❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ ❉❊ ▼❆❘■◆●➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆
❈r✐♣t♦❣r❛✜❛ ❈❧áss✐❝❛✱ ▼❛tr✐③❡s ❡ ❚❡❝♥♦❧♦❣✐❛✳
●✐❧❜❡rt♦ ❆♣❛r❡❝✐❞♦ ❚❡♥❛♥✐
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❘♦❜❡rt♦ ❚❡✐①❡✐r❛ Pr✐♠♦
❯◆■❱❊❘❙■❉❆❉❊ ❊❙❚❆❉❯❆▲ ❉❊ ▼❆❘■◆●➪ ❈❊◆❚❘❖ ❉❊ ❈■✃◆❈■❆❙ ❊❳❆❚❆❙ ❉❊P❆❘❚❆▼❊◆❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆ ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆
❈r✐♣t♦❣r❛✜❛ ❈❧áss✐❝❛✱ ▼❛tr✐③❡s ❡ ❚❡❝♥♦❧♦❣✐❛✳
●✐❧❜❡rt♦ ❆♣❛r❡❝✐❞♦ ❚❡♥❛♥✐
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡✳
❖r✐❡♥t❛❞♦r
Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❘♦❜❡rt♦ ❚❡✐①❡✐r❛ Pr✐♠♦
❆❣r❛❞❡❝✐♠❡♥t♦s
➚ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛ ▲❡✐❧✐❛♥❡✱ à ♠✐♥❤❛ ❛♠❛❞❛ ✜❧❤❛ P❛✉❧❛ ❡ ❛ t♦❞♦s ♠❡✉s ❛♠✐❣♦s ♣♦r ♠❡ ❞❛r❡♠ ♦ s✉♣♦rt❡ ♥❡❝❡ssár✐♦ ♣❛r❛ ❡st❛ ❝❛♠✐♥❤❛❞❛✳
❆♦s ♠❡✉s ♣❛✐s✱ ♣♦r t❡r❡♠✲♠❡ ❞❛❞♦ ❡❞✉❝❛çã♦✱ ✈❛❧♦r❡s ❡ ♣♦r t❡r❡♠✲♠❡ ❡♥s✐♥❛❞♦ ❛ ❛♥❞❛r✳ ➚ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ ❞❡ ▼❛r✐♥❣á✱ s❡✉ ❝♦r♣♦ ❞♦❝❡♥t❡✱ ❞✐r❡çã♦ ❡ ❛❞♠✐♥✐str❛çã♦ q✉❡ ♦♣♦rt✉♥✐③❛r❛♠ ❛ ❥❛♥❡❧❛ ♣♦r ♦♥❞❡ ✈✐s❧✉♠❜r♦ ✉♠ ❤♦r✐③♦♥t❡ s✉♣❡r✐♦r✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢✳ ❉r✳ ▼❛r❝♦s ❘♦❜❡rt♦ ❚❡✐①❡✐r❛ Pr✐♠♦✱ ♣❡❧♦ ❣r❛♥❞❡ s✉♣♦rt❡ ♣r❡st❛❞♦✱ ♣❡❧♦s ♠♦♠❡♥t♦s ❞❡ r❡✢❡①ã♦ ♣r♦♣♦r❝✐♦♥❛❞♦s✱ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ ♣❡❧♦ ✐♥❝❡♥t✐✈♦✳ ❆♦ ❈♦♦r❞❡♥❛❞♦r ❞♦ P❘❖❋▼❆❚ ❞❛ ❯❊▼ Pr♦❢✳ ❉r✳ ▲❛❡rt❡ ❇❡♠♠ ♣❡❧♦ ✐♥❝❡♥t✐✈♦ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✳
❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚ ❞❛ ❯❊▼ ♣♦r ♠❡ ♣r♦♣♦r❝✐♦♥❛r❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♥ã♦ ❛♣❡♥❛s r❛❝✐♦♥❛❧✱ ♠❛s ❛ ♠❛♥✐❢❡st❛çã♦ ❞♦ ❝❛rát❡r ❡ ❛❢❡t✐✈✐❞❛❞❡s ❞❛ ❡❞✉❝❛çã♦ ♥♦ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦✳
❆♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ♣❡❧❛ ❛❥✉❞❛ ♥❛s ❞ú✈✐❞❛s ❡ ♣❡❧❛ ♠♦t✐✈❛çã♦ ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡str❛❞♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♠♦s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s ❡ ❞❡ ❝r✐♣t♦❛♥á❧✐s❡ ❝❧áss✐❝♦s ✉t✐❧✐✲ ③❛♥❞♦ á❧❣❡❜r❛ ♠❛tr✐❝✐❛❧✳ ❆♣ós ❢❛③❡r♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ à ❝r✐♣t♦❣r❛✜❛ ❝❧áss✐❝❛✱ ✐♥tr♦❞✉③✐♠♦s ♦s ♠ét♦❞♦s ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❝✐❢r❛s ❞❡ ❍✐❧❧✱ ♠♦str❛♠♦s ❝♦♠♦ ❝r✐♣t♦❣r❛❢❛r ❡ ❞❡s❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s ✉s❛♥❞♦ ❡ss❛s ❝✐❢r❛s✳ ❆✐♥❞❛✱ ♠♦str❛♠♦s q✉❛♥❞♦ ❡ ❝♦♠♦ é ♣♦ssí✈❡❧ q✉❡❜r❛r ❡ss❡s ♠ét♦❞♦s ❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❢❡rr❛♠❡♥t❛ ❝♦♠♣✉t❛❝✐♦♥❛❧✱ ❛✉①✐❧✐❛r ♥❛ r❡❛❧✐③❛çã♦ ❞❡ ❝á❧❝✉❧♦s ❜❛s❡❛❞♦ ♥❛ ❧✐♥❣✉❛❣❡♠ ❞❡ ♣r♦❣r❛♠❛çã♦ ❏✉❧✐❛ ❡ ♥♦ ❛♠❜✐❡♥t❡ ✐♥t❡r❛t✐✈♦ ❏✉❧✐❛❇♦①✳ ❆❧é♠ ❞✐ss♦✱ ❞❡st❛❝❛♠♦s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s ♠❛t❡♠át✐❝♦s q✉❡ ❥✉st✐✜❝❛♠ ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞❡ss❡s ♠ét♦❞♦s✱ t❛✐s ❝♦♠♦✿ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❝♦♥❣r✉ê♥❝✐❛s ❡ ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s✳
❆❜str❛❝t
❚❤✐s ✇♦r❦ ♣r❡s❡♥ts ❝❧❛ss✐❝ ❝r✐♣t♦❣r❛♣❤② ❛♥❞ ❝r②♣t❛♥❛❧②s✐s ♠❡t❤♦❞s ❜❛s❡❞ ♦♥ ♠❛tr✐① ❛❧❣❡❜r❛✳ ❆❢t❡r ❛ ❜r✐❡❢ ✐♥tr♦❞✉❝t✐♦♥ ❛❜♦✉t ❝❧❛ss✐❝ ❝r✐♣t♦❣r❛♣❤② ✇❡ ✐♥tr♦❞✉❝❡ t❤❡ ♠❡t❤♦❞s ❦♥♦✇♥ ❛s t❤❡ ❍✐❧❧✬s ❝✐♣❤❡rs✳ ❲❡ ❛❧s♦ s❤♦✇ ✇❤❡♥ ❛♥❞ ❤♦✇ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❜r❡❛❦ t❤✐s ♠❡t❤♦❞ ❛♥❞ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❝♦♠♣✉t✐♥❣ t♦♦❧ ❢♦r ❜❛s❡❞ ♦♥ ❏✉❧✐❛ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ❛♥❞ t❤❡ ✐♥t❡r❛❝t✐♦♥ ❡♥✈✐r♦♠❡♥t ❏✉❧✐❛❇♦①✳ ❋✉rt❤❡r♠♦r❡✱ ✇❡ ❢♦❝✉s t❤❡ ♠❛✐♥ ♠❛t❤❡♠❛t✐❝❛❧ r❡s✉❧ts t❤❛t ❥✉st✐❢② t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ t❤❡s❡ ♠❡t❤♦❞s✱ s✉❝❤ ❛s✿ ❞✐✈✐s✐❜✐❧✐t②✱ ❝♦♥❣r✉❡♥❝❡s ❛♥❞ ❧✐♥❡❛r ❝♦♥❣r✉❡♥❝❡s✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ❈♦♥❣r✉ê♥❝✐❛s ✷
✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✸ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✹ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✺ ❙✐st❡♠❛s ❞❡ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s 2×2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷ ■♥tr♦❞✉çã♦ à ❈r✐♣t♦❣r❛✜❛ ❈❧áss✐❝❛ ✶✾
✷✳✶ ❱✐sã♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✶ ▼ét♦❞♦s ❞❡ ❈❤❛✈❡ ❙✐♠étr✐❝❛ ❡ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✷ ❈✐❢r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✸ ❈♦♠♣r✐♠❡♥t♦ ❞❛ ❝❤❛✈❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✷ ❆❧❣✉♥s ❙✐st❡♠❛s ❈r✐♣t♦❣rá✜❝♦s ❈❧áss✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸ ❈r✐♣t♦❣r❛✜❛✱ ▼❛tr✐③❡s ❡ ❚❡❝♥♦❧♦❣✐❛ ✷✾
✸✳✶ ■♥tr♦❞✉çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✷ ❈♦♥❣r✉ê♥❝✐❛s ❡ ▼❛tr✐③❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ❈✐❢r❛ ❞❡ ❍✐❧❧ ❞❡ ♦r❞❡♠n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
✸✳✸✳✶ ❖ ♣r♦❝❡ss♦ ❞❡ ❡♥❝r✐♣t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸✳✷ ❖ ♣r♦❝❡ss♦ ❞❡ ❉❡s❝r✐♣t♦❣r❛✜❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✸✳✸ ◗✉❡❜r❛♥❞♦ ❛ ❈✐❢r❛ ❞❡ ❍✐❧❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✹ ❚❡❝♥♦❧♦❣✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸
✹ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✹✼
❆ ▲✐♥❣✉❛❣❡♠ ❏✉❧✐❛ ❡ ❏✉❧✐❛❇♦① ✹✽
❆✳✷✳✸ ▼❛tr✐③❡s ❡ ❈♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
■♥tr♦❞✉çã♦
