ♠❡r❝❛❞♦ ❞♦ ❧✐✈r♦ ❝✉❥♦ ♦❜❥❡t✐✈♦ ❡r❛ ❝r✐❛r ✉♠ s✐st❡♠❛ ✐♥t❡r♥❛❝✐♦♥❛❧ ❞❡ ♥✉♠❡r❛çã♦ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❧✐✈r♦s✳ ❉❡ss❡ ♠♦❞♦✱ ✐♥✐❝✐♦✉✲s❡ ❛ ❜✉s❝❛ ❞❡ ✉♠ s✐st❡♠❛ q✉❡ ♣✉❞❡ss❡ ✐❞❡♥t✐✜❝❛r ❝❛❞❛ ❧✐✈r♦✱ ♦✉ s❡❥❛✱ ❝❛❞❛ ❧✐✈r♦ ❞❡✈❡r✐❛ ♣♦ss✉✐r ✉♠ ♥ú♠❡r♦ ú♥✐❝♦ ❡ ✉♥✐✈❡rs❛❧✳ ▲♦❣♦✱ ❡r❛ ♥❡❝❡ssár✐♦ ♦ ❝♦♥tr♦❧❡ ❞❡ t♦❞♦ ♦ ❡st♦q✉❡ ❡ ✐ss♦ s❡r✐❛ ❢❡✐t♦ ♣❡❧♦ ✉s♦ ❞❡ ❝♦♠♣✉t❛❞♦r❡s q✉❡ ♣r♦❝❡ss❛ss❡♠ t♦❞❛s ❛s ✐♥❢♦r♠❛çõ❡s✳ ❊♠ 1970 ❢♦✐ ❝r✐❛❞♦ ❡ ❛♣r♦✈❛❞♦ ♦ ■❙❇◆ q✉❡ é r❡❣✉❧❛♠❡♥t❛❞♦ ♣❡❧❛ ■❙❖ ✭■♥t❡r♥❛t✐♦♥❛❧ ❖r❣❛♥✐③❛t✐♦♥ ❢♦r ❙t❛♥❞❛r❞✐③❛t✐♦♥✮✳ ❆té ♦ ✜♠ ❞❡ 2006 ♦ ■❙❇◆ ❡r❛ ❝♦♠♣♦st♦ ♣♦r 10 ❞í❣✐t♦s ✜❝❛♥❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ISBN − 10✳ ❆ ♣❛rt✐r ❞❡ 01 ❞❡ ❏❛♥❡✐r♦ ❞❡ 2007✱ ♣❛ss♦✉ ❛ ❝♦♥t❡r 13 ❞í❣✐t♦s✱ ♣❛r❛ ❛✉♠❡♥t❛r ❛ ❝❛♣❛❝✐❞❛❞❡ ❞♦ s✐st❡♠❛ ❞❡✈✐❞♦ ❛♦ ❛✉♠❡♥t♦ ♥♦ ♥ú♠❡r♦ ❞❡ ♣✉❜❧✐❝❛çõ❡s✱ ✜❝❛♥❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ISBN − 13✳ ❊ ❛❣♦r❛ q✉❛s❡ t♦❞♦s ♦s ♣❛ís❡s ❞♦ ♠✉♥❞♦ ✉t✐❧✐③❛♠ ♦ s✐st❡♠❛ ■❙❇◆ ♣❛r❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❡ ♣✉❜❧✐❝❛çõ❡s✱ ❛tr✐❜✉✐♥❞♦ ✉♠ ♥ú♠❡r♦ ú♥✐❝♦ ♣❛r❛ ❝❛❞❛ ❡❞✐çã♦✳ ✸✳✹✳✷ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ISBN − 10❄ ❖ ISBN − 10 é s❡♠♣r❡ ♣r❡❝❡❞✐❞♦ ❞❛s ❧❡tr❛s ■❙❇◆ ❡ é ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ♣❛rt❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ✈❛r✐á✈❡❧✱ q✉❡ ❞❡✈❡♠ s❡r s❡♣❛r❛❞❛s ♣♦r ❤í❢❡♥✳ ❈❛❞❛ ✉♠❛ ❞❡ss❛s ✾✷

(300) ♣❛rt❡s r❡♣r❡s❡♥t❛ r❡s♣❡❝t✐✈❛♠❡♥t❡✿ ♣❛ís ❞❡ ♦r✐❣❡♠ ❞♦ ♣r♦❞✉t♦✱ ❡♠♣r❡s❛ ❢❛❜r✐❝❛♥t❡ ✭❡❞✐t♦r❛✮✱ tít✉❧♦ ❡ ❞✐❣✐t♦ ✈❡r✐✜❝❛❞♦r✳ P♦r ❡①❡♠♣❧♦✱ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❞❡ ❢♦r♠❛ ❣❡♥❡r❛❧✐③❛❞❛ ♦ s❡❣✉✐♥t❡ ❝ó❞✐❣♦ ❜❛rr❛s ❞❡ ✉♠ ❧✐✈r♦ ■❙❇◆ XX − Y Y Y − ZZZZ − W ✳ ❖s ❝ó❞✐❣♦s ❞❡ ❜❛rr❛ ■❙❇◆ s❡r✈❡♠ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ♦s ❧✐✈r♦s ❞❡ ✉♠❛ ❢♦r♠❛ ❝❧❛r❛♠❡♥t❡ ♦r❣❛♥✐③❛❞❛✱ ❡♥tã♦ ❞❡♥tr♦ ❞♦ ❝ó❞✐❣♦✱ t❡♠♦s✿ • ■❞❡♥t✐✜❝❛❞♦r ❞❡ ●r✉♣♦✱ P❛ís ♦✉ ➪r❡❛ ■❞✐♦♠át✐❝❛ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦s 2 ♣r✐♠❡✐r♦s ❞í❣✐t♦s✱ q✉❡ ♥♦ ♥♦ss♦ ❡①❡♠♣❧♦ é ✏ XX ✑✳ ❚♦❞♦s ♦s ✐❞❡♥t✐✜❝❛❞♦r❡s ❞❡ ❣r✉♣♦ sã♦ ❛tr✐❜✉í❞♦s ♣❡❧❛ ❆❣ê♥❝✐❛ ■♥t❡r♥❛❝✐♦♥❛❧ ❞♦ ■❙❇◆✱ ❡♠ ❇❡r❧✐♠✳ • ■❞❡♥t✐✜❝❛❞♦r ❞❡ ❊❞✐t♦r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦s 3 s❡❣✉✐♥t❡s ❞í❣✐t♦s ✏Y Y Y ✑✱ ❣❡r❛❧♠❡♥t❡ é ✐♥❞✐❝❛❞❛ ❛ ❡①❛t❛ ✐❞❡♥t✐✜❝❛çã♦ ❞❛ ❡❞✐t♦r❛ ❡ s❡✉ ❡♥❞❡r❡ç♦✳ ❖s ♣r❡✜①♦s ❞❡ ❡❞✐t♦r❛s sã♦ ❛tr✐❜✉í❞♦s ♣❡❧❛ ❆❣ê♥❝✐❛ ■❙❇◆ ❞♦ ❣r✉♣♦ r❡s♣♦♥sá✈❡❧ ♣❡❧❛ ❣❡stã♦ ❞♦ s✐st❡♠❛ ■❙❇◆ ♥♦ ♣❛ís✱ r❡❣✐ã♦ ♦✉ ❣r✉♣♦ ✐❞✐♦♠át✐❝♦ ♦♥❞❡ ♦ ❡❞✐t♦r é ❜❛s❡❛❞♦ ♦✜❝✐❛❧♠❡♥t❡✳ • ■❞❡♥t✐✜❝❛❞♦r ❞❡ ❚ít✉❧♦ r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦s ♣ró①✐♠♦s • ❉í❣✐t♦ ✈❡r✐✜❝❛❞♦r r❡♣r❡s❡♥t❛❞♦ ♣❡❧♦ ú❧t✐♠♦ ❞í❣✐t♦ ❞♦ ❝ó❞✐❣♦ ✏ W ✑ ❡ q✉❡ ♣♦❞❡ s❡r ❞❡ 0 ❛ 9 ❡ ♥♦ ❝❛s♦ ❞♦ ♥ú♠❡r♦ ♦ ❝ó❞✐❣♦ ■❙❇◆ 13✱ 10 4 ❞í❣✐t♦s ✏ ZZZZ ✑✳ é r❡♣r❡s❡♥t❛❞♦ ♣♦r X✱ ♥❡ss❡ ❝❛s♦ t❡r❡♠♦s q✉❡ tr❛t❛r❡♠♦s ♥❛ s❡q✉ê♥❝✐❛✳ ❈♦♠♦ é ❣❡r❛❞♦ ❡ss❡ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r❄ ❈♦♥s✐❞❡r❡ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✳ ❉✐str✐❜✉❛ ♦s ♥♦✈❡ ♣r✐♠❡✐r♦s ❞í❣✐t♦s a1 ✱ a2 ✱ a3 ✱ a4 ✱ a5 ✱ a6 ✱ a7 ✱ a8 ❡ a9 ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {10, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✶✵✳ a1 a2 a3 a4 a5 a6 a7 a8 a9 10 9 8 7 6 5 4 3 2 10a1 9a2 8a3 7a4 6a5 5a6 4a7 3a8 2a9 ❚❛❜❡❧❛ ✸✳✶✵✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 9 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s S = 10a1 + 9a2 + 8a3 + 7a4 + 6a5 + 5a6 + 4a7 + 3a8 + 2a9 ✳ ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ 11✱ ♦ q✉♦❝✐❡♥t❡ ❞❡✈❡rá s❡r ✐♥t❡✐r♦ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ s❡rá s✉❜tr❛í❞♦ ❞❡ 11✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r a10 ✳ ❊ss❡ ❛ss✐♠✿ ✏ S ✑ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ 11✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ ♦❜t✐❞❛ é ✏ S ✑✱ ♦ ♥ú♠❡r♦ S + a10 ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 11✱ ♦✉ s❡❥❛✱ S + a10 ≡ 0 mod 11✳ ❖ r❡st♦ ♣♦❞❡ s❡r ♦ ♥ú♠❡r♦ 0 ♥♦ ❡♥t❛♥t♦✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá ♦ ♣ró♣r✐♦ ③❡r♦✳ ❈❛s♦✱ ♦ r❡st♦ s❡❥❛ ♦ ♥ú♠❡r♦ 1 ✭❝♦♠♦ 11 − 1 = 10✮ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá ♦ ♥ú♠❡r♦ 10 ❡♠ ❛❧❣❛r✐s♠♦ r♦♠❛♥♦✱ ♦✉ s❡❥❛✱ ♦ ❝❛r❛❝t❡r❡ X s❡rá ✐♥s❡r✐❞♦ ♥♦ ❧✉❣❛r ❞♦ ♥ú♠❡r♦ 10✱ ❥á q✉❡ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r é ❢♦r♠❛❞♦ ♣♦r ✉♠ ♠ú❧t✐♣❧♦ ❞❡ ú♥✐❝♦ ❞í❣✐t♦✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ✈❛♠♦s ❡❢❡t✉❛r ❛ ✈❡r✐✜❝❛çã♦ ❞❡ s❡❣✉r❛♥ç❛ ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ✉♠ ❧✐✈r♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ 3❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✱ q✉❡ ❝♦♥s✐st❡ ❡♠ r❡❛❧✐③❛r ❛s ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s ❛❜❛✐①♦ ♣❛r❛ ❝♦♥❢❡r✐r ♦ r❡s✉❧t❛❞♦ ❞❡❧❛s ❝♦♠ ♦ ❞í❣✐t♦ ❞❡ ✾✸

