ARITMÉTICA MODULAR, CÓDIGOS ELEMENTARES E CRIPTOGRAFIA

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■P❊

❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❊ ❚❊❈◆❖▲❖●■❆

❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲

❆r✐t♠ét✐❝❛ ▼♦❞✉❧❛r✱ ❈ó❞✐❣♦s ❊❧❡♠❡♥t❛r❡s ❡

❈r✐♣t♦❣r❛✜❛

❘❡❣❡♥❡ ❈❤❛✈❡s P✐♠❡♥t❡❧ P❡r❡✐r❛ ❇❛rr❡t♦

❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❙❊❘●■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❘❖❋▼❆❚✮✳

❖❘■❊◆❚❆❉❖❘✿ Pr♦❢✳ ❉r✳ ❏✳ ❆♥❞❡rs♦♥ ❱❛❧❡♥ç❛ ❈❛r❞♦s♦

❊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♦✳

❋■❈❍❆ ❈❆❚❆▲❖●❘➪❋■❈❆ ❊▲❆❇❖❘❆❉❆ 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✉❧♦✳

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

❆❣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❛♣❛ ❝♦♥❝❧✉✐❞❛ ♥❛ ♠✐♥❤❛ ✈✐❞❛✳

❘❡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❞✐❣♦ ❊❧●❛♠❛❧✳

❆❜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 ❝♦❞❡✱ ▼❍ ❝♦❞❡✱ ❊❧●❛♠❛❧ ❝♦❞❡✳

❙✉♠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♦ ❞♦ Pr♦❜❧❡♠❛ ❞❛ ▼♦❝❤✐❧❛ ✲

❉❡❝♦❞✐✜❝❛♥❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺ ✷✳✽ ❈ó❞✐❣♦ ❊❧●❛♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✽

✷✳✽✳✶ ❊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 ✾✾

■♥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❡

❜❡♠ ♥♦ ❛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♥❞✐❝❡ ❆✮✳

❈❛♣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♣♦✐s6 = 2_×3❀ 5_|10♣♦✐s10 = 5_×2❡ 1_|a (_∀a _∈Z)

♣♦✐sa = 1_×a✳ ◆♦ ❡♥t❛♥t♦✱ 0∤b✱ ❝♦♠ b₆= 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♦sk1 ❡ k2 ❝♦♠

b =k1a ❡ c=k2b✳

❙✉❜st✐t✉✐♥❞♦ ♦ ✈❛❧♦r ❞❡b ♥❛ ❡q✉❛çã♦ c=k2b t❡r❡♠♦s

c=k2k1a✱

♦ q✉❡ ✐♠♣❧✐❝❛ ❡①✐st✐r k=k1k2 ✐♥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♦X s♦❜r❡

✐♥❝♦♠❡♥s✉rá✈❡✐s ❡ ♦s três ú❧t✐♠♦s tr❛t❛♠ s♦❜r❡ ❣❡♦♠❡tr✐❛ ♥♦ ❡s♣❛ç♦✳ ❆❧é♠ ❞✐ss♦✱ ❡♥❝♦♥tr❛♠♦s t❛♠❜é♠ ✉♠❛ ❡①♣♦s✐çã♦ ❞❛ t❡♦r✐❛ ❞❛s ♣r♦♣♦rçõ❡s ♥✉♠ér✐❝❛ ♦✉ ♣✐t❛❣ór✐❝❛✳

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=k1c ❡ b=k2c✳

▼✉❧t✐♣❧✐❝❛♥❞♦✲s❡ ❡st❛s ❞✉❛s ❡q✉❛çõ❡s r❡s♣❡❝t✐✈❛♠❡♥t❡ ♣♦rm ❡ n t❡r❡♠♦s ma=mk1c ❡ nb=nk2c✳

❙♦♠❛♥❞♦✲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) a₆= 0 ❡ ad_|an _⇒ d_|n❀

(iv) 1_|n❀

(v) n_|0❀

(vi) d_|n ❡ n₆= 0 _{⇒ |}d_{| ≤ |}n_|❀

(vii) d_|n ❡ n_|d _{⇒ |}d_|=_|n_|❀

(viii) d_|n ❡ d₆= 0 _⇒ (n/d)_|n✳

❉❡♠♦♥str❛çã♦✳ (i)✿ ❈♦♠♦ n= 1n s❡❣✉❡ ❞❛ ❞❡✜♥✐ç❛õ q✉❡ n_|n✳

(ii)✿ ❙❡ d_|n ❡♥tã♦ n = cd♣❛r❛ ❛❧❣✉♠ ✐♥t❡✐r♦ c✳ ▲♦❣♦ an= cad✱ ♦ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