P❡ss♦❛s s❡♠♣r❡ t✐✈❡r❛♠ ❢❛s❝✐♥❛çã♦ ❡♠ ♠❛♥t❡r ✐♥❢♦r♠❛çã♦ ❡s❝♦♥❞✐❞❛ ❞❡ ♦✉tr♦s✳ ◗✉❛♥❞♦ ❝r✐❛♥ç❛s✱ ♠✉✐t♦ ❞❡ ♥ós tí♥❤❛♠♦s ♠❛♥❡✐r❛s s❡❝r❡t❛s ❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ❝♦❞✐✜❝❛❞❛s ♣❛r❛ ♥♦ss♦s ❛♠✐❣♦s ♠❛♥t❡♥❞♦ s❡❣r❡❞♦ ❞❡ ✐r♠ã♦s✱ ♣❛✐s ❡ ♣r♦❢❡ss♦r❡s✳ ❆ ❤✐stór✐❛ ♥♦s ♠♦str❛ ♠✉✐t♦s ❡①❡♠♣❧♦s ❞❡ ♣❡ss♦❛s ❡ ♥❛çõ❡s q✉❡ t❡♥t❛r❛♠ ♠❛♥t❡r ✐♥❢♦r♠❛çõ❡s ❧♦♥❣❡ ❞♦ ❛❧❝❛♥❝❡ ❞❡ ✐♥✐♠✐❣♦s✳ ❯♠❛ ❞❛s ❢♦r♠❛s ❞❡ s❡ ❢❛③❡r ✐ss♦ é ❛ ❝♦❞✐✜❝❛çã♦ ❞❛ ♠❡♥s❛❣❡♠✳ ❈♦♠ ❛ ❡✈♦❧✉çã♦ ❞❛ s♦❝✐❡❞❛❞❡✱ ♥♦✈♦s ❡ ♠❛✐s s♦✜st✐❝❛❞♦s ♠ét♦❞♦s ❞❡ ♣r♦t❡❣❡r ✐♥❢♦r♠❛✲ çã♦ ❛tr❛✈és ❞❛ ❝♦❞✐✜❝❛çã♦ s✉r❣✐r❛♠✳ ❆s té❝♥✐❝❛s ♥❡❝❡ssár✐❛s ♣❛r❛ ❝♦❞✐✜❝❛r ✐♥❢♦r♠❛çõ❡s ♣❡rt❡♥❝❡♠ ❛♦ ❝❛♠♣♦ ❞❡ ❡st✉❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝r✐♣t♦❣r❛✜❛✳ ◆❛t✉r❛❧♠❡♥t❡✱ ❥✉♥t♦ ❝♦♠ ❛ ❝r✐♣t♦❣r❛✜❛ s✉r❣✐✉ t❛♠❜é♠ ♦ ❝❛♠♣♦ ❞❡ ❡st✉❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❝r✐♣t♦❛♥á❧✐s❡ q✉❡ ❝♦♥s✐st❡ ❥✉st❛♠❡♥t❡ ♥♦ ♣r♦❝❡ss♦ ✐♥✈❡rs♦ ❞❛ ❝r✐♣t♦❣r❛✜❛✱ ♦✉ s❡❥❛✱ ❛s té❝♥✐❝❛s ♥❡❝❡s✲ sár✐❛s ♣❛r❛ ❞❡s❝♦❜r✐r ❛s ✐♥❢♦r♠❛çõ❡s ✈❡r❞❛❞❡✐r❛s ❝♦♥t✐❞❛s ❡♠ ♠❡♥s❛❣❡♥s ❝♦❞✐✜❝❛❞❛s✳ ❚❡❝♥✐❝❛♠❡♥t❡✱ ❛ ❝r✐♣t♦❧♦❣✐❛ é ♦ t❡r♠♦ q✉❡ ❡♥❣❧♦❜❛ t♦❞♦ ♦ ❡st✉❞♦ s♦❜r❡ ❝♦♠✉♥✐❝❛çã♦ ❡♠ ❝❛♥❛✐s ♥ã♦ s❡❣✉r♦s ❡ s❡✉s ♣r♦❜❧❡♠❛s r❡❧❛❝✐♦♥❛❞♦s✳ ❆ ❝r✐♣t♦❧♦❣✐❛ ♠♦❞❡r♥❛ é ✉♠ ❝❛♠♣♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦ q✉❡ ❡①✐❣❡ ❢♦rt❡ ❞♦♠í♥✐♦ ❞❡ ❝♦♥t❡ú❞♦ ♠❛t❡♠át✐❝♦ ❡ ❝♦♠♣✉t❛✲ ❝✐♦♥❛❧✳
◆❡ss❡ tr❛❜❛❧❤♦✱ tr❛t❛r❡♠♦s ❞❛ ♣❛rt❡ ❞❛ ❝r✐♣t♦❧♦❣✐❛ ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❝❧áss✐❝❛✳ ❖s ♠é✲ t♦❞♦s ❝r✐♣t♦❣rá✜❝♦s ❡ ❞❡ ❝r✐♣t♦❛♥á❧✐s❡ ❝❤❛♠❛❞♦s ❝❧áss✐❝♦s ❢♦r❛♠ ❞❡s❡♥✈♦❧✈✐❞♦s ❛♥t❡s ❞❛ ❡r❛ ❞♦ ❝♦♠♣✉t❛❞♦r ♣❡ss♦❛❧ ✭♣♦r ✈♦❧t❛ ❞❡ ✶✾✼✵✮ ❡ ❝♦♥t✐♥✉❛♠ ❛✐♥❞❛ s❡♥❞♦ ❛♣❧✐❝❛❞♦s ❡♠ ✐♥ú♠❡r❛s s✐t✉❛çõ❡s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ tr❛❜❛❧❤❛r❡♠♦s ❝♦♠ ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s ❡ ❞❡ ❝r✐♣t♦❛♥á❧✐s❡ ❝♦♥❤❡❝✐❞♦s ❝♦♠♦ ❝✐❢r❛s ❞❡ ❍✐❧❧✱ q✉❡ tê♠ ♣♦r ❜❛s❡ tr❛♥s❢♦r♠❛çõ❡s ♠❛tr✐❝✐❛✐s✳ ❆♣r❡s❡♥t❛r❡♠♦s ✈ár✐♦s ❡①❡♠♣❧♦s q✉❡ ♣♦❞❡♠ s❡r ✉t✐❧✐③❛❞♦s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ❡ ❝♦♠♦ ♦ ✉s♦ ❞❡ ❢❡rr❛♠❡♥t❛s t❡❝♥♦❧ó❣✐❝❛s ♣♦❞❡ ♥♦s ❛✉①✐❧✐❛r ♥❡ss❛ t❛r❡❢❛✳
❯♠ ❞♦s r❡q✉✐s✐t♦s ❜ás✐❝♦s ♣❛r❛ ♦s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s sã♦ ❛s ✐❞❡✐❛s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ❝♦♥❣r✉ê♥❝✐❛s✳ ❆ ✐❞❡✐❛ ❜ás✐❝❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ é ❜❡♠ s✐♠♣❧❡s✳ ❉❛❞♦ ❞♦✐s ♥ú♠❡r♦ ✐♥t❡✐r♦s
a❡b✱ ❞✐③❡♠♦s q✉❡aé ❝♦♥❣r✉❡♥t❡ ❛b♠ó❞✉❧♦m❡ ❡s❝r❡✈❡♠♦sa≡b(mod m)✱ s❡m|(a−b)✳
❆ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m ♥♦s ♣❡r♠✐t❡ ❞❡✜♥✐r ♦ ❝♦♥❥✉♥t♦ Zm = {0,1,2, . . . , m−1}✱
❞❡♥♦♠✐♥❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦m✭ ♦s ❡❧❡♠❡♥t♦s ❞❡ss❡ ❝♦♥❥✉♥t♦
sã♦ ♦s ♣♦ssí✈❡✐s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ✐♥t❡✐r♦ a ♣♦r m✮✳ ◆♦ ❝❛♣ít✉❧♦ ✶ ❡st✉❞❛r❡♠♦s
♦s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ❝♦♥❣r✉ê♥❝✐❛s✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ❡ ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞♦s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s ❡ ❞❡
✶
❝r✐♣t♦❛♥á❧✐s❡✳
❈r✐♣t♦❣r❛✜❛ é ♦ ❡st✉❞♦ ❞♦s ♠ét♦❞♦s ❞❡ ❡♥✈✐❛r ♠❡♥s❛❣❡♥s ❞❡ ♠❛♥❡✐r❛ ❞✐s❢❛rç❛❞❛ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ s♦♠❡♥t❡ ♦ ❞❡st✐♥❛tár✐♦ ♦r✐❣✐♥❛❧ ♣♦ss❛ r❡♠♦✈❡r ♦ ❞✐s❢❛r❝❡ ❡ ❧❡r ❛ ♠❡♥s❛❣❡♠✳ ❈♦♠ ❛ ❡✈♦❧✉çã♦ ❞❛ s♦❝✐❡❞❛❞❡✱ ♠ét♦❞♦s ♠❛✐s s♦✜st✐❝❛❞♦s ❞❡ ♣r♦t❡çã♦ ❞❡ ❞❛❞♦s ❢♦r❛♠ ❝r✐❛❞♦s✳ ❆ ♠❛✐♦r✐❛ ❞♦s ♠ét♦❞♦s ❝r✐♣t♦❣rá✜❝♦s ❡ ❞❡ ❝r✐♣t♦❛♥á❧✐s❡ ❡♥✈♦❧✈❡♠ ❣r❛♥❞❡ q✉❛♥✲ t✐❞❛❞❡ ❞❡ ♠❛t❡♠át✐❝❛✱ s✐♠♣❧❡s ❡ ❝♦♠♣❧❡①❛✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛s ♥♦çõ❡s ❜ás✐❝❛s s♦❜r❡ ❝r✐♣t♦❣r❛✜❛ ❡ ❝r✐♣t♦❛♥á❧✐s❡✱ ❞❡✜♥✐♥❞♦ ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ❡ t❡r♠♦s ✉t✐❧✐③❛❞♦s ❡ ❛♣r❡s❡♥t❛♥❞♦ ❛❧❣✉♥s ♠ét♦❞♦s s✐♠♣❧❡s ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡ ❞❡ ❝r✐♣t♦❛♥á❧✐s❡✳ ❯♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❛ ❝r✐♣t♦❣r❛✜❛ ❡ ❞❛ ❝r✐♣t♦❛♥á❧✐s❡ ❝❧áss✐❝❛✱ ❛s ❝✐❢r❛s ❞❡ ❍✐❧❧✱ sã♦ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡s ❛♦ ✉t✐❧✐③❛r❡♠ á❧❣❡❜r❛ ♠❛tr✐❝✐❛❧ ❡ ❝♦♥❣r✉ê♥❝✐❛s ♣❛r❛ ❝r✐❛r ♠ét♦❞♦s ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡ ❝r✐♣t♦❛♥á❧✐s❡ ❞❡ ♠❡♥s❛❣❡♥s✳ ❖ ♥♦♠❡ é ❡♠ r❡❢❡rê♥❝✐❛ ❛ ▲❡st❡r ❙✳ ❍✐❧❧ ✭✶✽✾✶✕✶✾✻✶✮ q✉❡ ✐♥tr♦❞✉③✐✉ ❡ss❡s ♠ét♦❞♦s ❡♠ ❬✹❪ ❡ ❬✺❪✳ ◆♦ ❝❛♣ít✉❧♦ ✸ tr❛t❛r❡♠♦s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ✉t✐❧✐③❛♥❞♦ ♠❛tr✐③❡s✱ ❞♦s ♠ét♦❞♦s ❞❡ ❝r✐♣t♦❣r❛✜❛ ❡ ❝r✐♣t♦❛♥á❧✐s❡ ✉s❛♥❞♦ ❛s ❝✐❢r❛s ❞❡ ❍✐❧❧✱ ❛♥❛❧✐s❛r❡♠♦s ❝♦♠♦ ❡ q✉❛♥❞♦ é ♣♦ssí✈❡❧ q✉❡❜r❛r ♦ ♠ét♦❞♦ ❡ ♠♦str❛r❡♠♦s ❝♦♠♦ ❢❡rr❛♠❡♥t❛s t❡❝♥♦❧ó❣✐❝❛s ♣♦❞❡♠ ♥♦s ❛✉①✐❧✐❛r ♥❛ r❡❛❧✐③❛çã♦ ❞❡ ❝á❧❝✉❧♦s q✉❡ s❡r✐❛♠ ❜❛st❛♥t❡ t❡❞✐♦s♦s s❡ ❢❡✐t♦s ❛ ♠ã♦✳
✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ❡ ❈♦♥❣r✉ê♥❝✐❛s
❚❡♦r✐❛ ❞♦s ♥ú♠❡r♦s✱ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✱ é ♦ ❡st✉❞♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ r❡♣r❡s❡♥t❛❞♦ ♣♦r Z✱ ❡ ❞❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ❢❛r❡♠♦s ✉♠❛ r❡✈✐sã♦ ❞❡
❛❧❣✉♥s tó♣✐❝♦s ❞❡ t❡♦r✐❛ ❡❧❡♠❡♥t❛r ❞♦s ♥ú♠❡r♦s q✉❡ ✉s❛r❡♠♦s ♣♦st❡r✐♦r♠❡♥t❡ ❡♠ ♥♦ss♦ tr❛❜❛❧❤♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ♥❛ ❛r✐t♠ét✐❝❛ ♠♦❞✉❧❛r ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐sN✳ ❖s r❡s✉❧t❛❞♦s ❡①✐❜✐❞♦s ❢♦r❛♠ ❡①tr❛í❞♦s✱ q✉❛s❡ ❡♠ s✉❛ t♦t❛❧✐❞❛❞❡✱
❞❡ ❬✸❪ ❡ ❬✼❪✳
✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡
❈♦♠♦ ❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦r ♦✉tr♦✱ q✉❛♥❞♦ ❡①✐st✐r✱ ♥❡♠ s❡♠♣r❡ é ❡①❛t❛✱ ❡①♣r❡ss❛✲s❡ ❡st❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❛tr❛✈és ❞❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✳