(301) s❡❣✉r❛♥ç❛✳ ❋✐❣✉r❛ ✸✳✾✿ ❈♦♥s✐❞❡r❡ ♦ ❝ó❞✐❣♦ ■❙❇◆ 85 − 16 − 01340 − 5✳ ❉✐str✐❜✉❛ ♦s ♥♦✈❡s ♣r✐♠❡✐r♦s ❞í❣✐t♦s 8✱ 5✱ 1✱ 6✱ 0✱ 1✱ 3✱ 4 ❡ 0 ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {10, 9, 8, 7, 6, 5, 4, 3, 2}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✶✶✳ 8 5 1 6 0 1 3 4 0 10 9 8 7 6 5 4 3 2 80 45 8 42 0 5 12 12 0 ❚❛❜❡❧❛ ✸✳✶✶✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 9 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 80 + 45 + 8 + 42 + 0 + 5 + 12 + 12 + 0 = 204. ❊ss❡ r❡s✉❧t❛❞♦ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 11✱ ♦ q✉♦❝✐❡♥t❡ é 18 ❡ ♦ r❡st♦ é 6✳ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ é ❡♥tã♦ s✉❜tr❛í❞♦ ❞❡ 11✱ ♦✉ s❡❥❛✱ 11 − 6 = 5✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r a10 = 5✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✜❝❛ ✈❡r✐✜❝❛❞♦ q✉❡ ♦ ❝ó❞✐❣♦ r❡❛❧♠❡♥t❡ ❡stá ❝♦rr❡t♦✳ ✸✳✹✳✸ ❈♦♠♦ é ❣❡r❛❞♦ ♦ ISBN − 13❄ ❖ ISBN − 13 é ❝♦♠♣♦st♦ ♣♦r ✶✸ ❞í❣✐t♦s ❡ ❢♦✐ ❣❡r❛❞♦ ❞❡✈✐❞♦ ❛♦ ❝r❡s❝❡♥t❡ ♥ú♠❡r♦ ❞❡ ♣✉❜❧✐❝❛çõ❡s✱ ❝♦♠ s✉❛s ❡❞✐çõ❡s ❡ ❢♦r♠❛t♦s✳ ◆❡ss❡ ❝ó❞✐❣♦✱ ♦s três ♣r✐♠❡✐r♦s ❞í❣✐t♦s r❡♣r❡s❡♥t❛♠ ♦ ♣❛ís ❞❡ r❡❣✐str♦ ❞♦ ♣r♦❞✉t♦✱ ♦s q✉❛tr♦ ❞í❣✐t♦s s❡❣✉✐♥t❡s ✐❞❡♥t✐✜❝❛♠ ♦ ❢❛❜r✐❝❛♥t❡✱ ♦s ♣ró①✐♠♦s ❝✐♥❝♦ ❞í❣✐t♦s ✐❞❡♥t✐✜❝❛♠ ♦ ♣r♦❞✉t♦ ❡ ♦ ú❧t✐♠♦✱ ❝♦♠♦ ❥á s❛❜❡♠♦s✱ é ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r ♦✉ ❞❡ ❝♦♥tr♦❧❡✳ ❖ ❝á❧❝✉❧♦ ❞♦ ❞í❣✐t♦ ❞❡ ✈❡r✐✜❝❛çã♦ é ❢❡✐t♦ ❞❡ ✉♠❛ ❢♦r♠❛ ❞✐❢❡r❡♥t❡✳ ❈♦♥s✐❞❡r❡ ♦ ❝ó❞✐❣♦ ■❙❇◆ XXX −Y Y −ZZZ −Y Y Y Y −W ✳ ❉✐str✐❜✉❛ ♦s ❞♦③❡ ♣r✐♠❡✐r♦s ❞í❣✐t♦s a1 ✱ a2 ✱ a3 ✱ a4 ✱ a5 ✱ a6 ✱ a7 ✱ a8 ✱ a9 ✱ a10 ✱ a11 ❡ a12 ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✶✷✳ ✾✹

(302) a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 1 3 1 3 1 3 1 3 1 3 1 3 1a1 3a2 1a3 3a4 1a5 3a6 1a7 3a8 1a9 3a10 1a11 3a12 ❚❛❜❡❧❛ ✸✳✶✷✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 12 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 1a1 + 3a2 + 1a3 + 3a4 + 1a5 + 3a6 + 1a7 + 3a8 + 1a9 + 3a10 + 1a11 + 3a12 . ❖ r❡s✉❧t❛❞♦ ❞❛ s♦♠❛ S é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 10✱ s❡♥❞♦ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ❝♦♠ 0 ≤ resto ≤ 9✳ ❖ r❡st♦ s❡rá s✉❜tr❛í❞♦ ❞❡ 10✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s✉❜tr❛çã♦ ❞❡t❡r♠✐♥❛ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r a13 ✳ ❊ss❡ ❞í❣✐t♦ é ♦ ♠❡♥♦r ✈❛❧♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ♣♦ssí✈❡❧✱ t❛❧ q✉❡ ❛♦ s❡r ❛❝r❡s❝❡♥t❛❞♦ à s♦♠❛ ♦❜t✐❞❛✱ ❣❡r❛ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ 10✱ ✐st♦ é✱ s❡ ❛ s♦♠❛ ♦❜t✐❞❛ é S ✱ ♦ ♥ú♠❡r♦ S + a13 ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ ❞❡ 10✱ ♦✉ s❡❥❛✱ S + a13 ≡ 0 mod 10✳ ❙❡♥❞♦ ♦ r❡st♦ ♦ ♥ú♠❡r♦ 0✱ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r s❡rá ♣ró♣r✐♦ ③❡r♦✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✜❝❛ ❞❡♠♦♥str❛❞♦ ❝♦♠♦ ♦ ISBN − 13 é ❣❡r❛❞♦✳ P❛r❛ ❡①❡♠♣❧✐✜❝❛r✱ ✈❛♠♦s ❡❢❡t✉❛r ❛ ✈❡r✐✜❝❛çã♦ ❞❡ s❡❣✉r❛♥ç❛ ❞❡ ✉♠ ❝ó❞✐❣♦ ❞❡ ✉♠ ❧✐✈r♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ 1❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ q✉❡ ❝♦♥s✐st❡ ❡♠ r❡❛❧✐③❛r ❛s ♦♣❡r❛çõ❡s ♠❛t❡♠át✐❝❛s ❛❜❛✐①♦ ♣❛r❛ ❝♦♥❢❡r✐r ♦ r❡s✉❧t❛❞♦ ❞❡❧❛s ❝♦♠ ♦ ❞í❣✐t♦ ❞❡ s❡❣✉r❛♥ç❛✳ ❋✐❣✉r❛ ✸✳✶✵✿ ❈♦♥s✐❞❡r❡ ♦ ❝ó❞✐❣♦ ■❙❇◆ 978−85−262−7731−1✳ ❉✐str✐❜✉❛ ♦s ❞♦③❡ ♣r✐♠❡✐r♦s ❞í❣✐t♦s 9✱ 7✱ 8✱ 8✱ 5✱ 3✱ 9✱ 9✱ 0✱ 2✱ 7 ❡ 5 ♥✉♠❛ t❛❜❡❧❛ ♦❜❡❞❡❝❡♥❞♦ ❛ ♦r❞❡♠✱ ❞❛ ❡sq✉❡r❞❛ ♣❛r❛ ❛ ❞✐r❡✐t❛✱ ❡ ♠✉❧t✐♣❧✐q✉❡ ♣❡❧♦s s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s {1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3}✱ ❝♦♠♦ ♠♦str❛ ❛ ❚❛❜❡❧❛ ✸✳✶✸✳ ❊♠ s❡❣✉✐❞❛✱ ❝❛❧❝✉❧❡ ♦ s♦♠❛tór✐♦ ❞❡ t♦❞❛s ❛s ♠✉❧t✐♣❧✐❝❛çõ❡s✱ q✉❡ r❡♣r❡s❡♥t❛r❡♠♦s ❛ss✐♠✿ S = 9 + 21 + 8 + 24 + 5 + 6 + 6 + 6 + 7 + 21 + 3 + 3 = 119. ❊ss❡ r❡s✉❧t❛❞♦ é ❡♥tã♦ ❞✐✈✐❞✐❞♦ ♣♦r 10✱ ♦ q✉♦❝✐❡♥t❡ é 11 ❡ ♦ r❡st♦ é 9✳ ❖ r❡st♦ ❞❛ ❞✐✈✐sã♦ é ❡♥tã♦ s✉❜tr❛í❞♦ ❞❡ 10✱ ♦✉ s❡❥❛✱ 10 − 9 = 1✱ ♦ r❡s✉❧t❛❞♦ ❞❡ss❛ s✉❜tr❛çã♦ ✾✺

(303) 9 7 8 8 5 2 6 2 7 7 3 1 1 3 1 3 1 3 1 3 1 3 1 3 9 21 8 24 5 6 6 6 7 21 3 3 ❚❛❜❡❧❛ ✸✳✶✸✿ Pr♦❞✉t♦ ❡♥tr❡ ♦s 12 ♣r✐♠❡✐r♦s ❞í❣✐t♦s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ♣❡s♦s ❞❡t❡r♠✐♥❛ ♦ ❞í❣✐t♦ ✈❡r✐✜❝❛❞♦r a13 = 1✳ ❉❡ss❛ ♠❛♥❡✐r❛✱ ✜❝❛ ✈❡r✐✜❝❛❞♦ q✉❡ ♦ ❝ó❞✐❣♦ r❡❛❧♠❡♥t❡ ❡stá ❝♦rr❡t♦✳ ✾✻

(304) ❆♣ê♥❞✐❝❡ ❆ ❇r❡✈❡ ❍✐stór✐❝♦ ❞❛ ❈r✐♣t♦❣r❛✜❛ ◆❡st❡ ❛♣ê♥❞✐❝❡✱ r❡❧❛t❡r❡♠♦s ✉♠❛ ❜r❡✈❡ ❤✐stór✐❛ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ❜❛s❡❛❞♦ ❡♠ ❛❧❣✉♥s ❛rt✐❣♦s ❝♦♠♦ ❬✶❪ ❡ ❬✼❪✳ ❙❡❣✉♥❞♦ ❡st✉❞♦s✱ ❛ ❝r✐♣t♦❣r❛✜❛ t❡✈❡ ✐♥í❝✐♦ ❝❡r❝❛ ❞❡ 1900a.C.✱ ♥♦ ❛♥t✐❣♦ ❊❣✐t♦✱ ♣❡❧♦ ❛rq✉✐t❡t♦ ❑❤♥✉♠❤♦t❡♣ ■■✱ q✉❡ s✉❜st✐t✉✐✉ tr❡❝❤♦s ❡ ♣❛❧❛✈r❛s ❞❡ ❞♦❝✉♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ♣♦r sí♠❜♦❧♦s ❞❡ ♠♦❞♦ ❛ ❞✐✜❝✉❧t❛r q✉❡ ❧❛❞rõ❡s ❝❤❡❣❛ss❡♠ ❛♦s t❡s♦✉r♦s ❞❡s❝r✐t♦s ♥❡ss❡s ❞♦❝✉♠❡♥t♦s✳ 600a.C.