(viii)✿ ❙❡ d_|n ❡♥tã♦ n = k1d ❡ ♣♦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✐❜✐❧✐❞❛❞❡✳

✶✳✷ ❆❧❣♦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_} ❡ B =_{2,4,6,8,10,_{· · · }} ♣♦ss✉❡♠ ❝♦♠♦ ♠❡♥♦r❡s ❡❧❡♠❡♥t♦s4 ❡ 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❚❡♦r❡♠❛ ✶✳✽✳ ❉❛❞♦s ❞♦✐s ✐♥t❡✐r♦s a ❡ d✱ d > 0✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ♣❛r ❞❡ ✐♥t❡✐r♦s q ❡ r t❛✐s q✉❡

a=qd+r, ❝♦♠ 0_≤r < d (r= 0 _⇔ d_|a) ✭✶✳✶✮

✭q é ❝❤❛♠❛❞♦ ❞❡ q✉♦❝✐❡♥t❡ ❡ r ❞❡ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡a ♣♦r d✮✳ ❉❡♠♦♥str❛çã♦✳

❊①✐stê♥❝✐❛✿ ❙❡❥❛ S ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ♥ã♦✲♥❡❣❛t✐✈♦s q✉❡ sã♦ ❞❛ ❢♦r♠❛ a₋dx✱ ❝♦♠ x_∈Z✱ ✐st♦ é✿

S=_{a₋dx:x_∈Z ❡ a₋dx_≥0_}.

❖ ❝♦♥❥✉♥t♦ S é ♥ã♦ ✈❛③✐♦✳ ❉❡ ❢❛t♦✱ s❡♥❞♦ d > 0✱ t❡♠♦s d _≥ 1 ❡✱ ♣♦rt❛♥t♦✱

❝♦♥s✐❞❡r❛♥❞♦x=_−|a_| ♦❜t❡♠♦s

a₋dx=a+d_|a_{| ≥}a+_|a_{| ≥}0.

▲♦❣♦✱a₋d(_−|a_|)_∈S✳ ❆❣♦r❛✱ s❡♥❞♦S ♥ã♦ ✈❛③✐♦✱ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✱ ❡①✐st❡ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ r ❞❡S ❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ q t❛❧ q✉❡

r_≥0 e r=a₋dq,

♦✉ s❡❥❛✱ a = dq+r ❝♦♠ ❛❧❣✉♠ q _∈ Z✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s r < d✳ ❉❡ ❢❛t♦✱ s❡ ❢♦ss❡ r_≥d t❡rí❛♠♦s

0_≤r₋d=a₋dq₋d=a₋d(q+ 1)< r.

❉❡ss❛ ❢♦r♠❛✱ ♦❜t❡rí❛♠♦s r₋d_∈S ❞❡ ♠♦❞♦ q✉❡ r ♥ã♦ s❡r✐❛ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ ❞❡ S✳ P♦rt❛♥t♦✱ t❡♠♦s ❣❛r❛♥t✐❞❛ ❛ ♣❛rt❡ ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ✭✶✳✶✮✳

❯♥✐❝✐❞❛❞❡✿ P❛r❛ ❞❡♠♦♥str❛r ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ q ❡ r✱ s✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡♠ ❞♦✐s ♦✉tr♦s ✐♥t❡✐r♦s q1 ❡ r1 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).

P♦r ♦✉tr♦ ❧❛❞♦✱ t❡♠♦s

−d <₋r_≤0 e 0 _≤r1 < d,

q✉❡ ✐♠♣❧✐❝❛

−d < r1−r < d,

♦✉ s❡❥❛✱

|r1−r|< d.