❉❡✜♥✐çã♦ ✶✳✶✳ ❉❛❞♦s a ❡ b ∈ Z✱ ❞✐③❡♠♦s q✉❡ a ❞✐✈✐❞❡ b s❡ ❡①✐st✐r ✉♠ ✐♥t❡✐r♦ d t❛❧
q✉❡ b = ad✳ ◆❡st❡ ❝❛s♦✱ t❛♠❜é♠ é ❞✐t♦ q✉❡ a é ❞✐✈✐s♦r ♦✉ ❢❛t♦r ❞❡ b✱ ♦✉ ❛✐♥❞❛✱ b é
✉♠ ♠ú❧t✐♣❧♦ ❞❡ a✳
❙❡ a ❞✐✈✐❞❡ b ❡s❝r❡✈❡♠♦s a|b✱ ❝❛s♦ ❝♦♥trár✐♦✱ ❡s❝r❡✈❡♠♦s a∤b✳
❊①❡♠♣❧♦ ✶✳✷✳ ❚❡♠♦s q✉❡
13|65, −5|15, 13|169, 6∤35❡ 0∤17 ❊①❡♠♣❧♦ ✶✳✸✳ ❖s ❞✐✈✐s♦r❡s ❞❡ 4 sã♦
±1,±2 ❡ ±4 Pr♦♣♦s✐çã♦ ✶✳✹✳ ❙❡ a, b, c∈Z ❝♦♠ a|b ❡♥tã♦ a|bc✳
Pr♦✈❛✿ ❈♦♠♦a|b✱ ❡①✐st❡ ♥❛t✉r❛❧d1 ∈Z t❛❧ q✉❡b=ad1✳ ❆ss✐♠✱ bc= (ad1)c=a(d1c)✱
❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
Pr♦♣♦s✐çã♦ ✶✳✺✳ ✭❚r❛♥s✐t✐✈✐❞❛❞❡✮ ❙❡ a, b, c∈Z ❝♦♠ a|b ❡ b|c ❡♥tã♦ a|c✳
❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✸
Pr♦✈❛✿ ❈♦♠♦ a | b ❡ b | c✱ ❡①✐st❡♠ d1 ❡ d2inZ t❛✐s q✉❡ b = ad1 ❡ c = bd2✳ ❆ss✐♠✱
c=bd2 = (ad1)d2 =a(d1d2) ❡✱ ♣♦rt❛♥t♦✱a |c✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
Pr♦♣♦s✐çã♦ ✶✳✻✳ ❙❡ a, b, c, m, n∈Z ❡ a |b ❡ a|c✱ ❡♥tã♦ a|(mb+nc)✳
Pr♦✈❛✿ ❈♦♠♦ a | b ❡ a | c ❡①✐st❡♠ ✐♥t❡✐r♦s d1 ❡ d2 t❛✐s q✉❡b =ad1 ❡ c =ad2✳ ❆ss✐♠✱
mb+nc =m(ad1) +n(ad2) =a(md1+nd2)❡✱ ♣♦rt❛♥t♦✱a |(mb+nc)✒ ❝♦♠♣❧❡t❛♥❞♦ ❛
❞❡♠♦♥str❛çã♦✳
❈♦r♦❧ár✐♦ ✶✳✼✳ ❙❡ a, b, c∈N ❝♦♠ a|b ❡ a |c❡♥tã♦ a |(b+c) ❡ a |(b−c)✳ ❆ ❞❡✜♥✐çã♦ ❛ s❡❣✉✐r é ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ✶✳✶✶✳
❉❡✜♥✐çã♦ ✶✳✽✳ ❙❡❥❛ x∈R✳ ❖ ♠❛✐♦r ✐♥t❡✐r♦ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ x é ✐♥❞✐❝❛❞♦ ♣♦r [x] ✳ ❊①❡♠♣❧♦ ✶✳✾✳
[2,4] = 2, [−3,2] =−4, [5] = 5,
3 2
= 1.
❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ q✉❡
Pr♦♣♦s✐çã♦ ✶✳✶✵✳ ❙❡ x∈R ❡♥tã♦ x−1<[x]≤x✳
◗✉❛♥❞♦ ♥ã♦ ❡①✐st✐r ❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ✈❡r❡♠♦s q✉❡✱ ❛✐♥❞❛ ❛ss✐♠✱ s❡rá ♣♦ssí✈❡❧ ❡❢❡t✉❛r ✉♠❛ ❞✐✈✐sã♦ ❝♦♠ r❡st♦✱ ❝❤❛♠❛❞❛ ❞❡ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✳
❚❡♦r❡♠❛ ✶✳✶✶✳ ✭❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦✮ ❙❡ a, b∈Z ❝♦♠ b >0✱ ❡♥tã♦ ❡①✐st❡♠ ✐♥t❡✐r♦s
ú♥✐❝♦s q, r t❛✐s q✉❡ a=bq+r✱ ❝♦♠ 0≤r < b✳
❖ ♥ú♠❡r♦ q ∈ Z é ❝❤❛♠❛❞♦ ❞❡ q✉♦❝✐❡♥t❡ ❡ ♦ ♥ú♠❡r♦ r ∈ Z é ❝❤❛♠❛❞♦ ❞❡ r❡st♦ ❞❛
❞✐✈✐sã♦✳
Pr♦✈❛✿ ❙❡❥❛♠ q = [a/b] ❡ r =a−b[a/b]✳ ❚❡♠♦s ❡♥tã♦ q✉❡ a =bq +r✳ ▼♦str❛r❡♠♦s
q✉❡ ♦ r❡st♦ r s❛t✐s❢❛③ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ 0≤r < b✳ P❛r❛ ✐ss♦✱ ♦❜s❡r✈❡ q✉❡
a/b−1<[a/b]≤a/b.
▼✉❧t✐♣❧✐❝❛♥❞♦ ❡ss❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r b✱ ♦❜t❡♠♦s
a−b < b[a/b]≤a.
▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r −1✱❡♥❝♦♥tr❛♠♦s
◆ú♠❡r♦s Pr✐♠♦s ✹
❆❞✐❝✐♦♥❛♥❞♦ a ❛ ❝❛❞❛ ♠❡♠❜r♦✱ ❝❤❡❣❛♠♦s ❛
0≤r=a−b[a/b]< b
P❛r❛ ♠♦str❛r q✉❡ ♦ q✉♦❝✐❡♥t❡ q ∈ Z ❡ ♦ r❡st♦ r ∈ Z sã♦ ú♥✐❝♦s✱ s✉♣♦♥❤❛♠♦s q✉❡
a=bq1+r1 ❡ a=bq2+r2✱ ❝♦♠ 0≤r1 < b ❡0≤r2 < b✳ ▲♦❣♦
0 = b(q1−q2) + (r1−r2).
❆ss✐♠✱ t❡♠♦s q✉❡
r2−r1 =b(q1−q2)
❉✐st♦ r❡s✉❧t❛ q✉❡ b ❞✐✈✐❞❡ r2 − r1✳ ❉❡s❞❡ q✉❡ 0 ≤ r1 < b ❡ 0 ≤ r2 < b✱ t❡♠♦s
−b < r2−r1 < b✳ ■st♦ ♠♦str❛ q✉❡ b ♣♦❞❡ ❞✐✈✐❞✐r r2−r1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r2−r1 = 0✱
♦✉✱ ❡♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ s❡ r2 =r1✳ ❉❡s❞❡ q✉❡bq1 =r1 =bq2+r2 ❡r1 =r2✱ ✈❡♠♦s q✉❡
q1 =q2✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❖❜✈✐❛♠❡♥t❡✱ a é ❞✐✈✐sí✈❡❧ ♣♦r b s❡✱ ❡ s♦♠❡♥t❡ s❡✱r = 0 ♥♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✳ ❊①❡♠♣❧♦ ✶✳✶✷✳ ❚❡♠♦s✿
❛✮ 133 = 21.6 + 7 ❜✮ −50 = 8.(−7) + 6
✶✳✷ ◆ú♠❡r♦s Pr✐♠♦s
◆❡st❛ s❡çã♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ r❡✈✐sã♦ s♦❜r❡ ♥ú♠❡r♦s ♣r✐♠♦s✱ ✉♠ ❞♦s ❝♦♥❝❡✐t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞❡ t♦❞❛ ❛ ▼❛t❡♠át✐❝❛✳
❉❡✜♥✐çã♦ ✶✳✶✸✳ ❯♠ ♥ú♠❡r♦ ♣r✐♠♦ p∈ Z é ✉♠ ♥ú♠❡r♦ ♠❛✐♦r q✉❡ ✶ q✉❡ é ❞✐✈✐sí✈❡❧
❛♣❡♥❛s ♣♦r ✶ ❡ ♣♦r ❡❧❡ ♠❡s♠♦✳
❯♠ ♥ú♠❡r♦ q✉❡ ♥ã♦ é ♣r✐♠♦ é ❝❤❛♠❛❞♦ ❝♦♠♣♦st♦✳ ❆ss✐♠✱ s❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦n > 1 é ❝♦♠♣♦st♦✱ ❡①✐st✐rá ✉♠ ❞✐✈✐s♦r ♥❛t✉r❛❧ n1 ❞❡ n t❛❧ q✉❡ n1 6= 1 ❡ n1 6= n✳ P♦rt❛♥t♦✱
❡①✐st✐rá ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ n2 t❛❧ q✉❡
n=n1×n2, ❝♦♠ 1< n1 < n ❡1< n1 < n
❊①❡♠♣❧♦ ✶✳✶✹✳ ❚❡♠♦s✿
◆ú♠❡r♦s Pr✐♠♦s ✺
❙❡a ❡b∈Z✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ ❡♥tã♦ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡a❡ ❞❡b é ✉♠
❝♦♥❥✉♥t♦ ❞❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s ✜♥✐t♦✱ s❡♠♣r❡ ❝♦♥t❡♥❞♦ ♦s ✐♥t❡✐r♦s +1 ❡ −1✳
❉❡✜♥✐çã♦ ✶✳✶✺✳ ❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s ✐♥t❡✐r♦s a ❡ b✱ ♥ã♦ ❛♠❜♦s
♥✉❧♦s✱ é ♦ ♠❛✐♦r ✐♥t❡✐r♦ q✉❡ ❞✐✈✐❞❡ ❛♠❜♦s a ❡ b✳ ❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ a ❡ b
s❡rá ✐♥❞✐❝❛❞♦ ♣♦r mdc(a, b)✳
❊①❡♠♣❧♦ ✶✳✶✻✳ ❖s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ ✷✹ ❡ ✽✹ sã♦ ±1,±2,±3,±4,±6❡ ±12✳ ❆ss✐♠
mdc(24,84) = 12
❉❡✜♥✐çã♦ ✶✳✶✼✳ ❙❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❞♦✐s ✐♥t❡✐r♦s a ❡ b✱ ♥ã♦ ♥✉❧♦s✱
❢♦r ✐❣✉❛❧ ❛ ✶✱ ❞✐r❡♠♦s q✉❡ a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳
❊①❡♠♣❧♦ ✶✳✶✽✳ ❉❡s❞❡ q✉❡ mdc(25,42) = 1 t❡♠♦s q✉❡ ✷✺ ❡ ✹✷ sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✳ ❚❡♦r❡♠❛ ✶✳✶✾✳ ❖ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ♦s ✐♥t❡✐r♦s a ❡ b✱ ♥ã♦ ❛♠❜♦s ♥✉❧♦s✱ é
♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✳
Pr♦✈❛✿ ❙❡❥❛ d♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡a❡b✳ ✭❊①✐st❡♠
t❛❧ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❞❡s❞❡ q✉❡ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s1.a+ 0.b
❡ (−1)a+ 0.b é ♣♦s✐t✐✈❛✮✳ ❊s❝r❡✈❡♠♦s
d=ma+nb,
♦♥❞❡ m ❡ n sã♦ ✐♥t❡✐r♦s✳ ▼♦str❛r❡♠♦s q✉❡ d|a ❡ d|b✳ P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ♥ós
t❡♠♦s
a=dq+r, 0≤r < d.
