❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
❯♠❛ ❆❜♦r❞❛❣❡♠ ❞❛ ❆r✐t♠ét✐❝❛
▼♦❞✉❧❛r ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦
†♣♦r
❚ér❝✐♦ ❞❛s ◆❡✈❡s ❆❧♠❡✐❞❛
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐✲ ♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❛❣♦st♦✴✷✵✶✹ ❏♦ã♦ P❡ss♦❛ ✲ P❇
†❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❢♦✐ r❡❛❧✐③❛❞♦ ❝♦♠ ❛♣♦✐♦ ❞❛ ❈❆P❊❙✱ ❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡
▼♦❞✉❧❛r ♥♦ ❊♥s✐♥♦ ❇ás✐❝♦
♣♦r
❚ér❝✐♦ ❞❛s ◆❡✈❡s ❆❧♠❡✐❞❛
❚r❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡ ❝✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛✳ ❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❯❜❡r❧❛♥❞✐♦ ❇❛t✐st❛ ❙❡✈❡r♦ ✲ ❯❋P❇
Pr♦❢❛✳❉r❛✳ ▼❛r✐❛ ■s❛❜❡❧❧❡ ❙✐❧✈❛ ✲ ❯❊P❇
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ q✉❡ ♠❡ ♣r♦♣✐❝✐♦✉ s❛ú❞❡ ❡ ❢♦rç❛ ❞❡ ✈♦♥t❛❞❡ ♣❛r❛ q✉❡ ❡✉ ♣✉❞❡ss❡ ❝❤❡❣❛r ❛té ❛q✉✐✳
❆ ♠✐♥❤❛ ♠ã❡✱ ❘♦s✐❧❞❛ ❞❛s ◆❡✈❡s ❆❧♠❡✐❞❛✱ ♣♦r t❡r t✐❞♦ ♣❛❝✐ê♥❝✐❛ ❡ ❞❡❞✐❝❛çã♦ ❡♠ ❧✐❞❛r ❝♦♠✐❣♦ ❡ ✐♥s✐st✐r ♥♦s ♠❡✉s ❡st✉❞♦s✳
▼✐♥❤❛ ❢❛❧❡❝✐❞❛ ❛✈ó✱ ▼❛r✐❛ ▲✉í③❛ ❆❧✈❡s ❞❛s ◆❡✈❡s✱ q✉❡ ❝✉✐❞❛✈❛ ❞❡ t♦❞♦s ♦s ♠❡✉s ✐r♠ã♦s ❡ ❛ ♠✐♠ ❡♥q✉❛♥t♦ ♠✐♥❤❛ ♠ã❡ tr❛❜❛❧❤❛✈❛✱ ♥♦s ❞❛♥❞♦ ❡❞✉❝❛çã♦ ❞♦♠ést✐❝❛ ❡ ❝✉✐❞❛❞♦s ❞❡ ✉♠❛ s❡❣✉♥❞❛ ♠ã❡✳
❆ ▼✐♥❤❛ ❡s♣♦s❛✱ ❆♥❛ ❑❛r❧❛ ▼✐r❛♥❞❛ ❞❡ ▲✉♥❛✱ q✉❡♠ ♠❡ ✐♥❝❡♥t✐✈♦✉ ❛ ❢❛③❡r ❡ ❛ ❝♦♥❝❧✉✐r ♦ Pr♦❢♠❛t✱ ❡st❛♥❞♦ ❛♦ ♠❡✉ ❧❛❞♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞♦ ❝✉rs♦✳
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❇r✉♥♦ ❍❡♥r✐q✉❡ ❈❛r✈❛❧❤♦ ❘✐❜❡✐r♦✱ q✉❡ ♠❡ ❛❝♦❧❤❡✉ ♥❡st❡ ♣r♦❥❡t♦✳
❆♦s ♠❡✉s ❛♠✐❣♦s ❡ ❝♦❧❡❣❛s q✉❡ ❡st✐✈❡r❛♠ ❝♦♠✐❣♦ ♥❡st❛ ❥♦r♥❛❞❛✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❛♠✐❣♦ ❋r❛♥❝✐s❝♦ ❞♦ ◆❛s❝✐♠❡♥t♦ ▲✐♠❛✱ q✉❡✱ ❛❧é♠ ❞❡ ♠❡ ✐♥❝❡♥t✐✈❛r✱ t❡✈❡ ♣❛❝✐ê♥❝✐❛ ♣❛r❛ ❛❥✉❞❛r ♥❛ ❝♦♥str✉çã♦ ❞❡st❡ ♣r♦❥❡t♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s q✉❡ ❢❛③❡♠ ♦ Pr♦❢♠❛t ❛❝♦♥t❡❝❡r✱ ♣♦✐s s❡♠ ❡❧❡s ♥ã♦ ❤❛✈❡r✐❛ ❡st❡ s♦♥❤♦ ❞❡ ♠❡str❛❞♦ ♣r♦✜ss✐♦♥❛❧✐③❛♥t❡❀ s❡✐ q✉❡ s❡ ❡s❢♦rç❛r❛♠ ♣❛r❛ q✉❡ t♦❞♦s s❛íss❡♠ ❝♦♠ ♠❛✐s ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦✳
❆♦ ♠❡✉ ❢❛❧❡❝✐❞♦ ♣❛✐✱ ❋❡r♥❛♥❞♦ ❏♦sé ❇❛rr❡t♦✱ s❡♠♣r❡ s❡ ♣r❡♦❝✉♣♦✉ ❝♦♠ ♦s ❡st✉❞♦s ❞❡ t♦❞♦s ❡♠ ♠✐♥❤❛ ❝❛s❛✳
❆♦s ❝♦❧❡❣❛s ❞❡ tr❛❜❛❧❤♦ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❏♦ã♦ ❏♦sé ❞❛ ❈♦st❛ q✉❡ s❡♠♣r❡ ❡st❛✈❛♠ ❞❛♥❞♦ ❢♦rç❛ ♣❛r❛ q✉❡ ❢♦ss❡ ♣♦ssí✈❡❧ ❛ ❝♦♥❝❧✉sã♦ ❞♦ Pr♦❢♠❛t✳
❆♦s ♠❡✉ ❛❧✉♥♦s q✉❡ ♣❛rt✐❝✐♣❛r❛♠ ❞❛s ❛✉❧❛s✱ ♠❡s♠♦ ♦s q✉❡ t✐✈❡r❛♠ ❞❡ ❞❡s✐st✐r ♣♦r ♠♦t✐✈♦s ❛❞✈❡rs♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❡❧❡s✿ ❆♥❛ ❇❡❛tr✐③ ▼❡sq✉✐t❛ ❞❛ ❙✐❧✈❛✱ ▲❡tí❝✐❛ ❘❛q✉❡❧ ▼❡❞❡✐r♦s ❞❡ ▲✐♠❛ ❇❛r❜♦s❛✱ ❘✐❦❡❧♠❡ ❊s❝ó❝✐♦ P❡r❡✐r❛ ❞❛ ❙✐❧✈❛ ❡ ❱✐✈✐❛♥❡ ❱❡ríss✐♠♦ ❞❡ P❛✐✈❛✱ q✉❡ ❢♦r❛♠ ❛té ♦ ✜♠✱ ♠❡s♠♦ ❝♦♠ t♦❞❛s ❛s ❞✐✜❝✉❧❞❛❞❡s✳
❉❡❞✐❝♦ ❡st❡ tr❛❜❛❧❤♦ ❛ ♠✐♥❤❛ ❡s♣♦s❛ q✉❡ ❡stá ❣rá✈✐❞❛ ❞❡ ✉♠❛ ♠❡♥✐♥❛ ❛ q✉❡♠ t❛♥t♦ ❛♠♦✱ ▲✉í③❛ ❑❛r❧❛ ▼✐r❛♥❞❛ ❞❡ ❆❧♠❡✐❞❛✱ ♠✐♥❤❛ ♠ã❡ q✉❡ ❡st❡✈❡ s❡♠♣r❡ ❝♦♠✐❣♦✳
❘❡s✉♠♦
❊st❡ tr❛❜❛❧❤♦ ❝♦♥s✐st❡ ❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞❡ ❆r✐t♠ét✐❝❛ ✈♦❧t❛❞❛ ♣❛r❛ ❛s ❖❧✐♠♣í❛❞❛s ❇r❛s✐❧❡✐r❛s ❞❡ ▼❛t❡♠át✐❝❛ ❝♦♠ ❛❧✉♥♦s ❞❛ ❘❡❞❡ Pú❜❧✐❝❛ ❞❡ ❊♥✲ s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ❛♣r❡s❡♥t❛♥❞♦ ❛ t❡♦r✐❛ ❝♦♠♦ ♦s t❡♦r❡♠❛s✱ ❞❡✜♥✐çõ❡s✱ ❝♦r♦❧ár✐♦s ❡ ♣r♦♣♦s✐çõ❡s❀ ♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ♣♦ss❛ ❝♦♥str✉✐r ❡ ❝♦♠♣r❡❡♥❞❡r r❡❣r❛s ❞❡ ❞✐✈✐s✐❜✐❧✐✲ ❞❛❞❡✱ ❛ss✐♠ ❝♦♠♦ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ♣♦❞❡♠ s❡ ❛♣r❡s❡♥t❛r ♥♦ ❞✐❛ ❛ ❞✐❛ ♦✉ ♦s ♣r♦♣♦st♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
✳✳✳
❙✉♠ár✐♦
✶ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s ✹
✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✶✳✶ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✶✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✶✳✸ P❛r✐❞❛❞❡ ❞❡ ✉♠ ◆ú♠❡r♦ ■♥t❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✶✳✶✳✹ ❈♦♥❥✉♥t♦ ❞♦s ❉✐✈✐s♦r❡s ❞❡ ✉♠ ◆ú♠❡r♦ ■♥t❡✐r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✺ ❉✐✈✐s♦r❡s ❈♦♠✉♥s ❞❡ ❉♦✐s ◆ú♠❡r♦s ■♥t❡✐r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✳✻ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✶✳✼ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✶✳✽ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✶✳✾ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷ ❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✸ ❈♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✶✳✸✳✶ Pr♦♣r✐❡❞❛❞❡s ❞❛s ❈♦♥❣r✉ê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✶✳✸✳✷ ❆❧❣✉♠❛s ❘❡❣r❛s ❞❡ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷ ❆r✐t♠ét✐❝❛ ♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ✹✷
✷✳✶ ❆ ❆r✐t♠ét✐❝❛ ❡ ♦ P❈◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✷✳✷ ❆ ❆r✐t♠ét✐❝❛ ❡ ❛ ❖❇▼❊P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✸ ❖ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❚r❛❜❛❧❤♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✷✳✹ ❉❛s ❆✉❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✹✳✶ ❉♦ Pr✐♠❡✐r♦ ▼♦♠❡♥t♦ ❞❛s ❆✉❧❛s ✲ ◗✉❡st✐♦♥ár✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✷✳✹✳✷ ❉♦ ❙❡❣✉♥❞♦ ▼♦♠❡♥t♦ ❞❛s ❆✉❧❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✷✳✹✳✸ ❉♦ ❚❡r❝❡✐r♦ ▼♦♠❡♥t♦ ✲ ❘❡❛♣❧✐❝❛çã♦ ❞♦ ◗✉❡st✐♦♥ár✐♦ ✳ ✳ ✳ ✳ ✳ ✹✽ ✷✳✺ ❉❛s ❇❛rr❡✐r❛s ♦✉ ❉✐✜❝✉❧❞❛❞❡s ❊♥❝♦♥tr❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✷✳✺✳✶ ❉❛ ❋❛❧t❛ ❞❡ ■♥t❡r❡ss❡ ❞♦ ❊st✉❞❛♥t❡ ❊♥✈♦❧✈✐❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✺✳✷ ❋❛❧t❛ ❞❡ ❊s♣❛ç♦ ❋ís✐❝♦ ❆❞❡q✉❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✷✳✺✳✸ ❋❛❧t❛ ❞❡ ❆♣♦✐♦ ❡ ■♥❝❡♥t✐✈♦ ❞♦ ●♦✈❡r♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✷✳✺✳✹ ❋❛❧t❛ ❞❡ ❇❛s❡ ❞♦s ❆❧✉♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✷✳✺✳✺ ❋❛❧t❛ ❞❡ ▼❛t❡r✐❛❧ ❉✐❞át✐❝♦ ❉✐s♣♦♥í✈❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✷✳✺✳✻ ❋✐❣✉r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