❆ss✐♠✱ d_|(r1−r)❡ |r1−r|< d✳ ▲♦❣♦✱ r1−r = 0✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ d 6= 0 ❡ ❛❣♦r❛

q1d =qd✱ s❡❣✉❡ q✉❡ q=q1✳ ▲♦❣♦✱ r1 =r ❡q1 =q✳

❈♦r♦❧ár✐♦ ✶✳✾ ✭❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ❞❡ ❊✉❝❧✐❞❡s✮✳ ❙❡a ❡ d sã♦ ❞♦✐s ✐♥t❡✐r♦s ❝♦♠ d₆= 0✱ ❡♥tã♦ ❡①✐st❡♠ ♥ú♠❡r♦s ✐♥t❡✐r♦s q ❡ r✱ ❡ sã♦ ú♥✐❝♦s✱ t❛✐s q✉❡

a=dq+r, 0_≤r <_|d_|.

❉❡♠♦♥str❛çã♦✳ ❙❡ d > 0✱ ❛ ❝♦♥❝❧✉sã♦ é ♦❜t✐❞❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✽✳ ❆❣♦r❛✱ s❡ d < 0

❡♥tã♦ |d_|>0✱ ♥♦✈❛♠❡♥t❡ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✽✱ ❡①✐st❡♠ ú♥✐❝♦s ✐♥t❡✐r♦s q1 ❡ r t❛✐s q✉❡

a =_|d_|q1+r, 0≤r <|d|.

◆❡st❡ ❝❛s♦✱ ♥♦t❡ q✉❡ |d_|=₋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 ♣♦rd✳

❊①❡♠♣❧♦ ✶✳✶✵✳ ❆❝❤❛r ♦ q✉♦❝✐❡♥t❡ q ❡ ♦ r❡st♦ r ♥❛ ❞✐✈✐sã♦ ❞❡ a=₋83 ♣♦r b = 12

q✉❡ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✳

❊❢❡t✉❛♥❞♦ ❛ ❞✐✈✐sã♦ ✉s✉❛❧ ❞♦s ✈❛❧♦r❡s ❛❜s♦❧✉t♦s ❞❡ a ❡ b✱ ♦❜t❡♠♦s

83 = 12_×6 + 11, ♦✉ ❛✐♥❞❛✱

−83 = 12_×(₋6)₋11.

❈♦♠♦ ♦ t❡r♠♦ r = ₋11 <0 ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ 0_≤ r <12✱ ❡♥tã♦ s♦♠❛♥❞♦ ❡

s✉❜tr❛✐♥❞♦ ♦ ✈❛❧♦rb = 12 ❛♦ 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✳

✶✳✸ ❖ ▼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♦ ♣r✐♠♦s ❡♥tr❡ s✐✳

❉❡✜♥✐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)✱dé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡a❡b❡✱ ♣❡❧❛ ❝♦♥❞✐çã♦

(2)✱dé ♦ ♠❛✐♦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✱ ❡♥tã♦ c ♣♦❞❡ s❡r ₋1,₋2,1,2✳ ◆♦ ❡♥t❛♥t♦✱ 2 é ♠á①✐♠♦ ❞✐✈✐s♦r

❝♦♠✉♠ ❞❡ 6❡ 8✳

▲♦❣♦✱ ♦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 ❡ 0❀ ✐ss♦ é

❛♣❡♥❛s ✉♠❛ ❝♦♥✈❡♥çã♦ ❛❞❡q✉❛❞❛✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s✿

• mdc(a,1) = 1❀

• s❡ a₆= 0✱ ❡♥tã♦ ♦mdc(a,0) =_|a_|❀

✷_{◆❛ ❧✐t❡r❛t✉r❛ é ❝♦♠✉♠ ✉s❛✲s❡ ❛ ♥♦t❛çã♦ ♣❛r❛mdc(a, b)s✐♠♣❧❡s♠❡♥t❡ ❝♦♠♦(a, b)}

• 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_|✳

❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛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_|✳

Pr♦♣♦s✐çã♦ ✶✳✶✹✳ ❙❡ a=bq+r ❡ d=mdc(a, b)✱ ❡♥tã♦d =mdc(b, r)✳ ❆❧é♠ ❞✐ss♦✱

s❡ d=mdc(b, r)✱ ❡♥tã♦ d=mdc(a, b)✳

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦ d = mdc(a, b)✱ ❡♥tã♦ d_|a ❡ d_|b✳ ❉❡ss❛ ú❧t✐♠❛ r❡❧❛çã♦ r❡s✉❧t❛ q✉❡d_|bq✳ ▲♦❣♦