❉❡st❛s ❡q✉❛çõ❡s ♥ós ✈❡♠♦s q✉❡
r=a−dq =a−q(ma+nb) = (1−qm)a−qnb.
■st♦ ♠♦str❛ q✉❡ ♦ ✐♥t❡✐r♦ r é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✳ ❉❡s❞❡ q✉❡ 0≤e < d✱ ❡ d é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✱ ❝♦♥❝❧✉í♠♦s q✉❡ r= 0✱ ❡
❛ss✐♠ d|a✳ ❉❡ ❢♦r♠❛ s✐♠✐❧❛r✱ ♠♦str❛✲s❡ q✉❡d|b✳
▼♦str❛♠♦s q✉❡ d✱ ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ q✉❡ é ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✱ é ✉♠
❞✐✈✐s♦r ❞❡ a ❡ ❞❡ b✳ ❘❡st❛ ♠♦str❛r q✉❡ d é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ ❞❡ b✳ P❛r❛
✐st♦✱ ❞❡✈❡♠♦s ♠♦str❛r q✉❡ q✉❛❧q✉❡r ❞✐✈✐s♦r ❝♦♠✉♠c ❞❡a ❡ ❞❡b ❞❡✈❡ ❞✐✈✐❞✐r d✳ ❉❡s❞❡
q✉❡ q✉❡ d = ma+ nb✱ s❡ c|a ❡ c|b✱ ♦ ❚❡♦r❡♠❛ ✶✳✻ ❣❛r❛♥t❡ q✉❡ c|d✱ ❡♥tã♦ c ≤ d✳
❈♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❈♦♥❣r✉ê♥❝✐❛ ✻
Pr♦✈❛✿ ❇❛st❛ ♣r♦✈❛r q✉❡✱ s❡ p|ab❡ p6 |a ❡♥tã♦ p|b✳ ▼❛s s❡ p|ab✱ ❡♥tã♦ ❡①✐st❡ c∈ Z t❛❧
q✉❡ ab=pc✳ ❈♦♠♦mdc(p, a) = 1 ❡♥tã♦✱ t❡♠♦s q✉❡ ❡①✐st❡♠m, n∈Z t❛✐s q✉❡
np+ma= 1.
▼✉❧t✐♣❧✐❝❛♥❞♦ ♣♦r b ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ✐❣✉❛❧❞❛❞❡ ❛❝✐♠❛✱ t❡♠♦s q✉❡
b=npb+mab.
❙✉❜st✐t✉✐♥❞♦ ab♣♦r pc ♥❡st❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ✱ t❡♠♦s q✉❡
b =npb+mpc=p(nb+mc),
❡✱ ♣♦rt❛♥t♦✱ p|b✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❚❡♦r❡♠❛ ✶✳✷✶✳ ✭❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛✮ ❚♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ n >1 ♦✉ é ♣r✐♠♦ ♦✉ s❡ ❡s❝r❡✈❡ ❞❡ ♠♦❞♦ ú♥✐❝♦ ✭❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❢❛t♦r❡s✮ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❞❡ ♥ú♠❡r♦s ♣r✐♠♦s✳
❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❆r✐t♠ét✐❝❛ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✼❪✱ ♣á❣✐♥❛ ✶✶✷✳
✶✳✸ ❈♦♥❣r✉ê♥❝✐❛
❆ ❧✐♥❣✉❛❣❡♠ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❝♦♠❡ç♦ ❞♦ ❙é❝✉❧♦ ❳■❳ ♣♦r ●❛✉ss ❡ é ❡①tr❡♠❛♠❡♥t❡ út✐❧ ♣❛r❛ ❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳ ❚r❛t❛✲s❡ ❞❛ r❡❛❧✐③❛çã♦ ❞❡ ✉♠❛ ❛r✐t♠ét✐❝❛ ❝♦♠ ♦s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛ ♣♦r ✉♠ ♥ú♠❡r♦ ✜①❛❞♦✳
❉❡✜♥✐çã♦ ✶✳✷✷✳ ❙❡❥❛ m ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♥ã♦ ♥✉❧♦✳ ❉✐③❡♠♦s q✉❡ ❞♦✐s ♥ú♠❡r♦s
✐♥t❡✐r♦s a ❡ b sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✱ s❡ m|(a−b)✳ ❙❡ a é ❝♦♥❣r✉❡♥t❡ ❛ b ♠ó❞✉❧♦ m✱ ❡s❝r❡✈❡♠♦s
a≡b(mod m).
❙❡ m ∤(a−b)✱ ❡s❝r❡✈❡♠♦s a 6≡b(mod m)✳
❊ss❛ ❞❡✜♥✐çã♦ é ❡q✉✐✈❛❧❡♥t❡ ❛ ❞✐③❡r q✉❡ ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b sã♦ ❝♦♥❣r✉❡♥t❡s
♠ó❞✉❧♦ m✱ s❡ ♦s r❡st♦s ❞❡ s✉❛s ❞✐✈✐sõ❡s ❡✉❝❧✐❞✐❛♥❛s ♣♦rm sã♦ ✐❣✉❛✐s✳
❈♦♥❣r✉ê♥❝✐❛ ✼
❛✮
21≡13(mod 2)
♣♦✐s ♦s r❡st♦s ❞❛ ❞✐✈✐sã♦ ❞❡ ✷✶ ❡ ❞❡ ✶✸ ♣♦r ✷ sã♦ ✐❣✉❛✐s ❛ ✶✳ ❜✮
22≡4(mod 9)
♣♦✐s 9|(22−4) = 18✳
Pr♦♣♦s✐çã♦ ✶✳✷✹✳ ❙❡ a ❡ b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❡♥tã♦ a ≡b(mod m)s❡ ❡✱ s♦♠❡♥t❡ s❡✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ a=b+km✳
Pr♦✈❛✿ ❙❡ a ≡ b(mod m)✱ ❡♥tã♦ m | (a −b)✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k
t❛❧ q✉❡ km = a −b✱ ✐st♦ é✱ a = b+km✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧
q✉❡ a = b+km✱ ❡♥tã♦ km = a−b✳ ❆ss✐♠ m | (a−b)✱ ❡ ♣♦rt❛♥t♦✱ a ≡ b(mod m)✱
❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
Pr♦♣♦s✐çã♦ ✶✳✷✺✳ ❙❡❥❛ m ∈Z✳ ❆ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐✲
❡❞❛❞❡s✿
✐✮ ✭❘❡✢❡①✐✈❛✮ ❙❡ a é ✉♠ ✐♥t❡✐r♦✱ ❡♥tã♦ a≡a(mod m)✳
✐✐✮ ✭❙✐♠étr✐❝❛✮ ❙❡ a ❡ b sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ a≡b(mod m) ✱ ❡♥tã♦ b≡a(mod m)✳ ✐✐✐✮ ✭❚r❛♥s✐t✐✈❛✮ ❙❡ a✱ b ❡ c sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ a ≡ b(mod m) ❡ b ≡ c(mod m)✱
❡♥tã♦ a≡c(mod m)✳
Pr♦✈❛✿
✐✮ a≡a(mod m)✱ ♣♦✐s m|0 = (a−a)✳
✐✐✮ ❙❡ a ≡ b(mod m) ✱ ❡♥tã♦ m | (a − b)✳ ❆ss✐♠✱ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ k t❛❧ q✉❡ km=a−b✳ ■st♦ ♠♦str❛ q✉❡(−k)m =b−a❡ ❡♥tã♦m |(b−a)✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡
b ≡a(mod m)✳
✐✐✐✮ ❙❡a≡b(mod m)❡b≡c(mod m)✱ ❡♥tã♦m|(a−b)❡m|(b−c)✳ ❆ss✐♠✱ ❡①✐st❡♠ ✐♥t❡✐r♦s k ❡l ❝♦♠ km=a−b ❡ lm=b−c✳ ❉❡st❛ ❢♦r♠❛✱
a−c= (a−b) + (b−c) = km+lm= (k+l)m.
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ m | (a−c) ❡ a ≡ c(mod m)✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❈♦♥❣r✉ê♥❝✐❛ ✽
❆ Pr♦♣♦s✐çã♦ ✶✳✷✺ ♠♦str❛ q✉❡✱ ✜①❛❞♦ m ∈ Z✱ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦ m ❢♦r♠❛ ✉♠❛
r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ Z ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡✱ ❛ss✐♠✱ ♦ ❝♦♥❥✉♥t♦ Z é ♣❛rt✐❝✐♦♥❛❞♦ ❡♠m ❞✐❢❡r❡♥t❡s ❝♦♥❥✉♥t♦s ❝❤❛♠❛❞♦s ❝❧❛ss❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦
♠ ♦✉ ❝❧❛ss❡s r❡s✐❞✉❛✐s ♠ó❞✉❧♦ m✱ ❝❛❞❛ q✉❛❧ ❝♦♥t❡♥❞♦ ♦s ✐♥t❡✐r♦s q✉❡ sã♦ ❝♦♥❣r✉❡♥t❡s
♠ó❞✉❧♦ m✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ m s❡rá r❡♣r❡s❡♥t❛❞♦ ♣♦r
Zm✳
❊①❡♠♣❧♦ ✶✳✷✻✳ ❆s ✺ ❝❧❛ss❡s ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ♠ó❞✉❧♦ ✺ sã♦ ❞❛❞❛s ♣♦r
· · · ≡ −10≡ −5≡0≡5≡10. . .(mod 5)
· · · ≡ −9≡ −4≡1≡6≡11. . .(mod 5)
· · · ≡ −8≡ −3≡2≡7≡12. . .(mod 5)
· · · ≡ −7≡ −2≡3≡8≡13. . .(mod 5)
· · · ≡ −6≡ −1≡4≡9≡14. . .(mod 5)
❙❡❥❛a∈Z✉♠ ✐♥t❡✐r♦✳ ❉❛❞♦ ♦ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦m >1✱ ♣❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ t❡♠♦s
q✉❡a=bm+r ❝♦♠0≤r≤m−1✳ ❉❛ ❡q✉❛çã♦a =bm+r✱ ✈❡♠♦s q✉❡a≡r(mod m)✳ ❆ss✐♠✱ q✉❛❧q✉❡r ✐♥t❡✐r♦ é ❝♦♥❣r✉❡♥t❡ ♠ó❞✉❧♦m ❛ ❛❧❣✉♠ ❞♦s ✐♥t❡✐r♦s 0,1, . . . , m−1✱ ♦ q✉❛❧ é ♦ r❡st♦ ❞❡ s✉❛ ❞✐✈✐sã♦ ♣♦r m✳ ❈♦♠♦ ♥ã♦ ❡①✐st❡♠ ❞♦✐s ✐♥t❡✐r♦s ❡♥tr❡ 0 ❡ m−1 ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ ♠ó❞✉❧♦ m✱ t❡♠♦s q✉❡ q✉❛❧q✉❡r ✐♥t❡✐r♦ é ❝♦♥❣r✉❡♥t❡ ♠ó❞✉❧♦ m ❛
❡①❛t❛♠❡♥t❡ ✉♠ ❞❡st❡s ✐♥t❡✐r♦s 0,1, . . . , m−1✳
❉❡✜♥✐çã♦ ✶✳✷✼✳ ❯♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ m ✐♥t❡✐r♦s {a1, a2, . . . , am} q✉❡ s❡ ai 6=aj ❡♥tã♦ ai 6≡aj(mod m).