■♥tr♦❞✉çã♦
❊st❡ tr❛❜❛❧❤♦ tr❛t❛ ❞❡ ✉♠❛ ♣❡sq✉✐s❛ ❞❡ ❝❛♠♣♦ s♦❜r❡ ❛ ❆r✐t♠ét✐❝❛ ❡ r❡s♦❧✉çõ❡s ❞❡ q✉❡stõ❡s ✈♦❧t❛❞❛s ♣❛r❛ ❛ ❖❧✐♠♣í❛❞❛ ❇r❛s✐❧❡✐r❛ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛s ❊s❝♦❧❛s Pú❜❧✐❝❛s ✭❖❇▼❊P✮ ✉s❛♥❞♦ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦✿ ❞❡✜♥✐çõ❡s✱ Pr♦♣♦s✐çõ❡s✱ ❚❡♦r❡♠❛s✱ ▲❡♠❛s ❡ ❈♦r♦❧ár✐♦s✳ ❆ ♣❡sq✉✐s❛ t❛♠❜é♠ ♠♦str❛ ❛ ❢❛❧t❛ ❞❡ ✐♥❝❡♥t✐✈♦ ❛♦s ♣r♦✜ss✐♦♥❛✐s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❛♦s ❡st✉❞❛♥t❡s✱ ❡♠❜♦r❛ t❡♥❤❛♠♦s ❡st✉❞❛♥t❡s ❝♦♠ ❣r❛♥❞❡s ♣♦t❡♥❝✐❛✐s ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❝♦♠ ❛ ▼❛t❡♠át✐❝❛✳
❆ ▼❛t❡♠át✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r ❛ ❆r✐t♠ét✐❝❛✱ s❡ ❛♣r❡s❡♥t❛ ❝♦♠ ✉♠❛ ❞❛s ♠❛✐♦r❡s ✈✐❧ãs✱ ❥✉♥t♦ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ♣♦rt✉❣✉ês✱ ♥♦s í♥❞✐❝❡s ❞❡ r❡♣r♦✈❛çã♦ ❡ ❛❜❛♥❞♦♥♦ ❡s❝♦❧❛r✳ ➱ ✉♠❛ ❞✐s❝✐♣❧✐♥❛ ♣♦✉❝♦ ❛♣r❡❝✐❛❞❛ ♣❡❧❛ ♠❛✐♦r✐❛ ❞♦s ❡st✉❞❛♥t❡s ♥❛s ❡s❝♦❧❛s✱ ❛♣❡s❛r ❞❡ s❡✉ ✉s♦ ❞✐ár✐♦ ❡♠ ♥♦ss❛s ✈✐❞❛s✿ ♣❛ss❛r ♦✉ r❡❝❡❜❡r ♦ tr♦❝♦ ❞❡ ✉♠❛ ❞❡✲ t❡r♠✐♥❛❞❛ ❝♦♠♣r❛✱ ♦ t❡♠♣♦ q✉❡ ❢❛❧t❛ ♣❛r❛ r❡❛❧✐③❛r t❛r❡❢❛s ♦✉ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♣r♦❣r❛♠❛ ❝♦♠❡ç❛r ♦✉ t❡r♠✐♥❛r✱ ❝♦♥t❛❣❡♠ ❡ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s✳ ❙❡♥❞♦ ❛ss✐♠✱ ❝❛❜❡ ❛♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛ ❡♥❝♦♥tr❛r ❢♦r♠❛s ❞❡ ❞❡s♣❡rt❛r ❛ ❝✉r✐♦s✐❞❛❞❡ ❡ ❛ ✈♦♥t❛❞❡ ❞♦ ❡st✉❞❛♥t❡ ❞❡ q✉❡r❡r ❛♣r❡♥❞❡r ❡ ❞❡ ❢❛③❡r ♥♦✈❛s ❞❡s❝♦❜❡rt❛s ♥❛ ár❡❛ ❞❛ ▼❛t❡♠á✲ t✐❝❛ ❡ ❛♦ ●♦✈❡r♥♦ ❞❡ ❞❛r ✐♥❝❡♥t✐✈♦ ❛♦s ❡st✉❞❛♥t❡s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❛ ▼❛t❡♠át✐❝❛✱ ♣♦✐s ❛q✉✐ s❡ tr❛t❛ ❞❛ ❞✐s❝✐♣❧✐♥❛ ❜ás✐❝❛ ❞❛s ❝✐ê♥❝✐❛s ❡ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ t❡❝♥♦❧ó❣✐❝♦✳
❖ tr❛❜❛❧❤♦ q✉❡ ❛q✉✐ s❡ ❛♣r❡s❡♥t❛ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ ❞♦✐s ❝❛♣ít✉❧♦s✱ ❡♠ q✉❡ ✐r❡♠♦s ❞❡♠♦♥str❛r t♦❞❛ ❛ t❡♦r✐❛ ♣❛ss♦ ❛ ♣❛ss♦ ❡ ❝✐t❛r ❡①❡♠♣❧♦s ❡✱ t❛♠❜é♠✱ ❞✐s❝✉t✐r ✉♠❛ ❛♥á❧✐s❡ ❞❛ ❛♣❧✐❝❛çã♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❞♦s ❛ss✉♥t♦s ❡♥✈♦❧✈✐❞♦s✳
❖ ❈❛♣ít✉❧♦ ✶ tr❛t❛ ❞❛ ▼❛t❡♠át✐❝❛ ❢♦r♠❛❧✱ ❞❡✜♥✐çõ❡s✱ ♥♦t❛çõ❡s✱ ❞❡♠♦♥str❛çõ❡s ❞❡ t❡♦r❡♠❛s✱ ❧❡♠❛s✱ ❝♦r♦❧ár✐♦s ❡ ♣r♦♣♦s✐çõ❡s✱ s❡❣✉✐❞♦ ❞❡ ❡①❡♠♣❧♦s✱ ❛ss✐♠ ❝♦♠♦ tr❛r❡♠♦s ❞❡♠♦♥str❛çõ❡s ❞❡ ❛❧❣✉♠❛s r❡❣r❛s ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ✉s❛♥❞♦ ❝♦♥❣r✉ê♥❝✐❛ ❡ ❞❡✐①❛♥❞♦ ❛ ✐❞❡✐❛ ♣❛r❛ q✉❡ s❡ ♣♦ss❛ ❝♦♥str✉✐r ♦✉tr❛s r❡❣r❛s✳
❖ ❈❛♣ít✉❧♦ ✷ tr❛t❛ ❞❡ r❡❧❛t♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ✉♠❛ t✉r♠❛ ❢♦r♠❛❞❛ ❡s♣❡❝✐✜❝❛✲ ♠❡♥t❡ ♣❛r❛ ❡st❡ tr❛❜❛❧❤♦✱ r❡❧❛t❛♥❞♦ ♦ ✐♥t❡r❡ss❡ ♦✉ ❛ ❢❛❧t❛ ❞❡❧❡ ♥♦s ❡st✉❞❛♥t❡s✱ ❢❛❧t❛ ❞❡ ✐♥❝❡♥t✐✈♦ ❡ ❞❡ ❛♣♦✐♦ ♣❛r❛ ❢♦r♠❛r ✉♠ ❣r✉♣♦ ❞❡ ❡st✉❞♦ ✈♦❧t❛❞♦ ♣❛r❛ ❛ ❖❇▼❊P ♦✉ q✉❛❧q✉❡r ♦✉tr❛ ✜♥❛❧✐❞❛❞❡✱ ❡ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❡st✉❞❛♥t❡ ❝♦♠♣❛r❛♥❞♦ s❡✉ ❞❡s❡♠♣❡♥❤♦ ❛♥t❡s✱ ❞✉r❛♥t❡ ❡ ❞❡♣♦✐s ❞❛s ❛✉❧❛s✳
❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❞✐s❝✉t✐r❡♠♦s ❛❝❡r❝❛ ❞♦s ❝♦♥❝❡✐t♦s✱ ❞❡✜♥✐çõ❡s ❡ ♣r♦♣♦s✐çõ❡s ❞❛ ❚❡♦r✐❛ ❞♦s ◆ú♠❡r♦s✱ ❜❛s❡❛❞♦s ♥❛s ♦❜r❛s ❞❡ [✶] ❡ [✸]✱ q✉❡ s❡r✈✐rã♦ ❞❡ s✉♣♦rt❡ ♥❛s
❛t✐✈✐❞❛❞❡s ♣r♦♣♦st❛s ♥♦ ❝❛♣ít✉❧♦ s✉❜s❡q✉❡♥t❡✳
✶✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡
❉❡✜♥✐çã♦ ✶ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s a ❡ b ❝♦♠ a 6= 0✱ ❞✐r❡♠♦s q✉❡ a ❞✐✈✐❞❡ b✱ ❡s❝r❡✈❡♥❞♦ a|b✱ q✉❛♥❞♦ ❡①✐st✐rq ∈Zt❛❧ q✉❡ b =a·q✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s t❛♠❜é♠
q✉❡ a é ✉♠ ❞✐✈✐s♦r ♦✉ ✉♠ ❢❛t♦r ❞❡b ♦✉✱ ❛✐♥❞❛✱ q✉❡ b é ♠ú❧t✐♣❧♦ ❞❡ a✳
◆♦t❡ q✉❡ ❛ ♥♦t❛çã♦ a | b ♥ã♦ r❡♣r❡s❡♥t❛ ♦♣❡r❛çã♦ ❡♠ Z✱ ♥❡♠ r❡♣r❡s❡♥t❛ ✉♠❛
❢r❛çã♦✳ ❚r❛t❛✲s❡ ❞❡ ✉♠❛ s❡♥t❡♥ç❛ q✉❡ ❞✐③ s❡r ✈❡r❞❛❞❡ q✉❡ ❡①✐st❡q∈Zt❛❧ q✉❡b =a·q✳ ❊①❡♠♣❧♦✿ ❉❛❞♦s3❡9✱ t❡♠♦s q✉❡3|9✱ ♣♦✐s ♣♦❞❡♠♦s t♦♠❛r3∈Zt❛❧ q✉❡9 = 3·3❀
❉❛❞♦s3 ❡18✱ t❡♠♦s q✉❡ 3|18✱ ♣♦✐s ♣♦❞❡♠♦s t♦♠❛r 6∈Z t❛❧ q✉❡ 18 = 3·6❀
❉❛❞♦s2 ❡4✱ t❡♠♦s q✉❡ 2|4✱ ♣♦✐s ♣♦❞❡♠♦s t♦♠❛r2∈Z t❛❧ q✉❡ 4 = 2·4❀
❉❛❞♦s2 ❡6✱ t❡♠♦s q✉❡ 2|6✱ ♣♦✐s ♣♦❞❡♠♦s t♦♠❛r3∈Z t❛❧ q✉❡ 6 = 2·3✳
❆ ♥❡❣❛çã♦ ❞❛ s❡♥t❡♥ç❛ a |b é r❡♣r❡s❡♥t❛❞❛ ♣♦r a∤ b q✉❡ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ q ∈Z t❛❧ q✉❡ b=a·q✳
❊①❡♠♣❧♦✿ ❉❛❞♦s3❡5✱ t❡♠♦s q✉❡3∤5✱ ♣♦✐s t♦♠❛♥❞♦1∈Z✱ t❡♠♦s q✉❡5>3·1❡✱
t♦♠❛♥❞♦ 2∈Z✱ t❡♠♦s q✉❡5<3·2✳ ❈♦♠♦ s❛❜❡♠♦s q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♥ú♠❡r♦
✐♥t❡✐r♦ ❡♥tr❡ 1❡ 2✱ t❛❧ q✉❡ 5 = 3·q✱ t❡♠♦s q✉❡ 3∤5✳
❙❡a|b❀ s❡❥❛q∈Zt❛❧ q✉❡b =a·q✳ ❖ ♥ú♠❡r♦ ✐♥t❡✐r♦q é ❝❤❛♠❛♥❞♦ ❞❡ q✉♦❝✐❡♥t❡ ❞❡ b ♣♦r a ❡ é ❞❡♥♦t❛❞♦ ♣♦r q= b
a✳ ❊①❡♠♣❧♦✿ ❉❛❞♦s 0 ❡1✱ t❡♠♦s q✉❡ 0
1 = 0
❉❛❞♦s0 ❡2✱ t❡♠♦s q✉❡ 0 2 = 0
❉❛❞♦s18 ❡3✱ t❡♠♦s q✉❡ 18 3 = 6✳
Pr♦♣♦s✐çã♦ ✶✳✶ ❙❡❥❛♠ a, b∈Z∗ ❡ c∈Z ✳ ❚❡♠✲s❡ q✉❡✿ ✭✐✮ 1|c✱ −1|c✱ a|a ❡ a|0;∀a6= 0❀
✭✐✐✮ s❡ a |b ❡ b|c✱ ❡♥tã♦ a |c❀
✭✐✐✐✮ s❡ a|1 ❡♥tã♦ a=±1❀
✭✐✈✮ a|b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a| −b✳ ❉❡♠♦♥str❛çã♦✿
✭✐✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❞❛❞♦s 1, c ∈ Z✱ t❡♠♦s q✉❡ 1 | c s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r k ∈Z✱ t❛❧ q✉❡c= 1·k✱ ❧♦❣♦ c=k✳
❉❡ ♠❡s♠♦ ♠♦❞♦✱ ❞❛❞♦s −1, c ∈Z✱ t❡♠♦s q✉❡−1|c s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r t ∈Z✱ t❛❧ q✉❡ c= (−1)·t, ♦✉ s❡❥❛✱ c=−t ❞♦♥❞❡ t❡♠♦s t=−c✳
❉❛❞♦ ♦ ♥ú♠❡r♦ a∈Z✱ t❡♠♦s q✉❡ ❞❡ ❢❛t♦ a|a✱ ♣♦✐s1∈Z é t❛❧ q✉❡ a= 1·a✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s a, 0∈Z✱ t❡♠♦s q✉❡a|0✱ ♣♦✐s ❝♦♠♦0∈Z t❡♠♦s0 = a·0✳
✭✐✐✮ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❞❛❞♦s a, b e c∈Z ❡♥tã♦a |b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r q1 ∈Z t❛❧ q✉❡
b =a·q1 ✭✶✳✶✮
❡ b|c s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐rq2 ∈Z t❛❧ q✉❡ c=b·q2✳
❙✉❜st✐t✉✐♥❞♦ ❡♠ ✭✶✳✶✮ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♥❛ s❡❣✉♥❞❛✱ t❡♠♦s c= (a·q1)·q2.