✱ ❊♠ ♦s ❤❡❜r❡✉s ❝r✐❛r❛♠ ❛❧❣✉♥s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s✱ ♥♦♠❡❛❞♦ ♣♦r ❆t❜❛s❤✱ q✉❡ ❝♦♥s✐st❡ ❞❡ ✉♠❛ tr♦❝❛ s✐♠♣❧❡s ❡♥tr❡ ❛s ❧❡tr❛s ❞♦ ❤❡❜r❛✐❝♦✱ ♣♦r ♦r❞❡♠ ✐♥✈❡rs❛✳ ❊♠ 480a.C.✱ ❍❡ró❞♦t♦✱ ♥♦ ❧✐✈r♦ ✏❆s ❍✐stór✐❛s✑✱ r❡❧❛t❛ ♠❡♥s❛❣❡♥s t❛t✉❛❞❛s ♥❛s ❝❛❜❡ç❛s r❛s♣❛❞❛s ❞❡ ❡s❝r❛✈♦s ♣❛r❛ s❡r❡♠ ❡s❝♦♥❞✐❞❛s ♣❡❧♦s ❝❛❜❡❧♦s✱ ♠❡♥s❛❣❡♥s ❡s❝r✐t❛s ❡♠ t❛❜✉❧❡t❛s✱ ♦❝✉❧t❛s s♦❜ ❝❛♠❛❞❛s ❞❡ ❝❡r❛✱ ♦✉ ♠❡s♠♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ t✐♥t❛s ✐♥✈✐sí✈❡✐s ♥❛ ❡s❝r✐t❛ s♦❜r❡ ❛ ❝❛s❝❛ ❞❡ ♦✈♦s ❝♦③✐❞♦s✱ ♠❡♥s❛❣❡♠ ❝♦❧♦❝❛❞❛ ❞❡♥tr♦ ❞♦ ❡stô♠❛❣♦ ❞❡ ❛♥✐♠❛✐s ❞❡ ❝❛ç❛✳ ❊ss❡s t✐♣♦s ❞❡ tr❛s♠✐ssã♦ ❞❡ ♠❡♥s❛❣❡♥s r❡❝❡❜❡♠ ♦ ♥♦♠❡ ❞❡ ❡st❡❣❛♥♦❣r❛✜❛✱ q✉❡ ❞✐❢❡r❡♥t❡♠❡♥t❡ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ❛ ♠❡♥s❛❣❡♠ ♠❛♥té♠ s✉❛ ❢♦r♠❛ ❡ ❜❛s❡✐❛✲s❡ ♥♦ ❢❛t♦ ❞❡ ✉♠ ✐♥t❡r❝❡♣t♦r ♥ã♦ s❛❜❡r ❞❛ ❡①✐stê♥❝✐❛ ❞❛ ♠❡♥s❛❣❡♠✳ ❊♠ 475a.C.✱ s✉r❣✐✉ ♦ ♣r✐♠❡✐r♦ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ❞❡ ✉s♦ ♠✐❧✐t❛r✱ ♦ ❙❝②t❛❧❡ ♦✉ ❇❛st✐ã♦ ❞❡ ▲✐❝✉r❣♦✱ q✉❡ é ✉♠ t✐♣♦ ❞❡ ❝ó❞✐❣♦ ❞❡ tr❛♥s♣♦s✐çã♦✱ ✉t✐❧✐③❛❞♦ ♣❡❧♦ ❣❡♥❡r❛❧ ❡s♣❛rt❛♥♦ P❛s❛♥✐✉s✱ q✉❡ ❝♦♥s✐st❡ ❡♠ ❡s❝r❡✈❡r ❛ ♠❡♥s❛❣❡♠ ♥✉♠❛ t✐r❛ ❡str❡✐t❛ ❞❡ ❝♦✉r♦ ♦✉ ♣❡r❣❛♠✐♥❤♦ q✉❛♥❞♦ ❡st❛ ❡stá ❡♥r♦❧❛❞❛ ❡♠ t♦r♥♦ ❞❡ ✉♠ ❜❛stã♦ ❞❡ ♠❛❞❡✐r❛✳ ❆ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧ é ❡s❝r✐t❛ ♥♦ s❡♥t✐❞♦ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❜❛stã♦ ❡✱ ♣♦rt❛♥t♦✱ q✉❛♥❞♦ ❛ t✐r❛ é ❞❡s❡♥r♦❧❛❞❛ ♦❜té♠✲s❡ ❛ ♠❡♥s❛❣❡♠ ❝♦❞✐✜❝❛❞❛✳ P❛r❛ ✈♦❧t❛r ❛ ♦❜t❡r ❛ ♠❡♥s❛❣❡♠ ♦r✐❣✐♥❛❧✱ ❞❡✈❡✲s❡ ❡♥r♦❧❛r ♦✉tr❛ ✈❡③ ❛ t✐r❛ ♥✉♠ ❜❛stã♦ ❝♦♠ ♦ ♠❡s♠♦ ♣❡rí♠❡tr♦ ❡ ❢♦r♠❛✳ ❆♣r♦①✐♠❛❞❛♠❡♥t❡ 300a.C.✱ s✉r❣✐✉✱ ♥❛ ❮♥❞✐❛✱ ✉♠ ❧✐✈r♦ ✐♥t✐t✉❧❛❞♦ ❆rt❤❛s❛str❛ ❛tr✐❜✉í❞♦ ❛ ❑❛✉t✐❧②❛ ♦♥❞❡ sã♦ r❡❢❡r✐❞♦s ♦s ♣r✐♠❡✐r♦s ♠ét♦❞♦s ❞❡ ❝r✐♣t♦❛♥á❧✐s❡✳ ❆❧❣✉♠ t❡♠♣♦ ❞❡♣♦✐s✱ s✉r❣✐✉ ♦ ❝ó❞✐❣♦ ❞❡ ❞❡s❧♦❝❛♠❡♥t♦ ❝r✐❛❞♦ ♣♦r ❏ú❧✐♦ ❈és❛r q✉❡ ❝♦♥s✐st✐❛ ❡♠ s✉❜st✐t✉✐r ❝❛❞❛ ❧❡tr❛ ♣❡❧❛ ❧❡tr❛ q✉❡ s❡ ❡♥❝♦♥tr❛ três ♣♦s✐çõ❡s ❞❡♣♦✐s ♥♦ ❛❧❢❛❜❡t♦✳ ❊♠ 1466✱ ▲❡♦♥ ❇❛tt✐st❛ ❆❧❜❡rt✐✱ ❡s❝r❡✈❡✉ ✉♠ ❡♥s❛✐♦✱ ♥♦ q✉❛❧ ♠❡♥❝✐♦♥❛ ✉♠ ❝ó❞✐❣♦ ❡♠ ❞✐s❝♦✱ ❝r✐❛♥❞♦ ❛ ♥♦çã♦ ❞❡ ❝ó❞✐❣♦ ♣♦❧✐❛❧❢❛❜ét✐❝♦✳ ❊♠ 1553✱ ●✐♦✈❛♥ ❇❛t✐st❛ ❇❡❧❛s♦ ✐♥✈❡♥t♦✉ ✉♠ s✐st❡♠❛ ❝r✐♣t♦❣rá✜❝♦ ♣♦❧✐❛❧❢❛❜ét✐❝♦ ❝❤❛♠❛❞♦ ❞❡ ❝ó❞✐❣♦ ❞❡ ❱✐❣❡♥èr❡✱ ♣♦r t❡r s✐❞♦ ❡rr♦♥❡❛♠❡♥t❡ ❛tr✐❜✉í❞♦ ❛ ❇❧❛✐s❡ ❞❡ ✾✼