❉❛❞♦ ✉♠ s✐st❡♠❛ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✱ q✉❛❧q✉❡r ✐♥t❡✐r♦ é ❝♦♥❣r✉❡♥t❡ ♠ó❞✉❧♦ m ❛
❡①❛t❛♠❡♥t❡ ✉♠ ✐♥t❡✐r♦ ❞❡ss❡ ❝♦♥❥✉♥t♦✳
❊①❡♠♣❧♦ ✶✳✷✽✳ ❖ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦ ♠♦str❛ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s{0,1,2, . . . , m−1}
é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✳
❋❛r❡♠♦s ❛❣♦r❛ ❛❧❣✉♠❛ ❛r✐t♠ét✐❝❛ ❝♦♠ ❝♦♥❣r✉ê♥❝✐❛s✱ s❡♥❞♦ q✉❡ ❡st❛s t❡♠ ♠✉✐t❛s ❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ♣♦ss✉✐✳ ❈♦♠❡ç❛♠♦s ♠♦str❛♥❞♦ q✉❡ ❛❞✐çã♦✱ s✉❜tr❛çã♦ ❡ ♠✉❧t✐♣❧✐❝❛çã♦ ❡♠ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ ♣r❡s❡r✈❛ ❛ ❝♦♥❣r✉ê♥❝✐❛✳
❚❡♦r❡♠❛ ✶✳✷✾✳ ❙❡ a✱ b✱ c ❡ m sã♦ ✐♥t❡✐r♦s✱ ❝♦♠ m >0 t❛✐s q✉❡ a≡b(mod m)✱ ❡♥tã♦
✐✮ a+c≡b+c(mod m)❀ ✐✐✮ a−c≡b−c(mod m)❀
❈♦♥❣r✉ê♥❝✐❛ ✾
Pr♦✈❛✿
✐✮ ❈♦♠♦ a ≡ b(mod m)✱ m | (a−b)✳ ❆❣♦r❛✱ ❞❡ (a+c)−(b+ 1) = a−b✱ ✈❡♠♦s
q✉❡ m | [(a+c)−(b+c)]✳ ❆ss✐♠ a+c≡b+c(mod m)✳ ✐✐✮ ❆♥á❧♦❣♦ ❛♦ ✐t❡♠ ✭✐✮✳
✐✐✐✮ ❖❜s❡r✈❡ q✉❡ac−bc=c(a−b)✳ ❉❡s❞❡ q✉❡m | (a−b)✱ s❡❣✉❡ q✉❡ m | c(a−b)✱
❡ ❛ss✐♠✱ ac≡bc(mod m)
✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❊①❡♠♣❧♦ ✶✳✸✵✳ ❉❡s❞❡ q✉❡ 18≡2(mod 8)s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✾ q✉❡
25 = 18 + 7≡2 + 7 = 9(mod8) 17 = 18−1≡2−1 = 1(mod8) 36 = 18×2≡2×2 = 4(mod8)
❯♠❛ ♣❡r❣✉♥t❛ ♥❛t✉r❛❧ q✉❡ ♣♦❞❡♠♦s ♥♦s ❢❛③❡r é✿ ❖ q✉❡ ❛❝♦♥t❡❝❡ q✉❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ sã♦ ✑❞✐✈✐❞✐❞♦s✑ ♣♦r ✉♠ ✐♥t❡✐r♦❄ ❖ ❡①❡♠♣❧♦ ❛ s❡❣✉✐r ♠♦str❛ q✉❡ ❛ ❝♦♥❣r✉ê♥❝✐❛ ♥❡♠ s❡♠♣r❡ é ♣r❡s❡r✈❛❞❛ q✉❛♥❞♦ ❛♠❜♦s ♦ ❧❛❞♦s ❞❛ ♠❡s♠❛ sã♦ ❞✐✈✐❞✐❞♦s ♣❡❧♦ ♠❡s♠♦ ♥ú♠❡r♦✳
❊①❡♠♣❧♦ ✶✳✸✶✳ ❚❡♠♦s q✉❡
2×7≡2×4(mod 6) ♠❛s 76≡ 4(mod 6).
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❛ ❧❡✐ ❞♦ ❝❛♥❝❡❧❛♠❡♥t♦ ♥ã♦ ✈❛❧❡ ♥❛ ❝♦♥❣r✉ê♥❝✐❛✱ ❛♦ ❝♦♥trár✐♦ ❞❛ ✐❣✉❛❧❞❛❞❡✳
❚❡♦r❡♠❛ ✶✳✸✷✳ ❙❡ a, b, c, m∈Z ❝♦♠ c6= 0 ❡ m >1sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ mdc(c, m) =
d✱ ❡♥tã♦ ac≡bc(mod m)✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a≡b(mod m/d)✳ Pr♦✈❛✿
❈♦♠♦ m/d ❡ c/d sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ t❡♠♦s q✉❡
ac≡bc(mod m)⇐⇒m|(b−a)c⇐⇒m/d|(b−a)c/d ⇐⇒m/d |(b−a)⇐⇒a≡b(mod m/d),
❈♦♥❣r✉ê♥❝✐❛ ✶✵
❊①❡♠♣❧♦ ✶✳✸✸✳ ❚❡♠♦s 5×10≡2×10(mod 15) ❡ mdc(10,15) = 5✳ ❊♥tã♦✱ 5×10
10 ≡
2×10 10 (mod
15
5 ), ♦✉ s❡❥❛✱ 5≡ 2(mod3).
❈♦r♦❧ár✐♦ ✶✳✸✹✳ ❙❡ a, b, c, m ∈ Z ❝♦♠ c 6= 0 ❡ m > 1 sã♦ t❛✐s q✉❡ mdc(c, m) = 1✱ ❡♥tã♦ ac≡bc(mod m)✱ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a ≡b(mod m)✳
❊①❡♠♣❧♦ ✶✳✸✺✳ ❈♦♠♦ 6×7≡1×7(mod 5)❡ mdc(5,7) = 1✱ t❡♠♦s
6×7 7 ≡
1×7
7 (mod 5), ♦✉ 6≡ 1(mod 5).
❚❡♦r❡♠❛ ✶✳✸✻✳ ❙❡ a, b, c, d, m ∈ Z sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ m > 0✱ a ≡ b(mod m) ❡
c≡d(mod m)✱ ❡♥tã♦✿
✐✮ a+c≡b+d(mod m) ✐✐✮ a−c≡b−d(mod m) ✐✐✐✮ ac≡bd(mod m)
Pr♦✈❛✿ ❉❡s❞❡ q✉❡ a ≡ b(mod m) ❡ c ≡ d(mod m)✱ t❡♠♦s m | (a−b) ❡ m | (c−d)✳ ❆ss✐♠✱ ❡①✐st❡♠ ✐♥t❡✐r♦s k ❡ l ❝♦♠ km=a−b ❡ lm=c−d✳ ❉✐ss♦ t❡♠♦s✿
✐✮ (a+c)−(b+d) = (a−b) + (c−d) =km+lm= (k+l)m✳
❆ss✐♠✱ m |[(a+c)−(b+d)]✱ ❡✱ ♣♦rt❛♥t♦✱ a+c≡b+d(mod m) ✐✐✮ (a−c)−(b−d) = (a−b)−(c−d) =km−lm= (k−l)m✳
❆ss✐♠✱ m |[(a−c)−(b−d)]✱❡✱ ♣♦rt❛♥t♦✱a−c≡b−d(mod m)
✐✐✐✮ ac−bd=ac−bc+bc−bd=c(a−b) +b(c−d) =ckm+blm=m(ck+bl)✳
❆ss✐♠ m|(ac−bd) ❡✱ ❧♦❣♦ac≡bd(mod m)✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❊①❡♠♣❧♦ ✶✳✸✼✳ ❚❡♠♦s q✉❡ 13≡8(mod 5)❡7≡2(mod5)✳ P❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✻ ✈❡♠♦s q✉❡✿
❛✮ 13 + 7≡8 + 2(mod 5)❀
❜✮ 13−7≡8−7(mod5)❀ ❝✮ 13×7≡8×2(mod5)✳
❚❡♦r❡♠❛ ✶✳✸✽✳ ❙❡ {r1, r2, . . . , rm} é ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m ❡ s❡
a é ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❝♦♠ mdc(a, m) = 1✱ ❡♥tã♦
{ar1+b, ar2+b, . . . , arm+b}
❈♦♥❣r✉ê♥❝✐❛ ✶✶
Pr♦✈❛✿ ■♥✐❝✐❛❧♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡♠ ✐♥t❡✐r♦s
ar1+b, ar2+b, . . . , arm+b
q✉❡ s❡❥❛♠ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳ P❛r❛ ✐st♦✱ ♦❜s❡r✈❡ q✉❡ s❡
arj +b ≡ark+b(mod m),
❡♥tã♦ ♣❡❧♦ ✐t❡♠ ✭✐✐✮ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✾✱ t❡♠♦s
arj ≡ark(mod m).
❉❡s❞❡ q✉❡ mdc(a, m) = 1✱ ♦ ❈♦r♦❧ár✐♦ ✶✳✸✹ ❣❛r❛♥t❡
rj ≡rk(mod m).
❈♦♠✱ ♣♦r ❞❡✜♥✐çã♦✱ rj 6≡rk(mod m)s❡ j 6=k ✱ ❝♦♥❝❧✉í♠♦s q✉❡ j =k✳
❉❡s❞❡ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s ❡♠ q✉❡stã♦ ❝♦♥s✐st❡ ❞❡ m ✐♥t❡✐r♦s ♥ã♦ ❝♦♥❣r✉❡♥t❡s
❡♥tr❡ s✐ ♠ó❞✉❧♦m✱ ❡st❡s ✐♥t❡✐r♦s ❞❡✈❡♠ ❢♦r♠❛r ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s ♠ó❞✉❧♦ m✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❚❡♦r❡♠❛ ✶✳✸✾✳ ❙❡ a, b, n ❡ m sã♦ ✐♥t❡✐r♦s t❛✐s q✉❡ n > 0, m > 0 ❡ a ≡ b(mod m) ❡♥tã♦ an ≡bn(mod m)✳
Pr♦✈❛✿ ❱❛♠♦s ❢❛③❡r ❛ ❞❡♠♦♥str❛çã♦ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n ∈N✳ ❙❡ n= 1✱ ♦ r❡s✉❧t❛❞♦ é
✐♠❡❞✐❛t♦✳
❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❥❛ ✈❡r❞❛❞❡✐r♦ ♣❛r❛ n ∈ N ❡ ♠♦str❡♠♦s q✉❡
♦ r❡s✉❧t❛❞♦ é ✈á❧✐❞♦ t❛♠❜é♠ ♣❛r❛ n + 1 ∈ N✳ ❉❡ ❢❛t♦✱ ♣❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱
an ≡bn(mod m)✳ ❈♦♠♦ a≡b(mod m) s❡❣✉❡ ❞♦ ❚❡♦r❡♠❛ ✶✳✸✻ q✉❡ana ≡bnb(mod m)✳
▲♦❣♦✱ an+1
≡bn+1
(mod m)✱ ♦ q✉❡ ❞❡♠♦♥str❛ ♥♦ss♦ r❡s✉❧t❛❞♦✳
❊①❡♠♣❧♦ ✶✳✹✵✳ ❚❡♠♦s q✉❡ 7≡2(mod 5)✳ ❊♥tã♦ ♦ ❚❡♦r❡♠❛ ✶✳✸✾ ♥♦s ❞✐③ q✉❡
73
≡23
(mod 5) ♦✉ 343≡8(mod 5).