❊♥tã♦
c=a·(q1·q2).
❈♦♠♦ q1 ❡q2 ∈Z✱ t❡♠♦s q✉❡ q1·q2 =q ∈Z✳ ❆ss✐♠ t❡♠♦s c=a·(q1·q2)
❡ c=a·q, ♦✉ s❡❥❛✱ a |c.
✭✐✐✐✮ ❉❛❞♦s ♦s ♥ú♠❡r♦s a, 1 ∈ Z✱ t❡♠♦s q✉❡ a | 1✱ s❡ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r q ∈ Z✱
t❛❧ q✉❡ 1 =a·q✳ ❈♦♠♦ a, q ∈ Z✱ t❡♠✲s❡ q✉❡ ♦✉ a = 1 ❡ q = 1 ♦✉ a = −1 ❡
q =−1✳
✭✐✈✮ ❚❡♠♦s q✉❡ s❡ a| b✱ ❡♥tã♦ ❡①✐st❡ c∈Z✱ t❛❧ q✉❡ b =a·c✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ♣♦r −1✱ t❡♠♦s−b =a·(−c)✱ ✐st♦ é✱a| −b✳
❖❜s❡r✈❛çã♦ ✶ ❖ ✐t❡♠ ✭✐✮ ❞❛ ♣r♦♣♦s✐çã♦ ❛❝✐♠❛ ♥♦s ❞✐③ q✉❡ t♦❞♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ é ❞✐✈✐sí✈❡❧ ♣♦r ±1 ❡✱ s❡ ♥ã♦ ♥✉❧♦✱ ♣♦r s✐ ♠❡s♠♦✳
Pr♦♣♦s✐çã♦ ✶✳✷ ❙❡ a, b, c, d ∈ Z✱ ❝♦♠ a 6= 0 ❡ c 6= 0✱ t❛❧ q✉❡ a | b ❡ c | d✱ ❡♥tã♦ a·c|b·d✳
❉❡♠♦♥str❛çã♦✿ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ ❞❛❞♦s a, b, c❡ d∈Z✱ t❡♠♦s q✉❡
a | b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r q1 ∈ Z❀ b = a·q1 ❡ c | d s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st✐r
q2 ∈Z✱ t❛❧ q✉❡ d=c·q2✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ❛s ❞✉❛s ❡q✉❛çõ❡s✱ t❡♠♦s✿
b·d= (a·q1)·(c·q2).
❆ss✐♠✱ ✉s❛♥❞♦ ❛s ♣r♦♣r✐❡❞❛❞❡s ❝♦♠✉t❛t✐✈❛ ❡ ❛ ❛ss♦❝✐❛t✐✈❛✱ t❡♠♦s
b·d= (a·c)·(q1·q2)
b·d= (a·c)·q. ❚❡♠♦s q✉❡ q1·q2 =q ∈Z✱ ♦✉ s❡❥❛✱
a·c|b·d.
❊♠ ♣❛rt✐❝✉❧❛r✱ s❡a |b✱ ❡♥tã♦ a·c|b·c✱ ♣❛r❛ t♦❞♦ c∈Z✱ c6= 0✳
❊①❡♠♣❧♦✿ ❙❛❜❡♠♦s q✉❡ 2|6✱ ♣♦✐s3∈Z é t❛❧ q✉❡ 6 = 2·3 ❡ 3|12✱ ♣♦✐s4∈Z é
t❛❧ q✉❡ 12 = 3·4✳ ❉❡ss❛ ❢♦r♠❛✱ s✉❜st✐t✉✐♥❞♦ ❛ ♣r✐♠❡✐r❛ ❡q✉❛çã♦ ♥❛ s❡❣✉♥❞❛✱ t❡♠♦s 3·2|6·12⇒6|72✳
Pr♦♣♦s✐çã♦ ✶✳✸ ❙❡❥❛♠ a, b, c∈Z✱ ❝♦♠ a6= 0✱ t❛❧ q✉❡ a|(b+c)✳ ❊♥tã♦ a|b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a|c✳
❉❡♠♦♥str❛çã♦✿ ❙❡ a | (b+c)✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡✱ t❡♠♦s q✉❡ ❡①✐st❡
✉♠ q1 ∈Z✱ t❛❧ q✉❡ (b+c) = a·q1✳
❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ a |b✱ ✐st♦ é✱ ❡①✐st❡ q2 ∈Z✱ t❛❧ q✉❡b =a·q2✳
❙✉❜st✐t✉✐♥❞♦ ❛ s❡❣✉♥❞❛ ❡q✉❛çã♦ ♥❛ ♣r✐♠❡✐r❛✱ t❡♠♦s
a·q2+c=a·q1.
❆ss✐♠✱
c=a·q1−a·q2,
♦✉ s❡❥❛✱
c=a·(q1−q2).
❊s❝r❡✈❡♥❞♦ q = (q2−q1)∈Z✱ t❡♠♦s✿
c=a·q. P♦rt❛♥t♦✱
a|c.
❙✉♣♦♥❤❛✱ ❛❣♦r❛✱ q✉❡a|c✳ ❉❡ss❛ ❢♦r♠❛✱ s❡ a|(b+c)✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ❞✐✈✐s✐❜✐✲
❧✐❞❛❞❡✱ ❡①✐st❡ ✉♠ q1 ∈Z✱ t❛❧ q✉❡ (b+c) = a·q1✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦a |c✱ ❡①✐st❡
t❛♠❜é♠ q3 ∈Z✱ t❛❧ q✉❡ c=a·q3✳
❙✉❜st✐t✉✐♥❞♦ ❡st❛ c♥❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r✱ t❡♠♦s
(b+a·q3) = a·q1,
✐st♦ é✱
b =a·q1−a·q3.
❉❡ss❛ ❢♦r♠❛✱
b =a·(q1−q3).
◆♦✈❛♠❡♥t❡✱ t❡♠♦s q✉❡ ❡①✐st❡ (q1−q3) = q′ ∈Z✱ t❛❧ q✉❡ b=a·q′,♦✉ s❡❥❛✱ a |b✳
❊①❡♠♣❧♦✿ ❖❜s❡r✈❡ q✉❡ 3 | 18✳ ❆ss✐♠✱ 3 | (6 + 12)✳ ❉❡ss❛ ❢♦r♠❛✱ 3 | 6✱ s❡✱ ❡
s♦♠❡♥t❡ s❡✱ 3 | 12✳ ❉❡ ❢❛t♦ t❡♠♦s q✉❡ 3 | 6✱ ♣♦✐s ❡①✐st❡ 2 ∈ Z, t❛❧ q✉❡ 6 = 3·2 ❡ 3|12✱ ♣♦✐s ❡①✐st❡4∈Z t❛❧ q✉❡ 12 = 3·4✳
Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛♠ a, b, c∈Z✱ ❝♦♠ a 6= 0✱ t❛❧ q✉❡ a |(b−c)✳ ❊♥tã♦ a|b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ a|c✳
❆ ❞❡♠♦♥str❛çã♦ ❞❡st❛ ♣r♦♣♦s✐çã♦ é ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✸ ❡ ❞♦ ✐t❡♠ ✭✐✈✮ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶✳
❊①❡♠♣❧♦✿ ❉❛❞♦s ♦s ♥ú♠❡r♦s 3, 6 ❡9 ∈Z✱ t❡♠♦s q✉❡ 3| (6−9)✳ ❊♥tã♦ 3| 6s❡✱
❡ s♦♠❡♥t❡ s❡✱ 3| 9✳ ❉❡ ❢❛t♦✱ 3| (6−9)✱ ✐st♦ é✱ 3| −3✱ ♣♦✐s ❡①✐st❡ −1∈ Z✱ t❛❧ q✉❡
−3 = 3·(−1) ❡ 3 | 6✱ ♣♦✐s ❡①✐st❡ 2 ∈ Z✱ t❛❧ q✉❡ 6 = 3·2✳ ❊♥tã♦ 3 | −9✱ ♦ q✉❡ é
✈❡r❞❛❞❡✱ ♣♦✐s ❡①✐st❡ −3∈Z✱ t❛❧ q✉❡ −9 = 3·(−3)✳
❉❛ ♠❡s♠❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ ❝♦♠♦3|(6−9)✱ ✐st♦ é✱ 3| −3✱ ♣♦✐s ❡①✐st❡ −1∈Z✱
t❛❧ q✉❡ −3 = 3·(−1) ❡ 3 | −9✱ ♣♦✐s ❡①✐st❡ −3 ∈ Z✱ t❛❧ q✉❡ −9 = 3·(−3)✱ ❡♥tã♦ 3|6✳
Pr♦♣♦s✐çã♦ ✶✳✺ ❙❡ a, b, c ∈ Z✱ ❝♦♠ a 6= 0✱ sã♦ t❛✐s q✉❡ a | b ❡ a | c✱ ❡♥tã♦
a|(xb±yc)✱ ♣❛rt❛ t♦❞♦ x, y ∈Z✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦a |b ❡a |c✱ ♣❡❧❛ ♣r♦♣♦s✐çã♦ ✶✳✻✱ t❡♠♦s q✉❡a|x·b❡a|y·c✳
❉❡ss❛ ❢♦r♠❛✱ ❡①✐st❡♠ q1, q2 ∈ Z✱ t❛✐s q✉❡✱ x·b = a·q1 ❡ y·c= a·q2 ❙♦♠❛♥❞♦
♦✉ s✉❜tr❛✐♥❞♦ ❡st❛s ❞✉❛s ❡q✉❛çõ❡s✱ t❡♠♦s
x·b±y·c=a·q±a·q′. ▲♦❣♦✱
x·b±y·c=a·(q±q′). ❈♦♠♦ ❡①✐st❡ (q±q′) =q
3 ∈Z✱ t❡♠♦s
x·b±y·c=a·q3.
P♦rt❛♥t♦✱
a|(x·b±y·c).