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 =r1q2+r2 (0≤r2 < r1) ✭✶✳✷✮

r1 =r2q3+r3 (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−1q+rn

rn−₁ =r_nq_n+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)✿

q1 q2 q3 qn qn+1

a b r1 r2 . . . rn−₁ rn

r1 r2 r3 r4 0

❆ t❛❜❡❧❛ s❡ tr❛❞✉③ ♥❛ s❡❣✉✐♥t❡ r❡❣r❛✿ ♣❛r❛ s❡ ✏❛❝❤❛r✑ ♦ mdc(a, b)✱ ❞✐✈✐❞✐✲s❡ a ♣♦r b ❡ ❡♥❝♦♥tr❛✲s❡ ♦ ✏♣r✐♠❡✐r♦✑ r❡st♦ r1✳ ❖ ✏s❡❣✉♥❞♦✑ r❡st♦r2 é ♦❜t✐❞♦ ♣❡❧❛ ❞✐✈✐sã♦ ❞❡ b

♣♦rr1✳ ❖ t❡r❝❡✐r♦ r❡st♦r3 é ♦❜t✐❞♦ ♣❡❧❛ ❞✐✈✐sã♦ ❞❡r1 ♣♦rr2✱ ❡ ❛ss✐♠ s✉❝❡ss✐✈❛♠❡♥t❡

❛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 + ✷

6 = 2_×3 + 0

▲♦❣♦✱2 = mdc(630,22)✳ ❯s✉❛❧♠❡♥t❡ ♣r♦❝❡❞❡✲s❡ ❛ss✐♠✿ 28 1 1 1 3 630 22 14 8 6 2

14 8 6 ✷ 0

P♦rt❛♥t♦✱ ♦ mdc(630,22) = 2✳

❖ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s t❛♠❜é♠ ♣♦❞❡ s❡r ✉s❛❞♦ ♣❛r❛ ❛❝❤❛r ❡①♣r❡ssã♦ ❞♦ mdc(a, b) = rn ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✱ ♦✉ s❡❥❛✱ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r

♥ú♠❡r♦s ✐♥t❡✐r♦sx ❡ y t❛✐s q✉❡

mdc(a, b) =ax+by. ✭✶✳✸✮

P❛r❛ ❡♥❝♦♥tr❛r ♦s ♥ú♠❡r♦sx ❡ y ❜❛st❛ ❡❧✐♠✐♥❛r s✉❝❡ss✐✈❛♠❡♥t❡ ♦s r❡st♦s rn−₁, rn−₂, . . . , r₃, r₂, r₁

❊①❡♠♣❧♦ ✶✳✶✻✳ ❆❝❤❛r ❡①♣r❡ssã♦ ❞♦ mdc(630,22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ 630

❡ 22✳ ❈♦♠♦ ♥♦ ❊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❡r2 = 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♦ 2 = mdc(630,22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ 630 ❡ 22♥ã♦ é ú♥✐❝❛✳ ❖❜s❡r✈❡✱ ♣♦r ❡①❡♠♣❧♦✱ q✉❡ s♦♠❛♥❞♦ ❡ s✉❜tr❛✐♥❞♦ ♦ ♣r♦❞✉t♦630_×22

❛♦ 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♦2 = mdc(630,22) ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r

❞❡ 630 ❡ 22✳

❉❡✜♥✐çã♦ ✶✳✶✼✳ ❉♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦sa❡bs❡ ❞✐③❡♠ ♣r✐♠♦s ❡♥tr❡ s✐ s❡mdc(a, b) = 1✳ ◆❡st❡ ❝❛s♦ ❞✐③✲s❡ t❛♠❜é♠ q✉❡ a é ♣r✐♠♦ ❝♦♠ b ♦✉ ✈✐❝❡✲✈❡rs❛✳