❉❡✜♥✐çã♦ ✶✳✹✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ m é ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠
✭♠♠❝✮ ❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s m1, m2, . . . , mk✱ ♥ã♦ ♥✉❧♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ s❡
✐✮ m é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ m1, m2, . . . , mk✱ ❡
❈♦♥❣r✉ê♥❝✐❛ ✶✷
■♥❞✐❝❛r❡♠♦s ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞♦s ♥ú♠❡r♦s m1, m2, . . . , mk ♣♦r
mmc(m1, m2, . . . , mk).
❚❡♦r❡♠❛ ✶✳✹✷✳ ❙❡ m1, m2, . . . , mk sã♦ ♥ú♠❡r♦s ♥❛t✉r❛✐s ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦
mmc(m1, m2, . . . , mk) = m1×me× · · · ×mk.
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❚❡♦r❡♠❛ é ✉♠❛ ❝♦♥s❡q✉ê♥❝✐❛ ✐♠❡❞✐❛t❛ ❞❛ ♣r♦♣♦s✐çã♦ ✺✳✸✳✶✱ ♣❛✲ ❣✐♥❛ ✻✸ ❞❡ ❬✸❪✳
❚❡♦r❡♠❛ ✶✳✹✸✳ ❙❡a≡b(mod m1), a≡b(mod m2), . . . , a≡b(mod mk)❝♦♠a, b, m1, m2, . . . mk
✐♥t❡✐r♦s ❡ m1, m2, . . . , mk ♣♦s✐t✐✈♦s✱ ❡♥tã♦
a≡b(mod mmc(m1, m2, . . . , mk)).
Pr♦✈❛✿ ❉❡s❞❡ q✉❡ a ≡b(mod m1), a ≡b(mod m2), . . . , a≡b(mod mk)✱ ♥ós t❡♠♦s q✉❡
m1 |(a−b), m2 |(a−b), . . . , mk |(a−b)✳ ❆ss✐♠
mmc(m1, m2, . . . , mk)|(a−b)
❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡
a≡b(mod mmc(m1, m2, . . . , mk))
✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❈♦r♦❧ár✐♦ ✶✳✹✹✳ ❙❡ a ≡ b(mod m1), a ≡ b(mod m2), . . . , a ≡ b(mod mk) ♦♥❞❡ a ❡ b
sã♦ ✐♥t❡✐r♦s ❡ m1, m2, . . . mk ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❡ ♣r✐♠♦s ❡♥tr❡ s✐✱ ❡♥tã♦
a ≡b(mod m1×m2× · · · ×mk),
❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
Pr♦✈❛✿ ❉❡s❞❡ q✉❡ m1, m2, . . . , mk sã♦ ♣r✐♠♦s ❡♥tr❡ s✐✱ t❡♠♦s✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✹✷✱ q✉❡
mmc(m1, m2, . . . , mk) = m1×m2× · · · ×mk
❆ss✐♠✱ ❞♦ ❚❡♦r❡♠❛ ✭✶✳✹✸✮✱ s❡❣✉❡ q✉❡
a≡b(mod m1×m2 × · · · ×mk),
❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s ✶✸
✶✳✹ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s
◆❡st❛ s❡çã♦✱ ❡st✉❞❛r❡♠♦s ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡st❡ tó♣✐❝♦ ♣♦❞❡ ✈✐r ❛ ❢❛❝✐❧✐t❛r ❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❞❡ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✳
❉❡✜♥✐çã♦ ✶✳✹✺✳ ❯♠❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❛ ❢♦r♠❛
ax≡b(mod m),
♦♥❞❡ x∈Z é ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ❞❡s❝♦♥❤❡❝✐❞♦✱ é ❝❤❛♠❛❞❛ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r
❡♠ ✉♠❛ ✈❛r✐á✈❡❧✳
❖❜s❡r✈❡ q✉❡ s❡ x = x0 é ✉♠❛ s♦❧✉çã♦ ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ax ≡ b(mod m)✱ ❡ s❡ x1 ≡
x0(mod m)✱ ❡♥tã♦ ax1 ≡ ax0 ≡ b(mod m)✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ x1 t❛♠❜é♠ é s♦❧✉çã♦✳
❆ss✐♠✱ s❡ ✉♠ ♠❡♠❜r♦ ❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ♠ó❞✉❧♦mé s♦❧✉çã♦ ❞❡ ✉♠❛ ❝♦♥❣r✉ê♥❝✐❛
❧✐♥❡❛r✱ ❡♥tã♦ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡st❛ ❝❧❛ss❡ t❛♠❜é♠ sã♦ s♦❧✉çõ❡s✳
❚❡♦r❡♠❛ ✶✳✹✻✳ ❙❡❥❛♠ a, b, m ∈ Z ✐♥t❡✐r♦s ❝♦♠ m > 0 ❡ mdc(a, m) = d✳ ❙❡ d ∤ b✱
❡♥tã♦ax ≡b(mod m)♥ã♦ t❡♠ s♦❧✉çõ❡s✳ ❙❡d|b✱ ❡♥tã♦ax≡b(mod m) t❡♠ ❡①❛t❛♠❡♥t❡
d s♦❧✉çõ❡s ♠ó❞✉❧♦ m ♥ã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐✳
Pr♦✈❛✿ ❆ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r ax ≡ b(mod m) é ❡q✉✐✈❛❧❡♥t❡ ❛ ✉♠❛ ❡q✉❛çã♦ ❡♠ ❞✉❛s ✈❛r✐á✈❡✐s ♥❛ ❢♦r♠❛ ax−my = b✳ ❖ ✐♥t❡✐r♦ x é ✉♠❛ s♦❧✉çã♦ ❞❡ ax ≡ b(mod m) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡y ∈Z❝♦♠ax−my =b✳ ❙❛❜❡♠♦s ❞❛ ♣r♦♣♦s✐çã♦ ✻✳✻✳✶✱ ♣á❣✐♥❛ ✻✻ ❞❡
❬✸❪ q✉❡ s❡ d∤|b✱ ♥ã♦ ❡①✐st❡♠ s♦❧✉çõ❡s ♣❛r❛ ❛ ❡q✉❛çã♦ ax≡b(mod m)✱ ❡♥q✉❛♥t♦ q✉❡ s❡
d|b✱ ❛ ❡q✉❛çã♦ ax−my =b t❡♠ ✐♥✜♥✐t❛s s♦❧✉çõ❡s ❞❛❞❛s ♣♦r x=x0+ (m/d)t, y=y0+ (a/d)t,
♦♥❞❡ x=x0 ❡y =y0 sã♦ s♦❧✉çõ❡s ♣❛rt✐❝✉❧❛r❡s ❞❛ ❡q✉❛çã♦✳ ❖s ✈❛❧♦r❡s ❞❡ x✱
x=x0+ (m/d)t,
sã♦ s♦❧✉çõ❡s ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r✳ P❛r❛ ❞❡t❡r♠✐♥❛r q✉❛♥t❛s s♦❧✉çõ❡s ♥ã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ ❡①✐st❡♠✱ ✈❡❥❛♠♦s ❛s ❝♦♥❞✐çõ❡s q✉❡ ❞❡s❝r❡✈❡♠ q✉❛♥❞♦ ❞✉❛s ❞❛s s♦❧✉çõ❡s x1 =
x0 + (md)t1 ❡ x2 = x0 + (m/d)t2 sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳ ❙❡ ❡st❛s ❞✉❛s s♦❧✉çõ❡s
sã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tã♦
x0+ (m/d)t1 ≡x0+ (m/d)t2(mod m).
❙✉❜tr❛✐♥❞♦ x0 ❞❡ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡st❛ ❝♦♥❣r✉ê♥❝✐❛✱ ❡♥❝♦♥tr❛♠♦s q✉❡
❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s ✶✹
❈♦♠♦ m/d|m t❡♠♦s mdc(m, m/d) =m/d ❡✱ ♣♦rt❛♥t♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✸✷ ✈❡♠♦s q✉❡
t1 ≡t2(mod d).
■st♦ ♠♦str❛ q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ s♦❧✉çõ❡s ♥ã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ é ♦❜t✐❞♦ t♦♠❛♥❞♦✲s❡ x=x0+ (m/d)t✱ ♦♥❞❡t ✈❛r✐❛ ❛tr❛✈és ❞❡ ✉♠ s✐st❡♠❛ ❝♦♠♣❧❡t♦ ❞❡ r❡sí❞✉♦s
♠ó❞✉❧♦ d✳ ❊ss❡ ❝♦♥❥✉♥t♦ é ❞❛❞♦ ♣♦r x = x0 + (m/d)t ♦♥❞❡ t ∈ {0,1,2, . . . , d−1}✱
❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❊①❡♠♣❧♦ ✶✳✹✼✳ ❱❛♠♦s ❡♥❝♦♥tr❛r t♦❞❛s ❛s s♦❧✉çõ❡s ♥ã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ ❞❡ 9x≡
12(mod15)✱
❙♦❧✉çã♦✿ ❈♦♠♦ mdc(9,15) = 3 ❡ 3| 12, ❡①✐st❡♠ ❡①❛t❛♠❡♥t❡ três s♦❧✉çõ❡s ♥ã♦ ❝♦♥❣r✉✲
❡♥t❡s ❡♥tr❡ s✐✳ P♦❞❡♠♦s ❡♥❝♦♥tr❛r ❡ss❛s s♦❧✉çõ❡s ♣r✐♠❡✐r♦ ❡♥❝♦♥tr❛♥❞♦ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❡✱ ❡♥tã♦✱ s♦♠❛♥❞♦ ♦s ♠ú❧t✐♣❧♦s ❝♦rr❡t♦s ❞❡ 15/3 = 5✳
P❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r✱ ♦❜s❡r✈❡ q✉❡ r❡s♦❧✈❡r ❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r 9x≡
12(mod15) é ♠❡s♠♦ q✉❡ r❡s♦❧✈❡r ❛ ❡q✉❛çã♦ 9x−15y= 12.
P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s✱ t❡♠♦s
15 = 9.1 + 6,
9 = 6.1 + 3,
6 = 3.2.
❆ss✐♠✱
3 = 9−6.1 = 9−(15−9.1) = 9.2−15.
❖ q✉❡ r❡s✉❧t❛
9.8−15.4 = 12,
❡ ✉♠❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r ❞❡ 9x−15y é ❞❛❞❛ ♣♦r
x0 = 8 ❡ y0 = 4.