❊①❡♠♣❧♦✿ ❚❡♠♦s q✉❡ 7|7 ❡7|14✱ ❡♥tã♦ 7|(7·x±14·y)✳
❱✐♠♦s✱ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✻✱ q✉❡ s❡ a | b ❡♥tã♦ a | bx✱ ✐st♦ é✱ s❡ 7 | 7✱ ❡♥tã♦ 7
❞✐✈✐❞✐rá t♦❞♦s ♦s ♠ú❧t✐♣❧♦s ❞❡ 7❀ ❞❛ ♠❡s♠❛ ❢♦r♠❛✱ 7| 14✱ ❡♥tã♦ 7 ❞✐✈✐❞✐rá t♦❞♦s ♦s
♠ú❧t✐♣❧♦s ❞❡ 14✳
Pr♦♣♦s✐çã♦ ✶✳✻ ❙❡ a, b∈Z✱ ❝♦♠ a6= 0✱ ❡ a|b✱ ❡♥tã♦ a|b·n,∀ n∈Z✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ a |b✱ ❡♥tã♦ ❡①✐st❡ q1 ∈Z t❛❧ q✉❡ b =a·q1✳ ▼✉❧t✐♣❧✐❝❛♥❞♦
❛ ú❧t✐♠❛ ✐❣✉❛❧❞❛❞❡ ♣♦r n✱ t❡♠♦s
b·n =a·q1·n ♦✉b·n =a·(q1·n).
❈♦♠♦ q1, n∈Z✱ t❡♠♦s q✉❡ ❡①✐st❡ q1 ·n=q∈Z✳ P♦rt❛♥t♦✱
b·n=a·q✱ ✐st♦ é✱a|b·n✳
❊①❡♠♣❧♦✿ ❙❡ 5|15✱ ❡♥tã♦ 5|30✱ ♣♦✐s 30 = 15·2✳
Pr♦♣♦s✐çã♦ ✶✳✼ ❙❡ a, b∈Z∗✱ t❡♠✲s❡ q✉❡ a|b✱ ❡♥tã♦ |a|6|b|✳
❉❡♠♦♥str❛çã♦✿ a | b s❡✱ ❡ s♦♠❡♥t❡ s❡✱ q ∈ Z∗✱ t❛❧ q✉❡ b = a·q, ❞❡ss❛ ❢♦r♠❛✱
|b| = |a·q|✱ ❛ss✐♠✱ |b| = |a| · |q| ❡ ❝♦♠♦ b 6= 0✱ t❡♠♦s q✉❡ q 6= 0✱ ❛ss✐♠ |q| > 1 ❡
♣♦rt❛♥t♦✿ |b|=|a| · |q|>|a| ·1>|a|✳ ❊①❡♠♣❧♦✿ ❈❧❛r❛♠❡♥t❡✱ ❛ r❡❝í♣r♦❝❛ ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ♥ã♦ é ✈á❧✐❞❛✱ ♣♦✐s✱ ♣♦r ❡①❡♠♣❧♦✱ |3|>|2|❀ ❡✱ ♥♦ ❡♥t❛♥t♦✱ 2∤3✳
◆♦t❡ q✉❡ ❛ r❡❧❛çã♦ ❞❡ ❞✐✈✐s✐❜✐❧✐❞❛❞❡ ❡♠ Z+∗ é ✉♠❛ r❡❧❛çã♦ ❞❡ ♦r❞❡♠✱ ♣♦✐s
✭✐✮ é r❡✢❡①✐✈❛✿ ∀a∈Z∗, a|a❀
✭✐✐✮ é ❛♥t✐ss✐♠étr✐❝❛✿ s❡ a|b ❡ b|a✱ ❡♥tã♦ a=b❀ ✭✐✐✐✮ é tr❛♥s✐t✐✈❛✿ s❡ a|b ❡b |c✱ ❡♥tã♦ a |c✳
✶✳✶✳✶ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛
❊✉❝❧✐❞❡s✱ ❡♠ s✉❛ ♦❜r❛ ❊❧❡♠❡♥t♦s✱ ♣r♦✈♦✉ q✉❡ ♠❡s♠♦ q✉❛♥❞♦a∤b✱ ❝♦♠a❡b∈N
é ♣♦ssí✈❡❧ ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡ b ♣♦r a ❝♦♠ ♦ r❡st♦✳
❚❡♦r❡♠❛ ✶✳✽ ✭❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛✮ ❙❡❥❛♠ a ❡ b ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s ❝♦♠ 0 < a < b✳ ❊①✐st❡♠ ❞♦✐s ú♥✐❝♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s q ❡ r t❛✐s q✉❡
b =a·q+r; r < a.
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡b > a❡ ❝♦♥s✐❞❡r❡✱ ❡♥q✉❛♥t♦ ✜③❡r s❡♥t✐❞♦✱ ♦s ♥ú♠❡r♦s b, b−a, b−2·a, . . . , b−n·a, . . .✳ ◆♦t❡ q✉❡ ♦ ❝♦♥❥✉♥t♦S ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❛❝✐♠❛ t❡♠ ✉♠ ♠❡♥♦r ❡❧❡♠❡♥t♦r =b−q·a✳ ❱❛♠♦s ♣r♦✈❛r q✉❡r t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ r❡q✉❡r✐❞❛✱ ♦✉ s❡❥❛✱ q✉❡ r < a✳ ❙❡ a|b✱ ❡♥tã♦ r = 0 ❡ ♥❛❞❛ ♠❛✐s t❡♠♦s ❛ ♣r♦✈❛r✳ ❙❡✱
♣♦r ♦✉tr♦ ❧❛❞♦✱ a ∤b✱ ❡♥tã♦ r6= 0✱ ❧♦❣♦✱ ❜❛st❛ ♠♦str❛r q✉❡ ♥ã♦ ♣♦❞❡ ♦❝♦rr❡r r >a✳ ❉❡ ❢❛t♦✱ s❡ ✐st♦ ♦❝♦rr❡ss❡✱ ❡①✐st✐r✐❛ ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧ c < r t❛❧ q✉❡ r = c+a✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡♥❞♦ r =c+a=b−q·a✱ t❡rí❛♠♦sc=b−(q+ 1)·a∈S✱ ❝♦♠ c < r✱ ❝♦♥tr❛❞✐çã♦ ❝♦♠ ♦ ❢❛t♦ ❞❡r s❡r ♦ ♠❡♥♦r ❡❧❡♠❡♥t♦ ❞❡S✳
P♦rt❛♥t♦✱ t❡♠♦s q✉❡ b=a·q+r ❝♦♠ r < a✱ ♦ q✉❡ ♣r♦✈❛ ❛ ❡①✐stê♥❝✐❛ ❞❡q ❡r✳ ❆❣♦r❛✱ ✈❛♠♦s ♣r♦✈❛r ❛ ✉♥✐❝✐❞❛❞❡✳
◆♦t❡ q✉❡✱ ❞❛❞♦s ❞♦✐s ❡❧❡♠❡♥t♦s ❞✐st✐♥t♦s ❞❡ S✱ ❛ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦ ♠❛✐♦r ❡ ♦ ♠❡♥♦r ❞❡ss❡s ❡❧❡♠❡♥t♦s✱ s❡♥❞♦ ✉♠ ♠ú❧t✐♣❧♦ ❞❡ a é ♣❡❧♦ ♠❡♥♦s a✳ ▲♦❣♦✱ s❡ r =b−a·q ❡ r′ =b−a·q′✱ ❝♦♠ r < r′ < a✱ t❡rí❛♠♦s r′−r >a✱ ♦ q✉❡ ❛❝❛rr❡t❛r✐❛r′ >r+a>a✱ ❛❜s✉r❞♦✳ P♦rt❛♥t♦✱ r =r′✳ ❉❛í s❡❣✉❡✲s❡ q✉❡ b−a·q=b−a·q′✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡
a·q =a·q′ ❡✱ ♣♦rt❛♥t♦✱ q=q′✳
✶✳✶✳✷ ❆❧❣♦r✐t♠♦ ❞❛ ❉✐✈✐sã♦
❚❡♦r❡♠❛ ✶✳✾ ❙❡ a ❡ b sã♦ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❝♦♠ b >0✱ ❡♥tã♦ ❡①✐st❡♠ ❡ sã♦
ú♥✐❝♦s ♦s ✐♥t❡✐r♦s q ❡ r q✉❡ s❛t✐s❢❛③❡♠ às ❝♦♥❞✐çõ❡s✿
a=b·q+r com 06r < b.
Pr✐♠❡✐r♦ ✈❛♠♦s ♠♦str❛r ❡①✐stê♥❝✐❛✳ ❙❡❥❛ S ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ♥ã♦ ♥❡❣❛t✐✈♦s q✉❡ sã♦ ❞❛ ❢♦r♠❛ a−b·x✱ ❝♦♠x∈Z✱ ✐st♦ é
S ={a−b·x | x∈Z, a−b·x>0}.
❊st❡ ❝♦♥❥✉♥t♦ S ♥ã♦ é ✈❛③✐♦ (S 6= 0)✱ ♣♦✐s✱ s❡♥❞♦ b >0✱ 06r < b✱ t❡♠♦s q✉❡ b>1
❡✱ ♣♦rt❛♥t♦✱ ♣❛r❛ x=−|a|✱ t❡♠♦s
a−b·x=a+b· |a|>a+|a|>0.
❆ss✐♠ s❡♥❞♦✱ ❡①✐st❡ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ r ❞❡ S t❛❧ q✉❡ r > 0 ❡ r = a−b ·a ♦✉ a=b·q+r✱ ❝♦♠q ∈Z✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡r < b✱ ♣♦✐s✱ s❡ ❢♦ss❡r >b✱ t❡rí❛♠♦s
06r−b =a−b·q=a−b·(q+ 1)< r. ■st♦ é✱ r ♥ã♦ s❡r✐❛ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ ❞❡S✳
❆❣♦r❛ ❛♠♦s ♠♦str❛r ✉♥✐❝✐❞❛❞❡ ❞❡ q ❡ r✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡♠ ❞♦✐s ♦✉tr♦s ♥ú✲ ♠❡r♦s ✐♥t❡✐r♦s q1 ❡ r1 t❛✐s q✉❡
a=b·q1+r1 ❡ 06r1 < b.
❊♥tã♦✱ t❡r❡♠♦s✿
b·q1+r1 =b·q+r,
❧♦❣♦
r1−r =b·q−b·q1.
❆ss✐♠ t❡♠♦s
r1−r=b·(q−q1).
P♦rt❛♥t♦✱
b |(r1−r).
P♦r ♦✉tr♦ ❧❛❞♦ t❡♠♦s✿
−b <−r 60e 06r1 < b,
♦ q✉❡ ✐♠♣❧✐❝❛✿
−b < r1−r < b isto: |r1−r|< b.
❆ss✐♠✱ b|(r1−r)✱ ♠❛s |r1−r|< b. ❈♦♠ ✐ss♦ r1−r = 0 ❡✱ ♣♦rt❛♥t♦✱
r1 =r ❡ q1 =q.
❊①❡♠♣❧♦✿ ❉❛❞♦s 3 ❡ 13✱ ♥ú♠❡r♦s ♥❛t✉r❛✐s✱ t❡♠♦s q✉❡ 3 ∤ 13✱ ♠❛s ❡①✐st❡♠ 4 ❡ 1 ∈ N✱ t❛❧ q✉❡ 13 = 3·4 + 1✳ ❉❛❞♦s ♦s ♥ú♠❡r♦s 4 ❡ 19✱ t❡♠♦s q✉❡ 4 ∤ 19✱ ♠❛s
❡①✐st❡♠ 4 ❡ 3∈N t❛❧ q✉❡ 19 = 4·4 + 3✳
❈♦r♦❧ár✐♦ ✶✳✾✳✶ ❙❡ a ❡ b sã♦ ❞♦✐s ✐♥t❡✐r♦s✱ ❝♦♠ b 6= 0✱ ❡①✐st❡♠ ❡ sã♦ ú♥✐❝♦s ♦s
✐♥t❡✐r♦s q ❡ r q✉❡ s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s
a=b·q+r com 06r <|b|.