❊①❡♠♣❧♦ ✶✳✶✽✳ ❉♦✐s ♥ú♠❡r♦s ❝♦♥s❡❝✉t✐✈♦s a ❡ a+ 1 sã♦ s❡♠♣r❡ ♣r✐♠♦s ❡♥tr❡ s✐✳

❉❡ ❢❛t♦✱ é ❝❧❛r♦ q✉❡1_|a ❡ 1_|(a+ 1)✳ ❆❣♦r❛✱ s❡ c_|a ❡ c_|(a+ 1)✱ ❡♥tã♦

c_|[(a+ 1)₋a], ♦✉ s❡❥❛✱c_|1✳

Pr♦♣♦s✐çã♦ ✶✳✶✾✳ ❙❡ d=mdc(a, b)✱ ❡♥tã♦ mdc(sa, sb) = _|s_|d✱ ♣❛r❛ t♦❞♦ s_∈Z✳ ❉❡♠♦♥str❛çã♦✳ ▼✉❧t✐♣❧✐q✉❡♠♦s ♣♦r |s_| ❝❛❞❛ ✉♠❛ ❞❛s ✐❣✉❛❧❞❛❞❡s ♦❜t✐❞❛s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♥♦ ♣r♦❝❡ss♦ ❞❛s ❞✐✈✐sõ❡s s✉❝❡ss✐✈❛s q✉❡ ❧❡✈❛ ❛d✱ ❛ ♣❛rt✐r ❞❡ _|a_| ❡|b_|✿

|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|r_n)q_n+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✐♦ ✶✳✷✵✳ ❙❡ a, b_∈ Z_{\ {}0_} ❡ d =mdc(a, b)✱ t❡♠♦s q✉❡ mdc(a/d, b/d) = 1✳

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ a/d ❡ b/d sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ ❉❡♠♦♥str❛çã♦✳ ❈♦♠♦

d=mdc(a, b) =mdc

da d, d

b d

=d_·mdc

a d,

b d

❡d₆= 0✱ ❡♥tã♦✿

mdc

a d,

b d

= 1.

❈♦r♦❧ár✐♦ ✶✳✷✶✳ ❙❡❥❛♠a, b, c_∈Z✳ ❙❡ a_|bc ❡ mdc(a, b) = 1✱ ❡♥tã♦ a_|c✳ ❉❡♠♦♥str❛çã♦✳ P♦r ❤✐♣ót❡s❡mdc(a, b) = 1. ❯s❛♥❞♦ ❛ Pr♦♣♦s✐çã♦ ✶✳✶✾✱ q✉❡

mdc(ac, bc) = _|c_|.

P♦r ❤✐♣ót❡s❡a_|bc✱ ❡ ♦❜✈✐❛♠❡♥t❡a_|ac✳ ❊♥tã♦✱ a_|mdc(ac, bc)✳ P♦rt❛♥t♦✱ a_|c✳

❈♦r♦❧ár✐♦ ✶✳✷✷✳ ❙❡❥❛♠ a, b, c _∈ Z✳ ❙❡ a ❡ b sã♦ ❞✐✈✐s♦r❡s ❞❡ c ❡ mdc(a, b) = 1✱

❡♥tã♦ ab_|c✳

❉❡♠♦♥str❛çã♦✳ ❉❡ mdc(a, b) = 1 ❞❡❝♦rr❡✱ ❡♠ ✈✐rt✉❞❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✾✱ q✉❡

mdc(ac, bc) = _|c_|✳ P♦r ❤✐♣ót❡s❡✱ a_|c❡ b_|c✳ ❆ss✐♠✱ ab_|cb ❡ ab_|ac. ▲♦❣♦ ab❞✐✈✐❞❡ mdc(ac, bc)✳ P♦rt❛♥t♦✱ ab_|c✳

❊①❡♠♣❧♦ ✶✳✷✸✳ P❛r❛ q✉❡ ✉♠ ♥ú♠❡r♦ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r6 é ♥❡❝❡ssár✐♦ ❡ s✉❢✉❝✐❡♥t❡

q✉❡ s❡❥❛ ❞✐✈✐sí✈❡❧ ♣♦r 2 ❡ ♣♦r 3 ♣♦✐s ♦ 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))

Pr♦✈❡♠♦s ❛ ♣r✐♠❡✐r❛ ❞❡ss❛s ✐❣✉❛❧❞❛❞❡s✳ ❙❡❥❛ d =mdc(a, b, c)✳ ❊♥tã♦ d_|a✱ d_|b ❡ d_|c✳ ❉❛s ❞✉❛s ♣r✐♠❡✐r❛s ❞❡ss❛s r❡❧❛çõ❡s s❡❣✉❡ q✉❡d_|mdc(a, b)✳ ❆ss✐♠✱

d_|mdc(a, b) ❡ d_|c.