❉♦ ❚❡♦r❡♠❛ ✶✳✹✻ ✈❡♠♦s q✉❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ ✸ s♦❧✉çõ❡s ♥ã♦ ❝♦♥❣r✉❡♥t❡s ❡♥tr❡ s✐ é ❞❛❞♦ ♣♦r
❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s ✶✺
❱❛♠♦s ❝♦♥s✐❞❡r❛r ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s ♥❛ ❢♦r♠❛
ax≡1(mod m).
❙❛❜❡♠♦s ❞♦ t❡♦r❡♠❛ ✶✳✹✻ q✉❡ ❡①✐st❡ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ❡st❛ ❝♦♥❣r✉ê♥❝✐❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ mdc(a, m) = 1 ❡ q✉❡✱ ♥❡st❡ ❝❛s♦✱ t♦❞❛s ❛s s♦❧✉çõ❡s sã♦ ❝♦♥❣r✉❡♥t❡s ♠ó❞✉❧♦ m✳
❉❡✜♥✐çã♦ ✶✳✹✽✳ ❉❛❞♦s ✐♥t❡✐r♦s a ❡ m ❝♦♠ m >0 ❡ mdc(a, m) = 1✱ ✉♠❛ s♦❧✉çã♦ ❞❡ ax≡1(mod m) é ❝❤❛♠❛❞♦ ✉♠ ✐♥✈❡rs♦ ❞❡ a ♠ó❞✉❧♦ m✳
❊①❡♠♣❧♦ ✶✳✹✾✳ ❱❛♠♦s ❡♥❝♦♥tr❛r t♦❞❛s ❛s s♦❧✉çõ❡s ❞❡ 7x≡1(mod 31)✳
❚❡♠♦s
mdc(7,31) = 1.
▲♦❣♦✱ ❡ss❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r ♣♦ss✉✐ s♦❧✉çã♦✳ P❛r❛ ❡♥❝♦♥tr❛r ✉♠❛ s♦❧✉çã♦✱ ❝♦♥s✐❞❡r❛✲ ♠♦s ❛ ❡q✉❛çã♦
7x−31y= 1,
❝✉❥❛ s♦❧✉çã♦ ♣❛rt✐❝✉❧❛r é ❞❛❞❛ ♣♦r x = 9✳ ❆ss✐♠✱ ❞❡s❞❡ q✉❡ ❛s s♦❧✉çõ❡s ❞❡ 7x≡1(mod31) s❛t✐s❢❛③❡♠
x≡9(mod31),
t❡♠♦s q✉❡ ✾ ❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ❝♦♥❣r✉❡♥t❡s ❛ ✾ ♠ó❞✉❧♦ ✸✶✱ sã♦ ✐♥✈❡rs♦s ❞❡ ✼ ♠ó❞✉❧♦ ✸✶✳
◗✉❛♥❞♦ ❝♦♥❤❡❝❡r♠♦s ✉♠ ✐♥✈❡rs♦ ❞❡a♠ó❞✉❧♦m✱ ♣♦❞❡♠♦s ✉sá✲❧♦ ♣❛r❛ r❡s♦❧✈❡r q✉❛❧q✉❡r
❝♦♥❣r✉ê♥❝✐❛ ♥❛ ❢♦r♠❛
ax≡b(mod m).
P❛r❛ ✐st♦✱ s❡❥❛ ¯a ♦ ✐♥✈❡rs♦ ❞❡ a ♠ó❞✉❧♦ m✱ ✐✳é✱ a¯a≡1(mod m)✳ ❊♥tã♦✱ s❡
ax≡b(mod m),
♠✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❡st❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦r a¯✱ t❡♠♦s
¯
a(ax)≡¯ab(mod m),
❡✱ ❞❡st❛ ❢♦r♠❛✱
x≡¯ab(mod m).
❊①❡♠♣❧♦ ✶✳✺✵✳ ❱❛♠♦s r❡s♦❧✈❡r ❛ ❝♦♥❣r✉ê♥❝✐❛ ❧✐♥❡❛r 7x≡22(mod 31)✳
❙♦❧✉çã♦✿ ❈♦♠♦ mdc(7,31) = 1✱ ❛ ❝♦♥❣r✉ê♥❝✐❛
❙✐st❡♠❛s ❞❡ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s 2×2 ✶✻
t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦✳ ❘❡s♦❧✈❡♥❞♦ ❛ ❝♦♥❣r✉ê♥❝✐❛✱ ❡♥❝♦♥tr❛♠♦s ¯7 = 9 ❡ ❛ss✐♠✱ ♠✉❧t✐✲ ♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦r ✾✱ ♦❜t❡♠♦s
x≡63x≡9×7x≡9×22(mod31),
✉ s❡❥❛✱
x≡198 ≡12(mod 31).
▲♦❣♦✱ 7x≡22(mod 31) t❡♠ ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦✱ ❛ s❛❜❡r✱ x= 12✳
Pr♦♣♦s✐çã♦ ✶✳✺✶✳ ❙❡❥❛ p ✉♠ ♣r✐♠♦✳ ❯♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ a é s❡✉ ♣ró♣r✐♦ ✐♥✈❡rs♦
♠ó❞✉❧♦ p s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a≡1(mod p) ♦✉ a≡ −1(mod p)✳ Pr♦✈❛✿ ❙❡ a ≡ 1(mod p) ♦✉ a ≡ −1(mod p)✱ ❡♥tã♦ a2
≡ 1(mod p) ❡ a é s❡✉ ♣ró♣r✐♦
✐♥✈❡rs♦ ♠ó❞✉❧♦ p✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ a é s❡✉ ♣ró♣r✐♦ ✐♥✈❡rs♦ ♠ó❞✉❧♦ p✱ s❡❣✉❡ q✉❡ a2
=a.a≡1(mod p)✳ ❆ss✐♠✱p|(a2
−1)✳ ❉❡s❞❡ q✉❡ a2
−1 = (a−1)(a+ 1)✱ ♦✉p|(a−1) ♦✉ p|(a+ 1)✳ ▲♦❣♦✱a ≡1(mod p)♦✉ a≡ −1(mod p)✳
✶✳✺ ❙✐st❡♠❛s ❞❡ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s
2
×
2
❈♦♥s✐❞❡r❛r❡♠♦s ❛❣♦r❛ s✐st❡♠❛s ❝♦♠ ❞✉❛s ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s ❡ ❞✉❛s ✐♥❝ó❣♥✐t❛s✱ t♦❞❛s ❝♦♠ ♦ ♠❡s♠♦ ♠ó❞✉❧♦✳ ◆♦ss♦ ♦❜❥❡t✐✈♦ é ❡st✉❞❛r q✉❛♥❞♦ s✐st❡♠❛s ❞❡st❛ ❢♦r♠❛ t❡♠ s♦❧✉çã♦ ❡ ❞❡t❡r♠✐♥á✲❧❛s✳
❚❡♦r❡♠❛ ✶✳✺✷✳ ❙❡❥❛♠ a, b, c, d, e, f, m ∈ Z t❛✐s q✉❡ m > 0 ❡ mdc(∆, m) = 1✱ ♦♥❞❡ ∆ = ad−bc✳ ❊♥tã♦✱ ♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s
ax+by≡e(mod m)
cx+dy≡f(mod m)
t❡♠ s♦❧✉çõ❡s ♠ó❞✉❧♦ m ❞❛❞❛s ♣♦r
x≡∆(¯ de−bf)(mod m)
y≡∆(¯ af−ce)(mod m),
♦♥❞❡ ∆¯ é ♦ ✐♥✈❡rs♦ ❞❡ ∆ ♠ó❞✉❧♦ m✳
Pr♦✈❛✿ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ❝♦♥❣r✉ê♥❝✐❛ ❞♦ s✐st❡♠❛ ♣♦r d ❡ ❛ s❡❣✉♥❞❛ ♣♦r b
♦❜t❡♠♦s
adx+bdy≡de(mod m)
❙✐st❡♠❛s ❞❡ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s 2×2 ✶✼
❊♥tã♦✱ s✉❜tr❛í♠♦s ❛ s❡❣✉♥❞❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❛ ♣r✐♠❡✐r❛✱ ♣❛r❛ ❡♥❝♦♥tr❛r q✉❡
(ad−bc)x≡de−bf(mod m),
♦✉
∆x≡de−bf(mod m).
❆❣♦r❛✱ ♠✉❧t✐♣❧✐❝❛♠♦s ❛♠❜♦s ♦s ❧❛❞♦s ❞❡st❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦r∆¯✱ ✉♠ ✐♥✈❡rs♦ ❞❡∆♠ó❞✉❧♦
m✱ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡
x≡∆(¯ de−bf)(mod m).
❉❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✱ ♠✉❧t✐♣❧✐❝❛♠♦s ❛ ♣r✐♠❡✐r❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦rc❡ ❛ s❡❣✉♥❞❛ ♣♦ra♣❛r❛
♦❜t❡r
acx+bcy ≡ce(mod m)
acx+ady ≡af(mod m).
❙✉❜tr❛✐♥❞♦ ❛ ♣r✐♠❡✐r❛ ❝♦♥❣r✉ê♥❝✐❛ ❞❛ s❡❣✉♥❞❛✱ ❡♥❝♦♥tr❛♠♦s
(ad−bc)y≡af −ce(mod m),
♦✉
∆y≡af−ce(mod m).
❋✐♥❛❧♠❡♥t❡✱ ♠✉❧t✐♣❧✐❝❛♠♦s ❛♠❜♦s ♦s ❧❛❞♦s ❞❡ss❛ ❝♦♥❣r✉ê♥❝✐❛ ♣♦r ∆¯ ♣❛r❛ ❝♦♥❝❧✉✐r q✉❡
y≡∆(¯ af−ce)(mod m).
P♦r ♦✉tr♦ ❧❛❞♦✱ s❡ t✐✈❡r♠♦s ✉♠ ♣❛r (x, y) ♥❛ ❢♦r♠❛
x≡∆(¯ de−bf)(mod m)
y ≡∆(¯ af −ce)(mod m),
❡♥tã♦
ax+by ≡a∆(¯ de−bf) +b∆(¯ af −ce)
≡∆(¯ ade−abf −abf −bce)
≡∆(¯ ad−bc)e
≡e(mod m) ❡
cx+dy ≡c∆(¯ de−bf) +d∆(¯ af −ce)
≡∆(¯ cde−bcf−adf −cde)
≡∆(¯ ad−bc)f
❙✐st❡♠❛s ❞❡ ❈♦♥❣r✉ê♥❝✐❛s ▲✐♥❡❛r❡s 2×2 ✶✽
❊①❡♠♣❧♦ ✶✳✺✸✳ ❱❛♠♦s r❡s♦❧✈❡r ♦ s✐st❡♠❛ ❞❡ ❝♦♥❣r✉ê♥❝✐❛s ❧✐♥❡❛r❡s
3x+ 4y≡5(mod13) 2x+ 5y≡7(mod13) .