❉❡♠♦♥str❛çã♦✿ ❝♦♠ ❡❢❡✐t♦✱ s❡ b > 0✱ ♥❛❞❛ ❤á q✉❡ ❞❡♠♦♥str❛r✱ ❡ s❡ b < 0✱ ❡♥tã♦
|b|>0✱ ❡ ♣♦r ❝♦♥s❡❣✉✐♥t❡ ❡①✐st❡♠ ❡ sã♦ ú♥✐❝♦s ♦s ✐♥t❡✐r♦s q1 ❡ r✱ t❛✐s q✉❡
a=|b| ·q1+r✱ ❝♦♠ 06r <|b|.
❖✉ s❡❥❛✱
|b|=−b. ▲♦❣♦✱
a=−b·q1+r,
❡♥tã♦
a =b·(−q1) +r
❡
06r <|b|. ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡ q=−q1✱ ♦✉ s❡❥❛✿
a=b·q+r e 06r <|b|.
❖s ✐♥t❡✐r♦s q ❡r ❝❤❛♠❛♠✲s❡✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♥❛ ❞✐✈✐sã♦ ❞❡ a
♣♦r b✳
❊①❡♠♣❧♦✿ ❊♥❝♦♥tr❡ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ❞❛ ❞✐✈✐sã♦ ❞❡ 19♣♦r −3✳
P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ ♥❛ ❞✐✈✐sã♦ ❞❡ 19 ♣♦r3∈Z+✱ t❡♠♦s
19 = 3·6 + 1. ❖ q✉❡ ✐♠♣❧✐❝❛✿ 19 =−3·(−6) + 1.
▲♦❣♦✱ q =−6 ❡ r= 1.
❊①❡♠♣❧♦✿ ❊♥❝♦♥tr❡ ♦ q✉♦❝✐❡♥t❡ ❡ ♦ r❡st♦ ♥❛ ❞✐✈✐sã♦ ❞❡ −19♣♦r 3✳
P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s ♥❛ ❞✐✈✐sã♦ ❞❡ 19♣♦r 3∈Z+✱ t❡♠♦s
▼✉❧t✐♣❧✐❝❛♥❞♦ t✉❞♦ ♣♦r −1✱ t❡♠♦s
−19 =−3·6−1.
❈♦♠♦ r=−1<0✱ ✐st♦ é✱ ♥ã♦ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ 06r <3✱ t❡♠♦s
−19 =−3·6−1 =−3·6−3 + 2. ❈♦♠ ✐ss♦✱ −19 = −3·7 + 2⇒ −19 = 3·(−7) + 2, ♦✉ s❡❥❛✿
q =−7 ❡ r= 2✳
❖❜s❡r✈❡ q✉❡bé ❞✐✈✐s♦r ❞❡as❡✱ ❡ s♦♠❡♥t❡ s❡✱r= 0✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦sa=b·q❡ ♦ q✉♦❝✐❡♥t❡q♥❛ ❞✐✈✐sã♦ ❡①❛t❛ ❞❡a♣♦rb✱ ✐♥❞✐❝❛❞♦ t❛♠❜é♠ ♣♦r a
b ♦✉a/b q = a
b =a/b
✱ ❡ ❧ê✲s❡ ✧a s♦❜r❡ b✧✳
✶✳✶✳✸ P❛r✐❞❛❞❡ ❞❡ ✉♠ ◆ú♠❡r♦ ■♥t❡✐r♦
◆❛ ❞✐✈✐sã♦ ❞❡ ✉♠ ✐♥t❡✐r♦ q✉❛❧q✉❡r a ♣♦r b = 2 ♦s ♣♦ssí✈❡✐s r❡st♦s sã♦ r = 0 ♦✉
r = 1✳ ❙❡ r = 0✱ ❡♥tã♦ ♦ ✐♥t❡✐r♦ a = 2·q é ❞❡♥♦♠✐♥❛❞♦ ♣❛r❀ ❡ s❡ r = 1✱ ❡♥tã♦ ♦
✐♥t❡✐r♦ s❡rá ❡s❝r✐t♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛ a= 2·q+ 1❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞❡♥♦♠✐♥❛❞♦
í♠♣❛r✳ ❖❜s❡r✈❡✲s❡ q✉❡✿
✭✐✮ ❙❡ a ❢♦r ♣❛r✱ ✐st♦ é✱ a= 2k✱ ❡♠ q✉❡ k∈Z✱ ❡♥tã♦ a2 é ♣❛r✳
✭✐✐✮ ❙❡ a ❢♦r í♠♣❛r✱ ✐st♦ é✱ a= 2k±1✱ ❡♠ q✉❡k ∈Z✱ ❡♥tã♦ a2 é í♠♣❛r✳
❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❉❡ ❢❛t♦✱ s✉♣♦♥❤❛ q✉❡ a = 2k, ❛ss✐♠ ✱ a2
= (2k)2
, ♦✉ s❡❥❛✱ a2
= 4k2
, ❧♦❣♦✱ a2
= 2·(2k2
)✳ P♦rt❛♥t♦ a2
= 2n, ❡♠ q✉❡ 2k2
=n∈Z
✭✐✐✮ ❙✉♣♦♥❤❛ q✉❡a= 2k±1,❞❡ss❛ ❢♦r♠❛✱a2
= (2k±1)2
,♦✉ s❡❥❛✱a2
= 4k2±
4k+1. ❆ss✐♠✱ t❡♠♦s q✉❡✱a2
= 2·(2k2
+2k)+1,✐st♦ é✱a2
= 2t+1,❡♠ q✉❡2k2
+2k=t∈Z. ❊①❡♠♣❧♦✿ ❖❜s❡r✈❡ q✉❡ ♦ q✉❛❞r❛❞♦ ❞❡ q✉❛❧q✉❡r ✐♥t❡✐r♦ í♠♣❛r é ❞❛ ❢♦r♠❛ 8k+ 1. P❡❧♦ ❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ t❡♠♦s q✉❡ ❞❛❞♦ ✉♠ ♥ú♠❡r♦ a ∈ Z✱ só t❡♠♦s q✉❛tr♦
♣♦ss✐❜✐❧✐❞❛❞❡s ❞❡ ❡s❝r❡✈❡r a ❡♠ ❢✉♥çã♦ ❞❡ 4✱ ♦✉ s❡❥❛✿
a= 4q, a= 4q+ 1, a= 4q+ 2 ♦✉a= 4q+ 3. ❉❡ss❛ ❢♦r♠❛✱ t❡♠♦s q✉❡
✐✬ ❚❡♠♦s q✉❡ a2
= (4q)2✱ ❛ss✐♠✱
a2
= 16q2✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠
a2
= 8·(2q2
)✳ ❉❡ss❛
❢♦r♠❛✱ ❝♦♠♦ (2q2
) = n∈Z✱ a2
✐✐✬ ❚❡♠♦s q✉❡ a2
= (4q+ 1)2
, ❛ss✐♠✱ a2
= 16q2
+ 8q+ 1 q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞❛
s❡❣✉✐♥t❡ ❢♦r♠❛✿ a2
= 8·(2q2
+q) + 1✳ ❉❡ss❛ ❢♦r♠❛✱ ❝♦♠♦(2q2
+q) =k ∈Z✱
a2
= 8k+ 1 q✉❡ é ✉♠ ♥ú♠❡r♦ í♠♣❛r❀
✐✐✐✬ ❚❡♠♦s q✉❡ a2
= (4q+ 2)2
,❛ss✐♠✱ a2
= 16q2
±16q+ 4✱ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞❛
s❡❣✉✐♥t❡ ❢♦r♠❛✿ a= 8(2·q2
+ 2·q) + 4 = 2[4(2·q2
+ 2·q) + 2]✱ ✐st♦ é✱ aé ♣❛r✳ ✐✈✬ ❚❡♠♦s q✉❡ a2
= (4q+ 3)2✱ ❛ss✐♠✱
a2
= 16q2
+ 24q+ 9✱ ✐st♦ é✱ a2
= 8·(2q2
) + 8·
(3q) + 8 + 1✱ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ a2
= 8·(2q2
+ 3q+ 1) + 1
a2
= 8k+ 1✱ ✐st♦ é✱ ❝♦♠♦(2q2
+ 3q+ 1) =k ∈Z✱ a2 é í♠♣❛r✳
❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ 7 ❡13 sã♦ ✐♥t❡✐r♦ í♠♣❛r❡s✱ ♣♦✐s✿ 72
= 49 = 8·6 + 1 132
= 169 = 8·21 + 1.
✶✳✶✳✹ ❈♦♥❥✉♥t♦ ❞♦s ❉✐✈✐s♦r❡s ❞❡ ✉♠ ◆ú♠❡r♦ ■♥t❡✐r♦
❉❡✜♥✐çã♦ ✷ ✿ ❙❡❥❛ a∈Z✱ ❞✐r❡♠♦s q✉❡ D(a) é ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞✐✈✐s♦r❡s ✐♥t❡✐r♦s ❞❡
a✱ s❡ ♣❛r❛ t♦❞♦ d∈D(a) d ❞✐✈✐❞✐r a✱ ✐st♦ é✱ D(a) = {x∈Z∗; x|a}✳
❊①❡♠♣❧♦✿ ❙❡❥❛ 2∈Z∗✱ ❡♥tã♦ D(2) = {−2, −1, 1, 2}✱ ♣♦✐s −2|2✱ −1|2✱ 1|2❡
2|2✳ ◆♦t❡✱ t❛♠❜é♠✱ q✉❡ ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠ ♦✉tr♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❡ ❞✐✈✐❞❡ 2✳
❊①❡♠♣❧♦✿ ❙❡❥❛9∈Z∗✱ ❡♥tã♦D(9) ={−9, −3, −1, 1, 3, 9}✱ ♣♦✐s−9|9✱−3|9✱
−1 | 9✱ 1 | 9✱ 3 | 9 ❡ 9 | 9✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛ q✉❡ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✱ ♥ã♦ ❡①✐st❡
♥❡♥❤✉♠ ♦✉tr♦ ♥ú♠❡r♦ ✐♥t❡✐r♦ q✉❡ ❞✐✈✐❞❛ 9✳
❖❜s❡r✈❛çã♦ ✷ ✿ ◆♦t❡ q✉❡✱ s❛❜❡♥❞♦ q✉❛✐s sã♦ ♦s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s ❞❡ ✉♠ ♥ú♠❡r♦ a ∈ Z✱ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ❞✐✈✐s♦r❡s ✐♥t❡✐r♦s ❜❛st❛ t♦♠❛r ♦s ❞✐✈✐s♦r❡s ♥❛t✉r❛✐s ❡ ♦s
s❡✉s s✐♠étr✐❝♦s✳
❖❜s❡r✈❛çã♦ ✸ D(a) = D(−a), ∀a ∈ Z✱ ♣♦✐s a = a·1 = (−a)·(−1)✳ 1, −1, a ❡
−a✱ sã♦ ❞✐t♦s ❞✐✈✐s♦r❡s tr✐✈✐❛✐s ❞❡ a✳
◗✉❛❧q✉❡r q✉❡ s❡❥❛ ♦ ✐♥t❡✐r♦ a 6= 0✱ s❡ x | a✱ ❡♥tã♦✿ −a 6 x 6 a, ♦✉ s❡❥❛✱ D(a) ⊂
[−a, a]✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡✱ ∀ a∈Z∗ ❡①✐st❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ❞✐✈✐s♦r❡s ❞❡ a✳
✶✳✶✳✺ ❉✐✈✐s♦r❡s ❈♦♠✉♥s ❞❡ ❉♦✐s ◆ú♠❡r♦s ■♥t❡✐r♦s
❉❡✜♥✐çã♦ ✸ ❉❛❞♦s a, b∈Z✱ ❝♦♠ ✉♠ ❞❡❧❡s ♥ã♦ ♥✉❧♦✱ ❞✐r❡♠♦s q✉❡ ♦ ♥ú♠❡r♦d∈Z∗ é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b s❡ d|a ❡ d|b✳
❊①❡♠♣❧♦✿ ❙❡❥❛♠ 3 ❡ 0∈Z✱ t❡♠♦s D(3) ={−3, −1, 1, 3} ❡ D(0) é ❢♦r♠❛❞♦ ♣♦r
t♦❞♦s ♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♥✉❧♦s✳ ◆♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ♦s ♥ú♠❡r♦s −3, −1, 1❡3
sã♦ ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ 3 ❡0✳
❙❡❥❛♠ −4 ❡ 12 ∈ Z✱ t❡♠♦s✿ D(−4) = {−4, −2, −1, 1, 2, 4} ❡ D(12) =
{−12, −6, −4, −3, −2, −1, 1, 2, 3, 4, 6, 12}✳ ◆♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ♦s ♥ú♠❡r♦s −4, −2, −1, 1, 2❡ 4 sã♦ ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡−4 ❡12✳
✶✳✶✳✻ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠
❉❡✜♥✐çã♦ ✹ ❉✐r❡♠♦s q✉❡ d ∈ N é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠✱ ♦✉ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r
❝♦♠✉♠ ✭♠❞❝✮✱ ❞❡ a ❡ b✱ ❡♠ q✉❡ ✉♠ ❞❡❧❡s s❡❥❛ ♥ã♦ ♥✉❧♦✱ s❡ ♣♦ss✉✐r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿
✐✮ d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b❀
✐✐✮ d é ❞✐✈✐sí✈❡❧ ♣♦r t♦❞♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ a ❡ b✳
❊①❡♠♣❧♦✿ ❚❡♠♦s q✉❡D(12) ={±1, ±2, ±3, ±4, ±6, ±12}❡D(8) ={±1, ±2,
±4, ±8} ❡ q✉❡ (12,8) = 4✱ ♣♦✐s ♦ ❝♦♥❥✉♥t♦ ❞♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ 12 ❡ 8 sã♦
{±1, ±2, ±4}❡ t♦♠❛♥❞♦ q✉❛❧q✉❡r ♦✉tr♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡❧❡s ✐rá ❞✐✈✐❞✐r4✱ ♦✉ s❡❥❛✱
s❡ x∈ {±1, ±2, ±4}✱ t❡♠♦s q✉❡ x|4✳
◆♦t❡ q✉❡−3, −1, 1 ❡3 sã♦ ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ 3 ❡ 0✱ ♦✉ s❡❥❛✱ t♦❞♦s ❡st❡s ♥ú✲
♠❡r♦s s❛t✐s❢❛③❡♠ ❛ ❝♦♥❞✐çã♦ ✐✮✳◆♦ ❡♥t❛♥t♦✱ ❛♣❡♥❛s ♦ 3s❛t✐s❢❛③ ❛s ❞✉❛s ❝♦♥❞✐çõ❡s✳
❆ ❝♦♥❞✐çã♦ ✭✐✐✮ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ✐✐✬✮ ❙❡cé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡b✱ ❡♥tã♦ c|d✳
P♦rt❛♥t♦✱ s❡dé ✉♠ ♠❞❝ ❞❡a❡b✱ ❡ s❡cé ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ss❡s ❞♦✐s ♥ú♠❡r♦s✱ ❡♥tã♦ c6d✳ ■st♦ ♥♦s ♠♦str❛ q✉❡ ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s é ❡❢❡t✐✲ ✈❛♠❡♥t❡ ♦ ♠❛✐♦r ❞❡♥tr❡ t♦❞♦s ♦s ❞✐✈✐s♦r❡s ❝♦♠✉♥s ❞❡ss❡s ♥ú♠❡r♦s✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ✐st♦ ♥♦s ♠♦str❛ q✉❡✱ s❡ d ❡ d′ sã♦ ❞♦✐s ♠❞❝ ❞❡ ✉♠ ♠❡s♠♦ ♣❛r ❞❡ ♥ú♠❡r♦s✱ ❡♥tã♦ d 6 d′ ❡ d′ 6 d✱ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ d= d′✳ ■st♦ é✱ ❤á ✉♥✐❝✐❞❛❞❡ ❞♦ ♠❞❝✳ ❖ ♠❞❝ ❞❡ a ❡ b s❡rá ❞❡♥♦t❛❞♦ ♣♦r (a, b)✳ ❈♦♠♦ ♦ ♠❞❝ ❞❡ a ❡ b ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ♦r❞❡♠ ❡♠ q✉❡ a ❡ b sã♦ t♦♠❛❞♦s✱ t❡♠♦s q✉❡ (a, b) = (b, a)✳
◆♦ ❚❡♦r❡♠❛ ✶✳✶✶ ❛ ❢r❡♥t❡ ♠♦str❛r❡♠♦s ❛ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❡ ❞♦✐s ♥ú♠❡r♦s a ❡ b ❡♠ q✉❡ ✉♠ ❞❡❧❡ é ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ P❛r❛ t❛♥t♦ ❢❛r❡♠♦s ❛❧❣✉♠❛s ♦❜s❡r✈❛çõ❡s ❛❜❛✐①♦✳
❆❣♦r❛ ✈❛♠♦s ♠♦str❛r ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❡ ❞♦✐sa ❡ b✱ ❡♠ q✉❡ ✉♠ ❞❡❧❡s ♥ã♦ s❡❥❛ ♥✉❧♦✳
❈♦♠♦✱ ♣♦r ❞❡✜♥✐çã♦✱a ❡ b sã♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s t❛✐s q✉❡ a6= 0 ♦✉b6= 0✱ t❡♠♦s✿
✭✐✮ ❙❡ ✉♠ ❞❡❧❡s ❢♦r ♥✉❧♦✿ (0, b) = |b| ♦✉(a,0) =|a|❀
✭✐✐✮ ❙❡ ✉♠ ❞❡❧❡s ❢♦r ✐❣✉❛❧ ❛ 1✿ (1, b) = 1 ♦✉ (a,1) = 1❀
✭✐✐✐✮ ❙❡ a=b✿ (a, a) = |a|❀
✭✐✈✮ ❙❡ a |b ❡♥tã♦ a= (a, b)✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ (a, b) =a ❡♥tã♦ a|b✳
P❛r❛ ♣r♦✈❛r ❛ ❡①✐stê♥❝✐❛ ❞♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s✱ ✉t✐❧✐③❛r❡♠♦s✱ ❡ss❡♥❝✐❛❧♠❡♥t❡✱ ♦ r❡s✉❧t❛❞♦ ❛❜❛✐①♦✳
❙❡a <0 t❡♠♦s q✉❡−a >0 ❡ −a|a✱ ♣♦✐s ❡①✐st❡ −1∈Z✱ t❛❧ q✉❡ a=−a·(−1)✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s❡ −a <0t❡♠♦s q✉❡ a >0 ❡ a| −a✳
❉❛ ♠❡s♠❛ ❢♦r♠❛ ♣♦❞❡♠♦s ❡st❡♥❞❡r ♦ ▲❡♠❛ ❞❡ ❊✉❝❧✐❞❡s ❛♣r❡s❡♥t❛❞♦✳
▲❡♠❛ ✶✳✶✵ ❙❡ ❡①✐st✐r (a, b−n·a)✱ ❡♥tã♦ (a, b) ❡①✐st❡ ❡ (a, b) = (a, b−n·a)✱ ♣❛r❛
a, b, n∈Z✳
❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ (a, b−n·a) = d∈N✱ ❡♥tã♦d|a ❡✱ ❝♦♠ ✐ss♦✱d |n·a❀ t❡♠♦s t❛♠❜é♠ q✉❡ d | b−n·a✳ P♦rt❛♥t♦✱ d | b−n·a+n·a ⇒ d | b✳ ❙✉♣♦♥❞♦✱ ❛❣♦r❛✱
q✉❡ cs❡❥❛ ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡a ❡ ❞❡b❀ ❧♦❣♦✱c|na❡✱ ❝♦♠ ✐ss♦✱ c|b−n·a✳ ❉❡ss❛
❢♦r♠❛✱ c|d✳ ▲♦❣♦✱ d= (a, b)✳ ❆ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❡ (a, b) ♣❛r❛ a ❡ b∈Z✱ ❡♠
q✉❡ a6= 0 ♦✉ b6= 0✱ ❞á✲s❡ ❞❛ ♠❡s♠❛ ❢♦r♠❛✿
✭✐✮ ❙❡ ✉♠ ❞❡❧❡s ❢♦r ♥✉❧♦✿ (0, b) = b✱ ♣❛r❛ b > 0 ♦✉ (0, b) = −b✱ ♣❛r❛ b < 0✱ s❡♥❞♦
b = 0✱ (a,0) =a✱ ♣❛r❛ a >0♦✉ (a,0) =−a✱ ♣❛r❛ a <0❀
✭✐✐✮ ❙❡ ✉♠ ❞❡❧❡s ❢♦r ✐❣✉❛❧ ❛ 1✿ (1, b) = 1 ♦✉ (a,1) = 1❀
✭✐✐✐✮ ❙❡ a=b✿ (a, a) = a✱ ♣❛r❛a >0 ♦✉(a, a) = −a✱ ♣❛r❛ a <0❀
✭✐✈✮ ❙❡ a | b ❡♥tã♦ (a, b) =a✱ ♣❛r❛ a > 0 ♦✉ s❡rá (a, b) = −a✱ ♣❛r❛ a < 0✳ ❚❡♠♦s
q✉❡ a é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ ❞❡ b✱ ❡✱ s❡ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ a ❡ b✱ ❡♥tã♦ c|a✱ ♦ q✉❡ ♥♦s ♠♦str❛ q✉❡ a= (a, b)✳
❚❡♦r❡♠❛ ✶✳✶✶ ❙❡ a ❡ b sã♦ ❞♦✐s ✐♥t❡✐r♦s ❡♠ q✉❡ ✉♠ ❞❡❧❡s ♥ã♦ é ♥✉❧♦ (a 6= 0 ♦✉
b 6= 0)✱ ❡♥tã♦ ❡①✐st❡ ❡ é ú♥✐❝♦ ♦ ♠❞❝ ❞❡ a ❡b❀ ❛❧é♠ ❞✐ss♦✱ ❡①✐st❡♠ ✐♥t❡✐r♦sx ❡ y t❛✐s q✉❡✿ (a, b) =a·x+b·y✱ ✐st♦ é✱ ♦ ♠❞❝ é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a ❡ b✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛S♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s ❞❛ ❢♦r♠❛a·u+b·v✱
❝♦♠ u, v ∈ Z✱ ✐st♦ é✿ S = {a·u+b·v | a·u+b·v > 0 ❡ u, v,Z}. ❊st❡ ❝♦♥❥✉♥t♦ S ♥ã♦ é ✈❛③✐♦✱ ♣♦✐s a ♦✉ b t❡♠ q✉❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ s✉♣♦♥❤❛ q✉❡ a 6= 0✱ ❡♥tã♦
✉♠ ❞♦s ✐♥t❡✐r♦s✿ a = a·1 +b·0✱ s❡ a > 0✱ ♦✉ −a = a·(−1) +b ·0✱ s❡ a < 0 é
♣♦s✐t✐✈♦ ❡ ♣❡rt❡♥❝❡ ❛ S✳ ▲♦❣♦✱ ♣❡❧♦ ✧Pr✐♥❝í♣✐♦ ❞❛ ❇♦❛ ❖r❞❡♥❛çã♦✧✱ ❡①✐st❡ ❡ é ú♥✐❝♦ ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ d∈S✳ ❊✱ ❝♦♥s♦❛♥t❡ ❛ ❞❡✜♥✐çã♦ ❞❡S✱ ❡①✐st❡♠ ✐♥t❡✐r♦s x ❡y t❛✐s q✉❡ d = a·x+b·y✳ P♦st♦ ✐ss♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ d = (a, b)✳ ❈♦♠ ❡❢❡✐t♦✱ ♣❡❧♦
❛❧❣♦r✐t♠♦ ❞❛ ❞✐✈✐sã♦✱ t❡♠♦s✿ a=d·q+r✱ ❝♦♠06r 6d ♦ q✉❡ ❞á
r=a−d·q =a−(a·x+b·y)·q =a(1−q·x) +b·(−qv˙)✱ ✐st♦ é✱ ♦ r❡st♦ r é ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ a❡ ❞❡ b✳ ❈♦♠06r6d ❡d >0é ♦ ❡❧❡♠❡♥t♦ ♠í♥✐♠♦ ❞❡
S✱ s❡❣✉❡✲s❡ q✉❡ r = 0 ❡ a =d·q✱ ♦✉ s❡❥❛✱ d |a✳ ❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♣r♦✈❛✲s❡ q✉❡
t❛♠❜é♠ d|b✳ P♦rt❛♥t♦✱d é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦s✐t✐✈♦ ❞❡ a❡ ❞❡ b✳ ❙✉♣♦♥❤❛ q✉❡ c é ✉♠ ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦s✐t✐✈♦ q✉❛❧q✉❡r ❞❡ a ❡ ❞❡ b✱ ♦✉ s❡❥❛✱ c|a ❡c|b✱ ❡♥tã♦
c| (a·x+b·y) ❡♥tã♦ c|d✱ ✐st♦ é✱ d é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦s✐t✐✈♦ ❞❡ a ❡ b✱ ♦✉ s❡❥❛✱ (a, b) =d=a·x+b·y✱ ❝♦♠ x, y ∈Z✳ ❖❜s❡r✈❛çã♦ ✹ ❊st❡ t❡♦r❡♠❛ ♠♦str❛ q✉❡✿
✭✐✮ (a, b)é ♦ ♠❡♥♦r ✐♥t❡✐r♦ ♣♦s✐t✐✈♦✱ t❛❧ q✉❡(a, b) = a·x+b·y✱ ✐st♦ é✱ ♦ ♠❞❝ ❞❡ a❡ b ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞♦s ♠❡s♠♦s ❡ q✉❡ ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ♥ã♦ é ú♥✐❝❛✱ ♣♦✐s t❡♠♦s✿
(a, b) = d=a·x+b·y. ❆ss✐♠
d=a·x+a·b·t−a·b·t+b·y. P♦rt❛♥t♦
d=a·(x+b·t) +b·(y−a·t), ∀t∈Z.