❙❡❥❛✱ ❛❣♦r❛✱ k ✉♠ ❞✐✈✐s♦r ❞❡ d1 = mdc(a, b) ❡ ❞❡ c✳ ❈♦♠♦ d1|a ❡ d1|b✱ ♣❡❧❛

tr❛♥s✐t✐✈✐❞❛❞❡ ❝♦♥❝❧✉✐✲s❡ q✉❡ k_|a✱ k_|b ❡ k_|c✳ ▲♦❣♦ k_|d ♣♦✐s d = mdc(a, b, c)✳ ❆

❞❡♠♦♥str❛çã♦ ✜❝❛ ❝♦♠♣❧❡t❛ ❝♦♥s✐❞❡r❛♥❞♦✲s❡ ❛ ✉♥✐❝✐❞❛❞❡ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✳

❊①❡♠♣❧♦ ✶✳✷✹✳ ❆❝❤❡♠♦s ♦ mdc(6,8,20)✳ ❯s❛♥❞♦ ♦ ❆❧❣♦rít♠♦ ❞❡ ❊✉❝❧✐❞❡s t❡♠♦s 1 3

8 6 2

✷ 0

▲♦❣♦✱mdc(2,20) = 2✱ ♣♦✐s 2_|20✳ ❊♥tã♦✱

mdc(6,8,20) = 2.

✶✳✹ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠

◆♦ ❝❛s♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ a ❞✐✈✐❞✐r ✉♠ ✐♥t❡✐r♦ b ❞✐③❡♠♦s t❛♠❜é♠ q✉❡ ♦ ♥ú♠❡r♦b é ▼ú❧t✐♣❧♦ ❞❡ a✳

❉❡✜♥✐çã♦ ✶✳✷✺✳ ❯♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ m é ❞✐t♦ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b q✉❛♥❞♦✿

✭✶✮ a_|m ❡ b_|m❀

✭✷✮ ♣❛r❛ q✉❛❧q✉❡r k > 0✱ s❡ a_|k ❡ b_|k, ❡♥tã♦ m_|k✳

❖❜s❡r✈❡✲s❡ q✉❡✱ ❡♠ ❧✐♥❣✉❛❣❡♠ ❧✐t❡r❛❧✱ ❛ ❝♦♥❞✐çã♦ (1) ❞❛ ❞❡✜♥✐çã♦ ❞✐③ q✉❡ m é ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡aq✉❛♥t♦ ❞❡ b❀ ❡♥q✉❛♥t♦ q✉❡ ❛ ❝♦♥❞✐çã♦(2) ❞✐③ q✉❡ t♦❞♦ ♠ú❧t✐♣❧♦

♣♦s✐t✐✈♦ ❞❡ a ❡ ❞❡ b é t❛♠❜é♠ ♠ú❧t✐♣❧♦ ❞❡ m✱ q✉❡ ❝❛r❛❝t❡r✐③❛ ❛ ♥♦♠❡♥❝❧❛t✉r❛ ✏♠í♥✐♠♦✑✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ s❡ m ❡ n s❛t✐s❢❛③❡♠ ❛ ❞❡✜♥✐çã♦✱ ❡♥tã♦ ♣❡❧❛ ❝♦♥❞✐çã♦ ✭✷✮ ❞❛ ❞❡✜♥✐çã♦ t❡♠♦s m_|n ❡ n_|m✳ ▲♦❣♦✱ m = n ❡ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ s❡ ❡①✐st✐r✱ ❞❡✈❡ s❡r ú♥✐❝♦✳ ❯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);

• mmc(_|a_|,_|b_|) =mmc(a, b);

P♦r ❡①❡♠♣❧♦✱ s❡❥❛♠ a=₋6 ❡b= 8✳ ■♥❞✐❝❛♥❞♦ ♣♦rMx ♦ ❝♦♥❥✉♥t♦ ❞♦s ♠✉❧t✐♣❧♦s

❞❡x_∈Z✱ t❡♠♦s

M−₆ ={· · · ,−12,−6,0,6,12,18,24,· · · } ❡ 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❀

✷✮ s❡ −6_|n ❡ 8_|c✱ ❡♥tã♦ c ❞❡✈❡ s❡r 24,48,72, ...✳ ◆♦ ❡♥t❛♥t♦✱ 24é ♠❡♥♦r ♠ú❧t✐♣❧♦