❚❡♠♦s ∆ =ad−bc= 3×5−4×2 = 7 ❡mdc(∆, m) =mdc(7,13) = 1✳ ❯♠ ✐♥✈❡rs♦ ❞❡
✼ ♠ó❞✉❧♦ ✶✸ é ✷ ❡✱ ❡♥tã♦
x≡∆¯ ×(de−bf) = 2×(5.5−4.7) =−6≡7(mod 13)
❡
✷ ■♥tr♦❞✉çã♦ à ❈r✐♣t♦❣r❛✜❛ ❈❧áss✐❝❛
❈✐❢r❛s sã♦ ❢♦r♠❛s ❞❡ tr❛♥s❢♦r♠❛r ✉♠❛ ♠❡♥s❛❣❡♠ ❞❡ t❡①t♦ ♣❧❛♥♦ ❡♠ ✉♠❛ ♠❡♥s❛❣❡♠ ❞❡ t❡①t♦ ❛❧t❡r❛❞❛ ❝❤❛♠❛❞❛ ❞❡ t❡①t♦ ❝✐❢r❛❞♦ ❞❡ ❢♦r♠❛ q✉❡ ❡st❛ s❡❥❛ ✐♥❞❡❝✐❢rá✈❡❧ ♣❛r❛ q✉❛❧q✉❡r ✉♠ q✉❡ ♥ã♦ ❝♦♥❤❡ç❛ ❛ r❡❣r❛ ❞❡ tr❛♥s❢♦r♠❛çã♦ ✲ ❝❤❛✈❡ ✶✳ ❖ ♣r♦❝❡ss♦ ❞❡
❝♦♥✈❡rt❡r ✉♠ t❡①t♦ ♣❧❛♥♦ ♣❛r❛ ✉♠ t❡①t♦ ❝✐❢r❛❞♦ é ❝❤❛♠❛❞♦ ❝r✐♣t♦❣r❛❢❛r✱ ❡ ♦ ♣r♦❝❡ss♦ ❝♦♥trár✐♦ é ❝❤❛♠❛❞♦ ❞❡s❝r✐♣t♦❣r❛❢❛r✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠❛ ❜r❡✈❡ ✐♥tr♦❞✉çã♦ ❛♦s ♠ét♦❞♦s ❝❧áss✐❝♦s ✭♣ré✲✶✾✼✵✮✳
✷✳✶ ❱✐sã♦ ●❡r❛❧
❊♠ ✉♠ ❝❡♥ár✐♦ s✐♠♣❧❡s ❞❡ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ❝♦♠♦ ♦ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✷✳✶ ✱ ❡①✐st❡♠ ❞♦✐s ❧❛❞♦s✱ q✉❡ ♥ós ❝❤❛♠❛r❡♠♦s ❞❡ ❆❧✐❝❡ ❡ ❇♦❜✷✱ q✉❡ ❞❡s❡❥❛♠ ❝♦♠✉♥✐❝❛r✲s❡ ❞❡ ♠❛♥❡✐r❛
s❡❣✉r❛✳ ❊✱ ❡♥tr❡ ❡❧❡s✱ ❡stá ❊✈❛✱ q✉❡ ❞❡s❡❥❛ ✐♥t❡r❝❡♣t❛r ❡ss❛ ❝♦♠✉♥✐❝❛çã♦✳
❋✐❣✉r❛ ✷✳✶✿ ❈❡♥ár✐♦ ❜ás✐❝♦ ❞❡ ❝♦♠✉♥✐❝❛çã♦✳
◗✉❛♥❞♦ ❆❧✐❝❡ ❞❡s❡❥❛ ❡♥✈✐❛r ✉♠❛ ♠❡♥s❛❣❡♠ s❡❝r❡t❛ ♣❛r❛ ❇♦❜✱ ❡❧❛ ❡♥❝r✐♣t❛ ♦ t❡①t♦
✶❊♠ s✐st❡♠❛s ♠♦❞❡r♥♦s ❞❡ ❝r✐♣t♦❣r❛✜❛✱ ❛ ❝❤❛✈❡ ♥ã♦ ♣r❡❝✐s❛ s❡r s❡❝r❡t❛✳ ❊♠ ✉♠ s✐st❡♠❛ ❝♦♠♦ ♦ ❘❙❆✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝❤❛✈❡ ♣♦❞❡ s❡r ❝♦♥❤❡❝✐❞❛ ♣✉❜❧✐❝❛♠❡♥t❡ ❞❡s❞❡ q✉❡ ❛ ♦❜t❡♥çã♦ ❞❛ ❝❤❛✈❡ ✐♥✈❡rs❛ ❛tr❛✈és ❞❡ss❛ s❡❥❛ ❡①tr❡♠❛♠❡♥t❡ ❞✐❢í❝✐❧✳
✷❱❛♠♦s ♠❛♥t❡r ♦s ♣❡rs♦♥❛❣❡♥s ✉s❛❞♦s ♥❛ r❡❢❡rê♥❝✐❛ ❬✽❪✱ ✈✐st♦ q✉❡ é ✉s✉❛❧ ♥♦s ♠❛✐s ❞✐✈❡rs♦s t❡①t♦s s♦❜r❡ ♦ ❛ss✉♥t♦✳
❱✐sã♦ ●❡r❛❧ ✷✵
✉s❛♥❞♦ ✉♠ ♠ét♦❞♦ ❝♦♠❜✐♥❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ❝♦♠ ❇♦❜✳ ◆♦r♠❛❧♠❡♥t❡✱ ♦ ♠ét♦❞♦ ❞❡ ❝r✐♣t♦❣r❛❢❛r ❛ ♠❡♥s❛❣❡♠ é ❝♦♥❤❡❝✐❞♦ ♣♦r ❊✈❛✳ ❖ q✉❡ ❡❧❛ ♥ã♦ ❝♦♥❤❡❝❡ é ❛ ❝❤❛✈❡ ❞❡ ❞❡s❝r✐♣t♦❣r❛❢❛r ❡✱ é ✐ss♦✱ q✉❡ ♠❛♥té♠ ❛ ♠❡♥s❛❣❡♠ ❡♠ s❡❣r❡❞♦✳ ◗✉❛♥❞♦ ❇♦❜ r❡❝❡❜❡ ❛ ♠❡♥s❛❣❡♠ ❝♦♠ ♦ t❡①t♦ ❝✐❢r❛❞♦✱ ❡❧❡ r❡❝✉♣❡r❛ ♦ t❡①t♦ ♣❧❛♥♦ ✉s❛♥❞♦ ❛ ❝❤❛✈❡ ❞❡ ❞❡s❝r✐♣✲ t♦❣r❛❢❛r✳
❊✈❛ ♣♦❞❡ t❡r q✉❛❧q✉❡r ✉♠ ❞♦s ♦❜❥❡t✐✈♦s ❛ s❡❣✉✐r✿
• ▲❡r ✉♠❛ ♠❡♥s❛❣❡♠✳
• ❊♥❝♦♥tr❛r ❛ ❝❤❛✈❡ ❡ ❧❡r t♦❞❛s ❛s ♠❡♥s❛❣❡♥s ❝r✐♣t♦❣r❛❢❛❞❛s ❝♦♠ ❛q✉❡❧❛ ❝❤❛✈❡✳ • ❆❞✉❧t❡r❛r ♠❡♥s❛❣❡♥s ❞❡ ❆❧✐❝❡ ❞❡ ❢♦r♠❛ q✉❡ ❇♦❜ r❡❝❡❜❛ ✉♠❛ ♠❡♥s❛❣❡♠ ❢❛❧s❛✳ • P❛ss❛r ♣♦r ❆❧✐❝❡✱ ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❇♦❜ ❛❝r❡❞✐t❡ q✉❡ ❡st❡❥❛ s❡ ❝♦♠✉♥✐❝❛♥❞♦ ❝♦♠
❆❧✐❝❡✳
❊✈❛ ♣♦❞❡ ❛❣✐r ❞❡ ✈ár✐❛s ♠❛♥❡✐r❛s ♣❛r❛ t❡♥t❛r ❛t✐♥❣✐r s❡✉s ♦❜❥❡t✐✈♦s✳ ❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ❡❧❛s ❞❡♣❡♥❞❡ ❞❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çã♦ q✉❡ ❡❧❛ ♣♦ss✉✐ q✉❛♥❞♦ ❡st✐✈❡r t❡♥t❛♥❞♦ ❞❡t❡r♠✐♥❛r ❛ ❝❤❛✈❡✳ P♦r ❡①❡♠♣❧♦✿
• ❊✈❛ ♣♦❞❡ t❡r ❛❝❡ss♦ s♦♠❡♥t❡ ❛♦ t❡①t♦ ❝r✐♣t♦❣r❛❢❛❞♦✳
• ❊✈❛ t❡♠ ❝ó♣✐❛ ❞❡ ✉♠ t❡①t♦ ❝r✐♣t♦❣r❛❢❛❞♦ ❡ ♦ t❡①t♦ ♣❧❛♥♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳
❊①❡♠♣❧♦ ✷✳✶✳ ❉✉r❛♥t❡ ❛ s❡❣✉♥❞❛ ❣✉❡rr❛ ♠✉♥❞✐❛❧✱ ♥♦ ❞❡s❡rt♦ ❞♦ ❙❛❛r❛✱ ✉♠ ♣♦st♦ ❆❧❡♠ã♦ ❡♥✈✐❛✈❛ t♦❞♦s ♦s ❞✐❛s ❛ ♠❡s♠❛ ♠❡♥s❛❣❡♠ ❝r✐♣t♦❣r❛❢❛❞❛ ❞✐③❡♥❞♦ q✉❡ ♥ã♦ ❤❛✈✐❛ ♥❛❞❛ ❞❡ ♥♦✈♦ ♣❛r❛ ✐♥❢♦r♠❛r✳ ❊♥tã♦✱ ❛ ❝❛❞❛ ❞✐❛✱ ♦s ❛❧✐❛❞♦s t✐♥❤❛♠ ❛❝❡ss♦ ❛ ✉♠❛ ❝ó♣✐❛ ❞❡ t❡①t♦ ❝r✐♣t♦❣r❛❢❛❞♦ ❡ ♦ t❡①t♦ ♣❧❛♥♦ ❝♦rr❡s♣♦♥❞❡♥t❡✳
❉❡ q✉❛❧q✉❡r ♠❛♥❡✐r❛✱ ♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡ ❊✈❛ é ❞❡s❝♦❜r✐r ❛ ❝❤❛✈❡ ♣❛r❛ ❝r✐♣t♦❣r❛❢❛r ❡ ❞❡s❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s ✈✐st♦ q✉❡ ✉♠ ❞♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ♣r✐♥❝í♣✐♦s ❞❡ s❡❣✉r❛♥ç❛ ✉s❛❞♦s ♥❛ ❝r✐♣t♦❣r❛✜❛ é ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❑❡r❝❦❤♦✛s ✸✿ ❞❡✈❡♠♦s s❡♠♣r❡ ❛ss✉♠✐r q✉❡
♦ ✐♥✐♠✐❣♦ ❝♦♥❤❡❝❡ ♦ ♠ét♦❞♦ s❡♥❞♦ ✉s❛❞♦✳
✷✳✶✳✶ ▼ét♦❞♦s ❞❡ ❈❤❛✈❡ ❙✐♠étr✐❝❛ ❡ ❞❡ ❈❤❛✈❡ Pú❜❧✐❝❛
❖s ♠ét♦❞♦s ♣❛r❛ ❝r✐♣t♦❣r❛❢❛r ❡ ❞❡s❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s ♣♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞♦s ❡♠ ♠ét♦❞♦s ❞❡ ❝❤❛✈❡ s✐♠étr✐❝❛ ❡ ♠ét♦❞♦s ❞❡ ❝❤❛✈❡ ♣ú❜❧✐❝❛✳
❈❤❛✈❡s ❙✐♠étr✐❝❛s✿ ❆s ❝❤❛✈❡s ♣❛r❛ ❝r✐♣t♦❣r❛❢❛r ❡ ❞❡s❝r✐♣t♦❣r❛❢❛r ♠❡♥s❛❣❡♥s sã♦ ❝♦✲ ♥❤❡❝✐❞❛s ♣❡❧♦ ❡♠✐ssár✐♦ ❡ r❡❝❡♣t♦r✳ ❊♠ ♠✉✐t♦s ❝❛s♦s✱ s❡♥❞♦ ❛ ♠❡s♠❛✳