✭✐✐✮ ❙❡d=a·r+b·s✱ ♣❛r❛ ❛❧❣✉♠ ♣❛r ❞❡ ✐♥t❡✐r♦sr❡s✱ ❡♥tã♦d♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦ ♠❞❝ ❞❡ a ❡ b✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ s❡✿
(a, b) = a·x+b·y, ♦✉ s❡❥❛✱
✐st♦ é✱
d=a·r+b·s, ❡♠ q✉❡
d=t·(a, b). P♦rt❛♥t♦✱
r =t·x ❡ s=t·y✳
❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡d∈Z✱ t❛❧ q✉❡d=a·x+b·y♥ã♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦ ♠❞❝ ❞❡ a ❡ b✱ ❡♥tr❡t❛♥t♦✱ s❡rá ✉♠ ♠ú❧t✐♣❧♦ ❞❡ (a, b)✳
✶✳✶✳✼ ❆❧❣♦r✐t♠♦ ❞❡ ❊✉❝❧✐❞❡s
❆q✉✐✱ tr❛r❡♠♦s ✉♠❛ ♣r♦✈❛ ❝♦♥str✉t✐✈❛ ❞❛ ❡①✐stê♥❝✐❛ ❞♦ ♠❞❝ ❞❡ a ❡ b✱ ❛tr✐❜✉í❞♦ ❛ ❊✉❝❧✐❞❡s ❡♠ s✉❛ ♦❜r❛ ❊❧❡♠❡♥t♦s✳
❉❛❞♦s a, b ∈ N✱ ♣♦❞❡♠♦s s✉♣♦r a 6b✳ s❡ a = 1 ♦✉ a =b✱ ♦✉ ❛✐♥❞❛ a | b✱ ❥á ✈✐♠♦s q✉❡ (a, b) = a✳ ❙✉♣♦♥❤❛♠♦s✱ ❡♥tã♦✱ q✉❡ 1 < a < b ❡ q✉❡ a ∤ b✳ ▲♦❣♦✱ ♣❡❧❛ ❞✐✈✐sã♦ ❡✉❝❧✐❞✐❛♥❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r b=a·q1+r1✱ ❝♦♠r1 < a✳
❚❡♠♦s ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿
✭✐✮ r1 |a✱ ❡♥tã♦✱r1 = (a, r1) = (a, b−q1·a) = (a, b)✱ ✭▲❡♠❛ ✶✳✶✵✮❀
✭✐✐✮ r1 ∤ a✱ ❡✱ ❡♠ t❛❧ ❝❛s♦✱ ♣♦❞❡♠♦s ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡ a ♣♦r r1✱ ♦❜t❡♥❞♦ a =
r1q2+r2✱ ❝♦♠ r2 < r1✳
◆♦✈❛♠❡♥t❡✱ ❞✉❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✿
✐✬✮ r2 | r1 ❡✱ ♥♦✈❛♠❡♥t❡✱ t❡rí❛♠♦s✿ r2 = (r1, r2) = (r1, a −r2 · q2) = (r1, a) =
(b−q1·a, a) = (a, b−q1·a) = (a, b)✱ ❡ ♣❛r❡♠♦s✱ ♣♦✐s t❡r♠✐♥❛ ♦ ❛❧❣♦r✐t♠♦✱ ♦✉
✐✐✬✮ r2 ∤ r1✱ ❡✱ ❡♠ t❛❧ ❝❛s♦✱ ♣♦❞❡♠♦s ❡❢❡t✉❛r ❛ ❞✐✈✐sã♦ ❞❡ r1 ♣♦r r2✱ ♦❜t❡♥❞♦ r1 =
r2·q3+r3✱ ❝♦♠r3 < r2✳
❊st❡ ♣r♦❝❡❞✐♠❡♥t♦ ♥ã♦ ♣♦❞❡ ❝♦♥t✐♥✉❛r ✐♥❞❡✜♥✐❞❛♠❡♥t❡✱ ♣♦✐s t❡rí❛♠♦s ✉♠❛ s❡q✉ê♥✲ ❝✐❛ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s a > r1 > r2 > r3 > . . . ♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ Pr✐♥❝✐♣✐♦ ❞❛ ❇♦❛
❖r❞❡♥❛çã♦ ♥♦s ♥❛t✉r❛✐s✳ ▲♦❣♦✱ ♣❛r❛ ❛❧❣✉♠ ♥✱ t❡♠♦s q✉❡ rn | rn−1✱ ♦ q✉❡ ✐♠♣❧✐❝❛
q✉❡ (a, b) = rn✳
❊①❡♠♣❧♦✿ ❱❛♠♦s ❝❛❧❝✉❧❛r (372,162)✳
(162,372) = (162,372 − 2· 162) = (162,48) = (48,162) = (48,162 −48· 3) = (48,18) = (18,48) = (18,48−18·2) = (18,12) = (12,18) = (12,18−12·1) = (12,6) = (6,12) = (6,12−6·2) = (6,0) = 6
✶✳✶✳✽ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝
❙❡❥❛♠ a, b ∈Z+✱ ❡♠ q✉❡ ✉♠ ❞❡❧❡s s❡❥❛ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✳ ❉❡✜♥✐♠♦s ♦ ❝♦♥❥✉♥t♦
J(a, b) ={x∈Z∗
+;∃ u, v ∈Z+, x=u·a−v·b}❡J(b, a) ={y ∈Z+∗;∃ u, v ∈Z+, y =
v·b−u·a}✳
▲❡♠❛ ✶✳✶✷ ❚❡♠✲s❡ q✉❡ J(a, b) =J(b, a)6=∅✳
❉❡♠♦♥str❛çã♦✿ ■♥✐❝✐❛❧♠❡♥t❡✱ ♠♦str❛r❡♠♦s q✉❡ ♦s ❞♦✐s ❝♦♥❥✉♥t♦s sã♦ ✐❣✉❛✐s✳ ❇❛st❛ ♠♦str❛r q✉❡ J(a, b)⊂J(b, a)✳ ❙❡❥❛ x∈J(a, b)✱ ❡♥tã♦ x=u·a−v·b✱ ❝♦♠u, v ∈Z✳
❉❡ss❛ ❢♦r♠❛✱ ❡①✐st❡♠λ, θ∈Z∗
+✱ t❛✐s q✉❡λ·a > v❡θ·b > u✳ ❚♦♠❛♥❞♦ρ=max{λ, θ}✱
t❡♠✲s❡ q✉❡ ρ·a > v ❡ ρ·b > u✳ ❉❡ss❛ ❢♦r♠❛✱
x=u·a−v·b. ❙♦♠❛♥❞♦ ❡ s✉❜tr❛✐♥❞♦ ρ·a·b✱ t❡♠♦s
x=u·a−ρ·a·b+ρ·a·b−v·b. ❆ss✐♠✱
x=−a·(ρ·b−u) +b·(ρ·a−v)
❡✱ ♣♦rt❛♥t♦✱
x=b·(ρ·a−v)−a·(ρ·b−u)∈J(b, a).
◆♦t❡ q✉❡ a∈ J(a, b)✱ ♣♦✐s ❜❛st❛ t♦♠❛r v = 0❡ u= 1 t❡r❡♠♦s x=u·a−b·v =a✱
♣♦rt❛♥t♦ J(a, b)6=∅✳
❚❡♦r❡♠❛ ✶✳✶✸ ❙❡❥❛♠ a, b∈Z∗
+✱ ❡ s❡❥❛ d=minJ(a, b)✳ ❚❡♠✲s❡ q✉❡✿
✐✮ d= (a, b)❀
✐✐✮ J(a, b) ={n·d;n∈Z∗+}✳
❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛ q✉❡ c ❞✐✈✐❞❛ a ❡ b❀ ❧♦❣♦✱ c | u·a ❡ c | v ·b✱ ♦✉ s❡❥❛✱ c|a·u−b·b✳ ❉❡ss❛ ❢♦r♠❛✱c|x,♣❛r❛ t♦❞♦ x∈J(a, b)❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ c|d✳
❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ q✉❡d∤x✳ P♦rt❛♥t♦✱ ♣❡❧❛ ❞✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛x=d·q+r✱ ❡♠
q✉❡06r < d✳ ❈♦♠♦ x=a·u−b·v ❡x=d·q+r✱ t❡♠♦s q✉❡d·q+r =a·u−b·v✱ ♦✉ s❡❥❛✱r =a·u−b·v−d·q✳ ❚❡♠♦s q✉❡ d∈J(a, b)✱ ♦✉ s❡❥❛✱ d=b·m−a·n✱ ♣❛r❛
❛❧❣✉♠ m, n∈Z∗
+✳ P♦rt❛♥t♦✿
r=a·u−b·v−b·m·q+a·n·q. ❆ss✐♠✱ t❡♠♦s
❉❡ss❛ ❢♦r♠❛
r =a·(u+n·q)−b·(v+q·m),
♦✉ s❡❥❛✱ r∈J(a, b)✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✱ ♣♦✐sd=minJ(a, b)❡r < d✳ ❊♠ ♣❛rt✐❝✉❧❛r d|a ❡ d|b✳
✭✐✐✮ ❉❛❞♦ q✉❡l·d=l·(n·a−m·b) = (l·n)·a−(l·m)·b∈J(a, b)é ❝❧❛r♦ q✉❡
{l·d;l∈Z∗+} ⊂J(a, b)✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❥á ♣r♦✈❛♠♦s q✉❡ t♦❞♦ x∈J(a, b)é t❛❧ q✉❡ d|x✱ ❡✱ ♣♦rt❛♥t♦✱ J(a, b)⊂ {l·d;l∈Z∗+} ❈♦r♦❧ár✐♦ ✶✳✶✸✳✶ ◗✉❛✐sq✉❡r q✉❡ s❡❥❛♠ a, b, n∈Z∗
+✱ (n·a, n·b) =n·(a, b)✳
❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ J(n·a, n·b) =n·J(a, b) = {n·x;x∈ J(a, b)}✳ ❈♦♠♦
d= (a, b) = minJ(a, b)✱ t❡♠♦s q✉❡ n·(a, b) =min [n·J(a, b)] =min J(n·a, n·b)∋
(n·a, n·b)✱ ♦✉ s❡❥❛✱ n·(a, b) = (n·a, n·b)✳ ❈♦r♦❧ár✐♦ ✶✳✶✸✳✷ ❉❛❞♦s a, b∈Z∗
+✱ t❡♠✲s❡ q✉❡
a
(a, b)
b
(a, b)
= 1✳
❉❡♠♦♥str❛çã♦✿ P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✷✱ t❡♠♦s
(a, b)·
a
(a, b),
b
(a, b)
=
(a, b)· a
(a, b),(a, b)·
b
(a, b)
= (a, b).