❝♦♠✉♠ ♣♦s✐t✐✈♦ ❞❡ −6❡ 8✳

▲♦❣♦✱ ♦mdc(₋6,8) = 24✳

◗✉❛♥t♦ à ❡①✐stê♥❝✐❛ ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ ❝♦♥s✐❞❡r❡♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♦ ❝❛s♦ a= 0 ❡ b q✉❛❧q✉❡r✳ ❉❡✈❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡mmc(0, b) = 0✳ ❉❡ ❢❛t♦✿

• 0_|0 ❡ b_|0 ✭♥♦t❡ q✉❡0 =b0✳✮

• 0_|m′ ❡ 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♦✳ ❱❛♠♦s ♣r✐♠❡✐r♦ ❝♦♥s✐❞❡r❛r a, b_∈N ❡d =mdc(a, b)✳ ❉❡s❞❡ q✉❡ d_|a ❡d_|b✱ t❡♠✲s❡d_|ab✳ ❊♥tã♦✱ m=ab/d_∈N✱ ❛❧é♠ ❞❡ a/d, b/d _∈N✳ ❆ss✐♠✿

• ❝♦♠♦

ab d =

ab d =m, ❡♥tã♦ a_|m✳ ❆♥❛❧♦❣❛♠❡♥t❡ s❡ ♠♦str❛ q✉❡ b_|m✳

• ❙❡❥❛ m′ ✉♠ ♠ú❧t✐♣❧♦ ❞❡

a ❡ ❞❡ b✳ ▲♦❣♦✱ ❡①✐st❡♠ r, s _∈ Z t❛✐s q✉❡ m′

= ar ❡ m′

=bs✳ ❊♥tã♦ ar =bs❡✱ ♣♦rt❛♥t♦✱ a dr=

b ds.

❉❛í s❡❣✉❡ q✉❡ a/d❞✐✈✐❞❡ (b/d)s✳ ❈♦♠♦ mdc(a/d, b/d) = 1✱ t❡♠♦s q✉❡(a/d)_|s ✭❈♦r♦❧ár✐♦ ✶✳✷✶ ✲ Pr♦♣♦s✐çã♦ ✶✳✶✾✮✳ ❆ss✐♠✱

s= a

dt ♣❛r❛ ❛❧❣✉♠ t_∈Z✳ ❉❡s❞❡ q✉❡ m′

=bs✱ ♦❜t❡♠♦s m′

=ba dt=

ab

d t=mt, ♦✉ s❡❥❛✱ m_|m′

.

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✐♦ ✶✳✷✼✳ ❙❡ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦ mmc(a, b) =_|ab_|✳ ❉❡♠♦♥str❛çã♦✳ ❉❡ ❢❛t♦✱ ❝♦♠♦d=mdc(a, b) = 1✱ ❡♥tã♦ mmc(a, b) = _|ab_|✳ ❖❜s❡r✈❛çã♦ ✶✳✷✽✳ ❙❡❥❛♠ a, b_∈ N_{\ {}0_}✳ P❡❧♦ q✉❡ ❢♦✐ ✈✐st♦✱ ab

d = m ∈ Ma∩Mb✳ ▼❛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✳

❊①❡♠♣❧♦ ✶✳✷✾✳ ❱❛♠♦s ✉s❛r ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♣❛r❛ ❛❝❤❛r mmc(20,8)✳

❈❛❧❝✉❧❛♥❞♦ ♦ ♠❞❝✱ t❡♠♦s

2 2 20 8 ✹

4 0

.

❊♥tã♦

mmc(20,8) = 20×8 4 = 40.

Pr♦♣♦s✐çã♦ ✶✳✸✵✳ ❙❡ m = mmc(a, b)✱ ❡♥tã♦ mmc(sa, sb) = _|s_|m ♣❛r❛ q✉❛❧q✉❡r s_∈Z✳

❉❡♠♦♥str❛çã♦✳ ◗✉❛♥❞♦ 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♠

é ✈❡r❞❛❞❡✐r♦✳

❙✉♣♦♥❤❛♠♦s a, b❡ s ♥ã♦ ♥✉❧♦s✳ ❊♥tã♦✱ ♣❡❧❛ ❞✉❛s ♣r♦♣♦s✐çõ❡s ❛♥t❡r✐♦r❡s✿

mmc(sa, sb) = |sasb|

mdc(sa, sb) =

s2_|ab|

|s_{| ·}mdc(a, b) =

|sab_|

mdc(a, b) =|s| ·mmc(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❡♠♣❧♦✿