(305) ❱✐❣❡♥èr❡ ❞✉r❛♥t❡ ♦ sé❝✉❧♦ ❳■❳✳ ❊ss❡ s✐st❡♠❛ é ❜❛s❡❛❞♦ ♥ã♦ ❛♣❡♥❛s ❡♠ ✉♠✱ ♠❛s s✐♠ ❡♠ 26 ❛❧❢❛❜❡t♦s ❝♦❞✐✜❝❛❞♦s✳ ❯t✐❧✐③❛ ♦ ❝ó❞✐❣♦ ❞❡ ❈és❛r ❞❡ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ♣❛r❛ ❝❛❞❛ ❛❧❢❛❜❡t♦✱ ❢♦r♠❛♥❞♦ ❛ss✐♠ ✉♠❛ t❛❜❡❧❛✱ ❝❤❛♠❛❞❛ ❞❡ ◗✉❛❞r❛❞♦ ❞❡ ❱✐❣❡♥èr❡✳ ❊ss❡ ❝ó❞✐❣♦ ❢♦✐ ❝♦♥s✐❞❡r❛❞♦ ✐♥❞❡❝✐❢rá✈❡❧ ❞✉r❛♥t❡ ♠✉✐t♦ t❡♠♣♦✱ s❡♥❞♦ q✉❡❜r❛❞♦ s♦♠❡♥t❡ ❡♠ 1854✳ ❉✉r❛♥t❡ ♦s sé❝✉❧♦s ❳❱■■■ ❡ ❳■❳✱ ❛ss✐st✐✉✲s❡ à ♣r♦❧✐❢❡r❛çã♦ ❞❡ ❈â♠❡r❛s ❊s❝✉r❛s✱ ❣❛❜✐♥❡t❡s ❞❡ ❡s♣✐♦♥❛❣❡♠✱ ♦♥❞❡ s❡ ✉t✐❧✐③❛✈❛ ❛ ❝r✐♣t♦❧♦❣✐❛ ♣❛r❛ ✜♥s ♠✐❧✐t❛r❡s ❡ ✜♥s ❝✐✈✐s✱ ♥♦♠❡❛❞❛♠❡♥t❡ ♣❛r❛ ❞❡❝♦❞✐✜❝❛r ♠❡♥s❛❣❡♥s ❞✐♣❧♦♠át✐❝❛s✳ ❉✉r❛♥t❡ ❛ Pr✐♠❡✐r❛ ●✉❡rr❛ ▼✉♥❞✐❛❧ ❛ss✐st❡✲s❡ ❛ ✉♠❛ ♣r♦❧✐❢❡r❛çã♦ ❞❡ s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ♣❛r❛ ✉s♦s ♠✐❧✐t❛r❡s✳ ❈♦♠♦ ❡①❡♠♣❧♦s✱ t❡♠♦s ♦ P❧❛②❢❛✐r ❡ ♦ ❆❉❋●❱❳✳ ❆♣ós ❡st❛ ❣✉❡rr❛ ❝♦♠❡ç❛♠ ❛ ❛♣❛r❡❝❡r ❛s ♣r✐♠❡✐r❛s ♠áq✉✐♥❛s ❝✐❢r❛♥t❡s q✉❡ ✉s❛♠ r♦t♦r❡s ♠❡❝â♥✐❝♦s✳ ❊♠ 1923✱ ❆rt❤✉r ❙❝❤❡r❜✐✉s ❞❡s❡♥✈♦❧✈❡ ♦ ❊◆■●▼❆✱ ♠áq✉✐♥❛ ❝✐❢r❛♥t❡ ✉t✐❧✐③❛❞❛ ♣❡❧♦s ❛❧❡♠ã❡s ❞✉r❛♥t❡ ❛ ❙❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧ ♣❛r❛ ❝♦♠✉♥✐❝❛çõ❡s ❝♦♠ ♦s s✉❜♠❛r✐♥♦s ❡ ♣❛r❛ ❞❡s❧♦❝❛r ❛s s✉❛s tr♦♣❛s✳ ❊♠ 1976✱ ❲❤✐t✜❡❧❞ ❉✐✣❡ ❡ ▼❛rt✐♥ ❍❡❧❧♠❛♥ ♣✉❜❧✐❝❛♠ ♦ ❛rt✐❣♦ ✏◆❡✇ ❉✐r❡❝t✐♦♥s✐♥ ❈r②♣t♦❣r❛♣❤✑✱ ♦♥❞❡ ✐♥tr♦❞✉③❡♠ ❛ ✐❞❡✐❛ ❞❡ ❝r✐♣t♦❣r❛✜❛ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ♥❡st❡ ❝❛s♦ ❜❛s❡❛❞❛ ♥♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✱ ❡ ❛✈❛♥ç❛♠ ❝♦♠ ❛ ✐❞❡✐❛ ❞❡ ❛✉t❡♥t✐❝❛çã♦ ✉t✐❧✐③❛♥❞♦ ❢✉♥çõ❡s ❞❡ ✉♠ só s❡♥t✐❞♦ ✭♦♥❡ ✇❛② ❢✉♥❝t✐♦♥s✮✳ ■♥s♣✐r❛❞♦s ♣♦r ❛q✉❡❧❡ ❛rt✐❣♦✱ ❘♦♥❛❧❞ ▲✳ ❘✐✈❡st✱ ❆❞✐ ❙❤❛♠✐r ❡ ▲❡♦♥❛r❞ ▼✳ ❆❞❧❡♠❛♥✱ ❞❡s❡♥✈♦❧✈❡♠ ✉♠ ❝ó❞✐❣♦ ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✱ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❘❙❆✱ q✉❡ t❛♠❜é♠ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❛ss✐♥❛t✉r❛s ❞✐❣✐t❛✐s✱ ❜❛s❡❛❞♦ ♥♦ ❝♦♥tr❛st❡ ❡♥tr❡ ❛ ❞✐✜❝✉❧❞❛❞❡ ❞❡ ❢❛t♦r✐③❛r ♥ú♠❡r♦s ❣r❛♥❞❡s ❡ ❛ r❡❧❛t✐✈❛ ❢❛❝✐❧✐❞❛❞❡ ❞❡ ✐❞❡♥t✐✜❝❛r ♥ú♠❡r♦s ♣r✐♠♦s ❣r❛♥❞❡s✳ ❊♠ 1984✱ ❚❛❤❡r ❊❧❣❛♠❛❧ ❞❡s❡♥✈♦❧✈❡ ♦ s✐st❡♠❛ ❊❧●❛♠❛❧ t❛♠❜é♠ ✉t✐❧✐③❛♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞♦ ❧♦❣❛r✐t♠♦ ❞✐s❝r❡t♦✳ ◆♦s ❛♥♦s ✾✵ ❛♣❛r❡❝❡♠ ❞✐✈❡rs♦s s✐st❡♠❛s ❝r✐♣t♦❣rá✜❝♦s ❡♠ ♣❛rt✐❝✉❧❛r ♦ ■❉❊❆ ✭■♥t❡r♥❛t✐♦♥❛❧ ❉❛t❛ ❊♥❝r②♣t✐♦♥ ❆❧❣♦r✐t❤♠✮ ❞❡ ❳✉❡❥✐❛ ▲❛✐ ❡ ❏❛♠❡s ▼❛ss❡②✱ q✉❡ ♣r❡t❡♥❞❡ s❡r ✉♠ s✉❜st✐t✉t♦ ❞♦ ❉❊❙✳ ❆ ❝r✐♣t♦❣r❛✜❛ q✉â♥t✐❝❛ é ✐♥tr♦❞✉③✐❞❛ ❡♠ 1990✳ 1991✱ ❛✐♥❞❛ ❖ P●P ✭Pr❡tt② ●♦♦❞ Pr✐✈❛❝②✮ ❞❡ P❤✐❧ ❩✐♠♠❡r♠❛♥♥✱ ❞❡s❡♥✈♦❧✈✐❞♦ ❡♠ é ✉♠ ❞♦s ♣r♦❣r❛♠❛s ♠❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ ♣r♦t❡❣❡r ❛ ♣r✐✈❛❝✐❞❛❞❡ ❞♦ ❡✲♠❛✐❧ ❡ ❞♦s ❛rq✉✐✈♦s ❣✉❛r❞❛❞♦s ♥♦ ❝♦♠♣✉t❛❞♦r ❞♦ ✉t✐❧✐③❛❞♦r✳ ❊♠ 1997✱ ♦ ◆■❙❚ s♦❧✐❝✐t♦✉ ♣r♦♣♦st❛s ♣❛r❛ ❛ s✉❜st✐t✉✐çã♦ ❞♦ ❉❊❙✳ ❊♠ 2000✱ ♦ ◆■❙❚ ❡s❝♦❧❤❡✉ ♦ ❘✐❥♥❞❛❡❧ ✭❞❡ ❡♥tr❡ ♦s ✜♥❛❧✐st❛s ❡st❛✈❛ ▼❆❘❙ ❞❛ ■❇▼✱ ❘❈✻ ❞❡ ❘❙❆ ▲❛❜♦r❛t♦r✐❡s✱ ❘✐❥♥❞❛❡❧ ❞❡ ❏♦❛♥ ❉❛❡♠❡♥ ❡ ❱✐♥❝❡♥t ❘✐❥♠❡♥✱ ❙❡r♣❡♥t ❞❡ ❆♥❞❡rs♦♥✱ ❇✐❤❛♠ ❡ ❑♥✉❞s❡♥✱ ❡ ♦ t✇♦✜s❤ ❞❡ ❇r✉❝❡ ❙❝❤♥❡✐❡r ❡ s✉❛ ❡q✉✐♣❡✮✱ ♣❛r❛ s❡r ♦ ♥♦✈♦ ❆❊❙ ✭❆❞✈❛♥❝❡❞ ❊♥❝r②♣t✐♦♥ ❙t❛♥❞❛r❞✮✳ ❙ó ❡♠ 2005 é q✉❡ ♦ ◆■❙❚ ✭◆❛t✐♦♥❛❧ ■♥st✐t✉t❡ ♦❢ ❙t❛♥❞❛r❞s ❛♥❞ ❚❡❝❤♥♦❧♦❣②✮✱ q✉❡ s✉❜st✐t✉✐✉ ♦ ◆❇❙✱ ♣✉❜❧✐❝❛ ✉♠ ♣❧❛♥♦ ❞❡ tr❛♥s✐çã♦ ❝♦♠ ❛ ❞✉r❛çã♦ ❞❡ ❞♦✐s ❛♥♦s✱ ♣❛r❛ q✉❡ ❛s 10 ✉t✐❧✐③❛r ♦ ❉❊❙ ❡ ♣❛ss❛ss❡♠ ❛ ✉t✐❧✐③❛r ♦ ❆❊❙✳ ✾✽ ❛❣ê♥❝✐❛s ❣♦✈❡r♥❛♠❡♥t❛✐s ❞❡✐①❛ss❡♠ ❞❡

(306) ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ❈r✐♣t♦❣r❛✜❛ ❡ ❙❡❣✉r❛♥ç❛✱ ❬✶❪ ❆❧♠❡✐❞❛✱ P✳ ❏✳✱ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❆✈❡✐r♦ ✭✷✵✶✷✮✳ ❬✷❪ ❆♥t♦♥✱ ❍✳ ❡ ❘♦rr❡s✱ ❈✳ ➪❧❣❡❜r❛ ▲✐♥❡❛r ❝♦♠ ❛♣❧✐❝❛çõ❡s✳ ❊❞✐t♦r❛ ❇♦♦❦♠❛♥✱ ✽❛ ❊❞✳✱ P♦rt♦ ❆❧❡❣r❡✱ ✭✷✵✵✶✮✳ ❬✸❪ ❇♦❧❞r✐♥✐✱ ❏✳ ▲✳✱ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ❊❞✐t♦r❛ ❍❛r❜r❛ ❧t❞❛✳✱ ✸❛ ❊❞✳✱ ❙ã♦ P❛✉❧♦✱ ✭✶✾✽✵✮✳ ❆❧✐❝❡ ❡ ❇♦❜✳ ❬✹❪ ❈❘❆❚❖✱ ◆✱✳ ❊①♣r❡ss♦ ✴ ❘❡✈✐st❛✱ 22 ❞❡ ❙❡t❡♠❜r♦✱ ♣♣✳ 120, (2001)✳ ❬✺❪ ❉♦♠✐♥❣✉❡s✱ ❍②❣✐♥♦ ❍✱ ❋✉♥❞❛♠❡♥t♦s ❞❡ ❛r✐t♠ét✐❝❛✱ 118 − ❊❞✐t♦r❛ ❆t✉❛❧✱ ❙ã♦ P❛✉❧♦✱ ✭✶✾✾✶✮✳ ❬✻❪ ❋❡rr❡✐r❛✱ ❆✳ ❇✳ ❞❡ ❍✳✱ ▼✐♥✐❛✉ré❧✐♦ ❙é❝✉❧♦ ❳❳■✳ ❊❞✐t♦r❛ ◆♦✈❛ ❋r♦♥t❡✐r❛✳✱ ❊❞✐çã♦ ❡s♣❡❝✐❛❧ ♣❛r❛ ♦ ❋◆❉❊✴P◆▲❉✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ✭✷✵✵✹✮✳ ❬✼❪ ❋r❡✐r❡✱ P✳ ❇✳ ❡ ❈❛st✐❧❤♦✱ ❏✳ ❊✳✱ ❆ ♠❛t❡♠át✐❝❛ ❞♦s ❝ó❞✐❣♦s ❝r✐♣t♦❣rá✜❝♦s✱ ❯♥✐✈❡rs✐❞❛❞❡ ❈❛tó❧✐❝❛ ❞❡ ❇r❛sí❧✐❛ ✭✷✵✵✺✮✳ ❬✽❪ ❋♦♥s❡❝❛✱ ❘✉❜❡♥s ❱✐❧❤❡♥❛✳✱ ❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❯❊P❆ ✴ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❙♦❝✐❛✐s ❡ ❊❞✉❝❛çã♦✱ ❇❡❧é♠✱ ✭✷✵✶✶✮✳ ❊❧❡♠❡♥t♦s ❞❡ ❛r✐t♠ét✐❝❛✱ ❬✾❪ ❍❡❢❡③✱ ❆✳✱ ❛ ❈♦❧❡çã♦ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✳ ❊❞✐t♦r❛ ❙❇▼✱ ✷ ❊❞✳✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ✭✷✵✵✻✮✳ ❬✶✵❪ ■❡③③✐✱ ● ❡ ❍❛③③❛♥✱ ❙✱ ❙❡q✉ê♥❝✐❛s✱ ♠❛tr✐③❡s✱ ❞❡t❡r♠✐♥❛♥t❡s ❡ s✐st❡♠❛s✱ ❛ ❈♦❧❡çã♦ ❋✉♥❞❛♠❡♥t♦s ❞❛ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✲ ❱♦❧✉♠❡ ✹✳ ❊❞✐t♦r❛ ❆t✉❛❧✱ ✻ ❊❞✳✱ ❙ã♦ P❛✉❧♦✱ ✭✷✵✵✶✮✳ ❬✶✶❪ P❛✐✈❛✱ ▼✳✱ ▼❛t❡♠át✐❝❛ ✲ ❱♦❧✉♠❡ ✷✳ ❊❞✐t♦r❛ ▼♦❞❡r♥❛✱ ✶❛ ❊❞✳✱ ❙ã♦ P❛✉❧♦✱ ✭✷✵✵✹✮✳ ❬✶✷❪ ▼✐❧✐❡s✱ ❈✳ P✳✱ ❆ ▼❛t❡♠át✐❝❛ ❞♦s ❈ó❞✐❣♦s ❞❡ ❇❛rr❛s✱ ❆rt✐❣♦✱ ❯❋●✭✷✵✵✻✮✳ ❖ q✉❡ é ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r❄ ✭❆ ♥♦çã♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ ❦ ❡ s✉❛s ❞✐✈❡rs❛s ❛♣❧✐❝❛çõ❡s ♥♦ ❝♦t✐❞✐❛♥♦✮ ✱ ❆rt✐❣♦✳ ❬✶✸❪ ❙á✱ ■✳ P✳ ❞❡✱ ❬✶✹❪ ❙❛♥t♦s✱ ❏✳ P✳ ❞❡ ❖✳✱ ■♥tr♦❞✉çã♦ à ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ❈♦❧❡çã♦ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛✳ ❆ss♦❝✐❛çã♦ ■♥st✐t✉t♦ ◆❛❝✐♦♥❛❧ ❞❡ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱✭✷✵✵✸✮✳ ✾✾

(307) ❬✶✺❪ ❙❤♦❦r❛♥✐❛♥✱ ❙✳✱ ❈r✐♣t♦❣r❛✜❛ ♣❛r❛ ✐♥✐❝✐❛♥t❡s✳ ❊❞✐t♦r❛ ❈✐ê♥❝✐❛ ▼♦❞❡r♥❛✱ ✷❛ ❊❞✳✱ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ✭✷✵✶✷✮✳ ❬✶✻❪ ◆❛s❝✐♠❡♥t♦✱ ❉✳ ❈✳ ❞♦✱ ❙❛♥t♦s✱ ▲✳ ❉✳ ❞♦s✱ P✐r❡s✱ ▲✳ ❱✳✱ ❖❧✐✈❡✐r❛✱ ❘✳ ❞❡✱ ❡ ❙❛♥t♦s✱ ❲✳ ❘✳✱ ❈◆P❏ ❄ ❈❛❞❛str♦ ◆❛❝✐♦♥❛❧ ❉❡ P❡ss♦❛ ❏✉rí❞✐❝❛✱ ❚r❛❜❛❧❤♦ ❝✐❡♥tí✜❝♦ ❛♣r❡s❡♥t❛❞♦ à ❞✐s❝✐♣❧✐♥❛ ❞❡ ❆❞♠✐♥✐str❛çã♦ ❞❡ s✐st❡♠❛ ❞❡ ✐♥❢♦r♠❛çã♦ ❣❡r❡♥❝✐❛❧✳ ❈❯❘■❚■❇❆ ✭✷✵✵✼✮✳ ❬✶✼❪ ❤tt♣✿✴✴✇✇✇✳✐♠♣♦st♦❞❡r❡♥❞❛✳♥❡t✴❤✐st♦r✐❛✴❤✐st♦r✐❛✲❞♦✲✐♠♣♦st♦✲❞❡✲ r❡♥❞❛✲♥♦✲❜r❛s✐❧✴✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 14/01/2014✳ ❖r❣❛♥✐③❛❞❛ ♣♦r ❙❛♥❞r❛ ✭✷✵✶✶✮✳ ❬✶✽❪ ❤tt♣✿✴✴✇✇✇✳❣❡r❛r❞♦❝✉♠❡♥t♦s✳❝♦♠✳❜r✴❄♣❣❂❡♥t❡♥❞❛✲❛✲❢♦r♠✉❧❛✲❞♦✲❝♣❢✱ ❝♦♥s✉❧t❛❞❛ ❡♠ 06/01/2014✳ ♣á❣✐♥❛ ❬✶✾❪ ❤tt♣✿✴✴✇✇✇✳❣❡r❛❞♦r❝♣❢✳❝♦♠✴✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 06/01/2014✳ ❬✷✵❪ ❤tt♣✿✴✴✇✇✇✳♣♦rt❛❧✳❢❛♠❛t✳✉❢✉✳❜r✴s✐t❡s✴❢❛♠❛t✳✉❢✉✳❜r✴✜❧❡s✴❆♥❡①♦s✴❇♦♦❦♣❛❣❡✴❢❛♠❛t✲ r❡✈✐st❛✲✶✸✲❛rt✐❣♦✲✸✲✵✳♣❞❢✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 14/01/2014✳ ❬✷✶❪ ❤tt♣✿✴✴✇✇✇✳♣♦rt❛❧✳❢❛♠❛t✳✉❢✉✳❜r✴s✐t❡s✴❢❛♠❛t✳✉❢✉✳❜r✴✜❧❡s✴❆♥❡①♦s✴❇♦♦❦♣❛❣❡✴❢❛♠❛t✲ r❡✈✐st❛✲✶✸✲❛rt✐❣♦✲✹✲✵✳♣❞❢✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 14/01/2014✳ ❬✷✷❪ ❤tt♣✿✴✴✇✇✇✳❛❝❡ss❛✳❝♦♠✴t❡❝♥♦❧♦❣✐❛✴❛rq✉✐✈♦✴s✉♣♦rt❡✴✷✵✶✶✴✵✷✴✶✻✲❝♣❢✴✱ ❝♦♥s✉❧t❛❞❛ ❡♠ 14/01/2014✳ ♣á❣✐♥❛ ❬✷✸❪ http : //www.gs1br.org/main.jsp?lumChannelId = 40288176383AC689013847EF 03635302✱ ✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 12/10/2014✳ ❖r❣❛♥✐③❛❞❛ ♣♦r ❆ss♦❝✐❛çã♦ ❇r❛s✐❧❡✐r❛ ❞❡ ❆✉t♦♠❛çã♦✳ ❬✷✹❪ http : //www.geradorcnpj.com/algoritmo − do − cnpj.htm✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 24/02/2014✳ ❬✷✺❪ ❤tt♣✿✴✴✇✇✇✳♥✉♠❛❜♦❛✳❝♦♠✳❜r✴❝r✐♣t♦❣r❛✜❛✴❝✐❢r❛s✴s✉❜st✐t✉✐❝♦❡s✴♠♦♥♦❛❧❢❛❜❡t✐❝❛s✴ s✐♠♣❧❡s✴✶✻✺✲❈♦❞✐❣♦✲❞❡✲❈❡s❛r✱ ♣á❣✐♥❛ ❝♦♥s✉❧t❛❞❛ ❡♠ 29/07/2014✳ ✶✵✵

(308)

Novo documento