❉❡✜♥✐çã♦ ✺ ❉♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a❡ b s❡rã♦ ❞✐t♦s ♣r✐♠♦s ❡♥tr❡ s✐✱ ♦✉ ❝♦♣r✐♠♦s✱ s❡ (a, b) = 1❀ ♦✉ s❡❥❛✱ s❡ ♦ ú♥✐❝♦ ❞✐✈✐s♦r ❝♦♠✉♠ ♣♦s✐t✐✈♦ ❞❡ ❛♠❜♦s é ♦ ♥ú♠❡r♦ 1✳
Pr♦♣♦s✐çã♦ ✶✳✶✹ ❉♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b sã♦ ♣r✐♠♦s ❡♥tr❡ s✐ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s n ❡ m t❛✐s q✉❡ m·a−n·b = 1✳
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ (a, b) = 1✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❡❧♦ t❡♦r❡♠❛ t❡♠♦s q✉❡
❡①✐st❡♠ ♥ú♠❡r♦s ♥❛t✉r❛✐s m ❡n✱ t❛✐s q✉❡m·a−n·b= (a, b) = 1✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱
s✉♣♦♥❤❛ q✉❡ ❡①✐st❛♠ ♥ú♠❡r♦s ♥❛t✉r❛✐sm❡nt❛✐s q✉❡ m·a−n·b = 1✳ ❙❡d= (a, b)✱
t❡♠♦s q✉❡ d|(m·a−n·b)✱ ♦ q✉❡ ♠♦str❛ q✉❡d|1✱ ❡✱ ♣♦rt❛♥t♦✱ d= 1✳ ❚❡♦r❡♠❛ ✶✳✶✺ ❙❡❥❛♠ a, b, c∈Z∗
+✳ ❙❡ a |b·c ❡ (a, b) = 1✱ ❡♥tã♦ a |c✳
❉❡♠♦♥str❛çã♦✿ ❙❡ a| b·c✱ ❡♥tã♦ ❡①✐st❡ e ∈Z✱ t❛❧ q✉❡ b·c=a·e✳ ❙❡ (a, b) = 1✱
❡♥tã♦ ❡①✐st❡♠ m, n∈Z∗
+✱ t❛❧ q✉❡ m·a−n·b= 1✳ ▼✉❧t✐♣❧✐❝❛♥❞♦ ♦s ❞♦✐s ♠❡♠❜r♦s
♣♦r ct❡♠♦s
❈♦♠♦ b·c=a·e✱ ♣♦❞❡♠♦s ❢❛③❡r
m·a·c−n·a·e=c, ♦✉ s❡❥❛✱
a·(m·c−n·e) =c.
P♦rt❛♥t♦✱ a |c.
❈♦r♦❧ár✐♦ ✶✳✶✺✳✶ ❉❛❞♦s a ∈ Z ❡ b, c ∈ Z∗✱ t❡♠♦s q✉❡ b | a ❡ c | a s❡✱ ❡ s♦♠❡♥t❡ s❡✱ b·c
(b, c) |a✳
❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠a∈Z❡b, c∈Z∗✳ ❙✉♣♦♥❤❛♠♦s q✉❡b|a❡c|a✱ ❞❡st❡ ♠♦❞♦ ❡①✐st❡♠ q1, q2 ∈Z✱ t❛✐s q✉❡a =b·q1 ❡ a=c·q2✳ ❆❧é♠ ❞✐ss♦✱ (b, c)|b ❡ (b, c)|c✳
❙❡♥❞♦ ❛ss✐♠✱ a
(b, c) =
b
(b, c)·q1 ❡
a
(b, c) =
c
(b, c)·q2✱ ♦✉ s❡❥❛✱
b
(b, c)·q1 =
c
(b, c)·q2✳
P❡❧♦ ❈♦r♦❧ár✐♦ ✶✳✶✸✳✷✱
b
(b, c),
c
(b, c)
= 1✱ ❧♦❣♦ b
(b, c) | q2 ❡
c
(b, c) | q1 ❡✱ ❝♦♠ ✐ss♦✱
c· b
(b, c) |c·q2 ❡ b·
c
(b, c) |b·q1✱ ♦✉ s❡❥❛✱
c·b
(b, c) |a ❡
b·c
(b, c) |a✳
P♦rt❛♥t♦✱
b·c
(b, c) |a.
❙✉♣♦♥❤❛♠♦s✱ ❛❣♦r❛✱ q✉❡ b·c
(b, c) | a, ♦✉ s❡❥❛✱ ❡①✐st❡q3 ∈ Z t❛❧ q✉❡ a =
b·c
(b, c) ·q3,
✐st♦ é✱ a=b·
c
(b, c)·q3
.❙❛❜❡♠♦s q✉❡
c
(b, c) ·q3
=q∈Z✱ ❧♦❣♦a=b·q, ♦✉ s❡❥❛✱ b |a.
❉❡ ♠❛♥❡✐r❛ ❛♥á❧♦❣❛✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡ra =c·
b
(b, c) ·q3
✱ ❛ss✐♠✱ a=c·q′,♦♥❞❡ b
(b, c)·q3 =q
′ ∈Z✱ ❡ ♣♦rt❛♥t♦ c|a ✳
✶✳✶✳✾ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠
❉❡✜♥✐çã♦ ✻ ❉✐r❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ m∈Né ✉♠ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ✭♠♠❝✮
❞❡ a ❡ ❞❡ b s❡ ♣♦ss✉✐r ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿ ✭✐✮ m é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ a ❡ ❞❡ b ❛♦ ♠❡s♠♦ t❡♠♣♦❀
✭✐✐✮ ❙❡ c é ✉♠ ♠ú❧t✐♣❧♦ ❞❡ a ❡ ❞❡ b✱ ❡♥tã♦ m|c✳
❊①❡♠♣❧♦✿ ❙❛❜❡♠♦s q✉❡ 12é ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ 2 ❡ ❞❡ 3✱ ♠❛s ♥ã♦ é ♦ ♠♠❝
❞❡st❡s ♥ú♠❡r♦s✱ ♣♦✐s 6∈N t❛♠❜é♠ é ♠ú❧t✐♣❧♦ ❡6|12✳
❙❡cé ✉♠ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡a ❡b✱ ❡♥tã♦✱ ❞♦ ✐t❡♠ ✭✐✐✮ ❞❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✱ t❡♠♦s q✉❡ m|c✳ P♦rt❛♥t♦m 6c✱ ♦ q✉❡ ♥♦s ❞✐③ q✉❡ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ s❡ ❡①✐st✐r✱ é ú♥✐❝♦ ❡ é ♦ ♠❡♥♦r ❞♦s ♠ú❧t✐♣❧♦s ❝♦♠✉♥s ❞❡ a ❡ b✳ ❉❡♥♦t❛r❡♠♦s ♦ ♠♠❝ ❞❡ a ❡ b✱ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ [a, b]✳
Pr♦♣♦s✐çã♦ ✶✳✶✻ ❉❛❞♦s ❞♦✐s ♥ú♠❡r♦s ♥❛t✉r❛✐s a ❡ b✱ t❡♠♦s q✉❡ [a, b] ❡①✐st❡ ❡ [a, b]·(a, b) =a·b.
❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡m= a·b
(a, b)✳ ▲♦❣♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡rm❞❛s s❡❣✉✐♥t❡s
❢♦r♠❛s✿
m=a· b
(a, b) ♦✉ m=b·
a
(a, b)✱
✐st♦ é✿
m =a· b
(a, b) =b·
a
(a, b).
❆ss✐♠✱ t❡♠♦s q✉❡ a | m ❡ b | m✱ ♣♦✐s a
(a, b) ❡
b
(a, b) ∈ Z✳ ❙❡❥❛ c ✉♠ ♠ú❧t✐♣❧♦
❝♦♠✉♠ ❞❡ a ❡b❀ ❧♦❣♦✱ c=n·a=n′ ·b✳ ❙❡❣✉❡ ❞❛í q✉❡
n· a
(a, b) =n
′· b
(a, b).
❈♦♠♦
a
(a, b),
b
(a, b)
= 1, t❡♠♦s q✉❡ a
(a, b) | n
′. ❆ss✐♠ ♦❜t❡♠♦s a·b
(a, b) | n
′ ·b, ♦✉
s❡❥❛✱ a·b
(a, b) |c. ▲♦❣♦ [a, b] =
a·b
(a, b) ♣♦r ❞❡✜♥✐çã♦ ❞❡ ♠♠❝✳
❈♦r♦❧ár✐♦ ✶✳✶✻✳✶ ❙❡ a ❡ b∈Z✱ t❛❧ q✉❡ (a, b) = 1✱ ❡♥tã♦ [a, b] =a·b✳
❉❡♠♦♥str❛çã♦✿ ❚❡♠♦s q✉❡[a, b]·(a, b) =a·b ❡ q✉❡(a, b) = 1✱ ❧♦❣♦✱ [a, b]·1 = a·b✱
❡♥tã♦ [a, b] =a·b✳
✶✳✷ ❊q✉❛çõ❡s ❉✐♦❢❛♥t✐♥❛s ▲✐♥❡❛r❡s
●❡♥❡r❛❧✐❞❛❞❡s
❖ t✐♣♦ ♠❛✐s s✐♠♣❧❡s ❞❡ ❡q✉❛çõ❡s ❞✐♦❢❛♥t✐♥❛s é ❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s X ❡ Y✿
a·X+b·Y =c ❖♥❞❡a✱b ❡c sã♦ ✐♥t❡✐r♦s ❞❛❞♦s✱ s❡♥❞♦ a, b6= 0✳
❚♦❞♦ ♣❛r ❞❡ ✐♥t❡✐r♦sX0, Y0 t❛✐s q✉❡a·X0+b·Y0 =c❞✐③✲s❡ ✉♠❛ s♦❧✉çã♦ ✐♥t❡✐r❛
♦✉ ❛♣❡♥❛s ✉♠❛ s♦❧✉çã♦ ❞❛ ❡q✉❛çã♦ a·X+b·Y =c✳
❈♦♥s✐❞❡r❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s✿
3·X+ 6·Y = 18
❚❡♠♦s✿
3·4 + 6·1 = 18 3·(−6) + 6·6 = 18 3·10 + 6·(−2) = 18
▲♦❣♦✱ ♦s ♣❛r❡s ❞❡ ✐♥t❡✐r♦s✿ 4 ❡ 1✱ −6 ❡ 6✱ 10 ❡ −2 sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ 3·X+ 6·Y = 18✳
❊①✐st❡♠ ❡q✉❛çõ❡s ❞✐♦❢❛♥t✐♥❛s ❧✐♥❡❛r❡s ❝♦♠ ❞✉❛s ✐♥❝ó❣♥✐t❛s q✉❡ ♥ã♦ tê♠ s♦❧✉çã♦✳ ❆ss✐♠✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r✿ 2·X+ 4·Y = 7 ♥ã♦ t❡♠ s♦❧✉çã♦✱
♣♦rq✉❡ 2·X + 4·Y é ✉♠ ✐♥t❡✐r♦ ♣❛r ♣❛r❛ q✉❛✐sq✉❡r q✉❡ s❡❥❛♠ ♦s ✈❛❧♦r❡s ✐♥t❡✐r♦s ❞❡X❡Y✱ ❡♥q✉❛♥t♦ q✉❡7é ✉♠ ✐♥t❡✐r♦ í♠♣❛r ✭♦❜s❡r✈❡✲s❡ q✉❡2 = (2,4)♥ã♦ ❞✐✈✐❞❡7✮✳
❉❡ ♠♦❞♦ ❣❡r❛❧✱ ❛ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r a·X+b·Y = c ♥ã♦ t❡♠ s♦❧✉çã♦ t♦❞❛s ❛s ✈❡③❡s q✉❡ d= (a, b) ♥ã♦ ❞✐✈✐❞❡c✳
❈♦♥❞✐çã♦ ❞❡ ❊①✐stê♥❝✐❛ ❞❡ ❙♦❧✉çã♦
❚❡♦r❡♠❛ ✶✳✶✼ ❆ ❡q✉❛çã♦ ❞✐♦❢❛♥t✐♥❛ ❧✐♥❡❛r a · x+b · Y = c t❡♠ s♦❧✉çã♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d ❞✐✈✐❞❡ c✱ s❡♥❞♦ d= (a, b)✳
