APLICAÇÃO DO MÍNIMO MÚLTIPLO COMUM GENERALIZADO NAS ONDAS DE PÊNDULOS

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛

Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠

▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧

❆♣❧✐❝❛çã♦ ❞♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠

❣❡♥❡r❛❧✐③❛❞♦ ♥❛s ♦♥❞❛s ❞❡ ♣ê♥❞✉❧♦s

❏❡❛♥ ❈❛r❧♦ ❞❡ ❙♦✉s❛ ❡ ❙✐❧✈❛

●♦✐â♥✐❛

(2)

TERMO DE CIÊNCIA E DE AUTORIZAÇÃO PARA DISPONIBILIZAR ELETRONICAMENTE OS TRABALHOS DE CONCLUSÃO DE CURSO NA BIBLIOTECA DIGITAL DA UFG

Na qualidade de titular dos direitos de autor, autorizo a Universidade Federal de Goiás (UFG) a disponibilizar, gratuitamente, por meio da Biblioteca Digital de Teses e Dissertações (BDTD/UFG), sem ressarcimento dos direitos autorais, de acordo com a Lei nº 9610/98, o do-cumento conforme permissões assinaladas abaixo, para fins de leitura, impressão e/ou down-load, a título de divulgação da produção científica brasileira, a partir desta data.

1. Identificação do material bibliográfico: Trabalho de Conclusão de Curso de Mestrado Profissional

2. Identificação do Trabalho

Autor : Jean Carlo de Sousa e Sila E-mail: [email protected]

Seu e-mail pode ser disponibilizado na página? [ x ]Sim [ ] Não Vínculo empregatício do autor Seletista

Agência de fomento: Nenhuma Sigla:

País: UF: CNPJ:

Título: Aplicação do mínimo múltiplo comum generalizado nas ondas de pêndulos Palavras-chave: _{Ondas de pêndulo, mmc, mmcg, comensurabilidade.}

Título em outra língua: Minimum application multiple widespread common in waves of pendulums

Palavras-chave em outra língua: Pendulum waves, lcm, lcmw, commensurability Área de concentração: Matemática do Ensino Básico

Data defesa:(dd/mm/aaaa) 30/03/2016 Programa de Pós-Graduação: PROFMAT

Orientador (a): Dr. Paulo Henrique de Azevedo Rodrigues E-mail: [email protected]

Co-orientador(a):* E-mail:

*Necessita do CPF quando não constar no SisPG

3. Informações de acesso ao documento:

Concorda com a liberação total do documento [ x ] SIM [ ] NÃO1

Havendo concordância com a disponibilização eletrônica, torna-se imprescindível o en-vio do(s) arquivo(s) em formato digital PDF ou DOC do trabalho de conclusão de curso.

O sistema da Biblioteca Digital de Teses e Dissertações garante aos autores, que os ar-quivos contendo eletronicamente as teses, dissertações ou trabalhos de conclusão de curso, antes de sua disponibilização, receberão procedimentos de segurança, criptografia (para não permitir cópia e extração de conteúdo, permitindo apenas impressão fraca) usando o padrão do Acrobat.

________________________________________ Data: _30_ / __03_ / _2016_ Assinatura do (a) autor (a)

Assinatura do (a) autor (a)

1 _{Neste caso o documento será embargado por até um ano a partir da data de defesa. A extensão deste prazo}

(3)

❏❡❛♥ ❈❛r❧♦ ❞❡ ❙♦✉s❛ ❡ ❙✐❧✈❛

❆♣❧✐❝❛çã♦ ❞♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠

❣❡♥❡r❛❧✐③❛❞♦ ♥❛s ♦♥❞❛s ❞❡ ♣ê♥❞✉❧♦s

❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ●♦✐ás✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ❇ás✐❝♦

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡ ❞❡ ❆③❡✈❡❞♦ ❘♦❞r✐❣✉❡s

●♦✐â♥✐❛

(4)

Ficha catalográfica elaborada automaticamente

com os dados fornecidos pelo(a) autor(a), sob orientação do Sibi/UFG.

Silva, Jean Carlo de Sousa e

Aplicação do mínimo múltiplo comum generalizado nas ondas de pêndulos [manuscrito] / Jean Carlo de Sousa e Silva. - 2016. iv, 43 f.: il.

Orientador: Prof. Dr. Paulo Henrique de Azevedo Rodrigues. Dissertação (Mestrado) - Universidade Federal de Goiás, Instituto de Matemática e Estatística (IME) , Programa de Pós-Graduação em Matemática, Goiânia, 2016.

Bibliografia. Anexos.

(5)

(6)

❚♦❞♦s ♦s ❞✐r❡✐t♦s r❡s❡r✈❛❞♦s✳ ➱ ♣r♦✐❜✐❞❛ ❛ r❡♣r♦❞✉çã♦ t♦t❛❧ ♦✉ ♣❛r❝✐❛❧ ❞❡st❡ tr❛❜❛❧❤♦ s❡♠ ❛ ❛✉t♦r✐③❛çã♦ ❞❛ ✉♥✐✈❡rs✐❞❛❞❡✱ ❞♦ ❛✉t♦r ❡ ❞❛ ♦r✐❡♥t❛❞♦r❛✳

(7)

(8)

❉❡❞✐❝♦ ❡ss❡ tr❛❜❛❧❤♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣♦✐s ❛ ❊❧❡ t♦❞❛ ❤♦♥r❛ ❡ ❣❧ór✐❛✳ ◆ã♦ ❡sq✉❡❝❡♥❞♦ ❞❡ t♦❞♦s ❛q✉❡❧❡s q✉❡ s❡ ❛❧❡❣r❛r❛♠ ❝♦♠✐❣♦ ❞❡s❞❡ ♦ í♥✐❝✐♦ ❛té ♥❡ss❡ ♠♦✲ ♠❡♥t♦ ❞❡ ❝♦♥❝❧✉sã♦ ❞♦ ♠❡str❛❞♦✳ ➚ ✈♦❝ês ♠❡✉s ❛♠✐❣♦s✱ ❢❛♠✐❧✐❛r❡s✱ ♠❡✉s q✉❡r✐❞♦s ♣❛✐s ❡ ❛ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛✳

(9)

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s ♣♦r t❡r ❝♦♥❝❡❞✐❞♦✲♠❡ ✈✐❞❛✱ s❛ú❞❡ ❛❧é♠ ❞❛ ♦♣♦rt✉✲ ♥✐❞❛❞❡ ❞❡ r❡❛❧✐③❛r ❡ss❡ ♠❡str❛❞♦✳ ❘❡❝♦♥❤❡ç♦ t❛♠❜é♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♠✐♥❤❛ ❛♠❛❞❛ ❡s♣♦s❛✱ ♠❡✉s ♣❛✐s✱ ♠✐♥❤❛ ✐r♠ã✱ ❞❡♥tr❡ ♦✉tr♦s q✉❡ ♠♦t✐✈❛r❛♠✲♠❡ ❡ ❝♦♠✲ ♣r❡❡♥❞❡r❛♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ t❛❧ ❝✉rs♦ ♣❛r❛ ♠✐♠✳ ◆ã♦ ♣♦ss♦ ❡sq✉❡❝❡r ❞♦ ❛♣♦✐♦ ❞♦s ❛♠✐❣♦s q✉❡ ❝♦♥tr✐✉❜✉✐r❛♠ ♠✉✐t♦ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ s♦♥❤♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦s ❛♠✐❣♦s ❞♦ P❘❖❋▼❆❚✱ q✉❡ ❛♣r♦①✐♠❛♠♦s ♠✉✐t♦ ♣♦r ❝♦♥t❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ♦❜❥❡t✐✈♦s t❡r❡♠ s❡ ❛♣r♦①✐♠❛❞♦s✳ ❆❣r❛❞❡ç♦ ♠✉✐t♦ ❛♦s ♣r♦❢❡ss♦r❡s✱ q✉❡ ❝♦♥❞✉③✐r❛♠✲♠❡ ❛ ✉♠❛ ❛✉t♦♥♦♠✐❛ ♥♦s ❡st✉❞♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r Pr♦❢❡ss♦r ❉r✳ P❛✉❧♦ ❍❡♥r✐q✉❡✱ q✉❡ ❝♦♠ ♠✉✐t❛ ♣❛❝✐ê♥❝✐❛ ❡ ❝♦♠♣❡tê♥❝✐❛ ♠❡ ❛✉①✐❧✐♦✉ ♥❛ ❝♦♥❢❡❝çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ P♦r ✜♠✱ q✉❡r♦ ❛❣r❛❞❡❝❡r à ❙❇▼ ❡ ❛ ❯❋● ♣♦r ❝r✐❛r❡♠ ❡ ♦❢❡r❡❝❡r❡♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ss❡ ♣r♦❣r❛♠❛ ❞❡ ♠❡str❛❞♦✱ ❝♦♥tr✐❜✉✐♥❞♦ ❛ss✐♠ ❝♦♠ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❛❝❛❞ê♠✐❝❛✳

(10)

❘❡s✉♠♦

❖s ♣ê♥❞✉❧♦s s✐♠♣❧❡s ♣♦ss✉❡♠ ♠♦✈✐♠❡♥t♦s ♦s❝✐❧❛tór✐♦s ❡ ♣❡r✐ó❞✐❝♦s✳ ❆ss✐♠✱ s❡ ❛♥❛❧✐✲ s❛r♠♦s ❛❧❣✉♥s ❞❡❧❡s ❝♦♠ ♠♦✈✐♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ❡❧❡s ♣♦❞❡rã♦ ✈♦❧t❛r ❛ s❡ ❡♥❝♦♥tr❛r ♥♦ ♠❡s♠♦ ♣♦♥t♦ s❡ s❡✉s ♣❡rí♦❞♦s ❢♦r❡♠ ❝♦♠❡♥s✉rá✈✐❡s✳ P❛r❛ ✐ss♦ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥✲ ❝❡✐t♦s ❜ás✐❝♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ss❡ ❢❡♥ô♠❡♥♦✳ ❋✐♥❛❧✐③❛♥❞♦ ❝♦♠ ✉♠❛ s✉❣❡stã♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞♦ ♠❡s♠♦✳

P❛❧❛✈r❛s✲❝❤❛✈❡

❖♥❞❛s ❞❡ ♣ê♥❞✉❧♦✱ ♠♠❝✱ ♠♠❝❣✱ ❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡✳

(11)

❆❜str❛❝t

❚❤❡ s✐♠♣❧❡ ♣❡♥❞✉❧✉♠s ❤❛✈❡ ♦s❝✐❧❧❛t♦r② ❛♥❞ ♣❡r✐♦❞✐❝ ♠♦✈❡♠❡♥ts✳ ❚❤✉s✱ ✐❢ ✇❡ ❛♥❛❧②③❡ s♦♠❡ ♦❢ t❤❡♠ ✇✐t❤ ✐♥❞❡♣❡♥❞❡♥t ♠♦✈❡♠❡♥ts✱ t❤❡② ♠❛② ❛❣❛✐♥ r❡t✉r♥ t♦ ❜❡ ✐♥ t❤❡ s❛♠❡ s♣♦t✱ ✐❢ ②♦✉r ♣❡r✐♦❞s ❛r❡ ❝♦♠❡♥s✉rá✈✐❡s✳ ❲❡ ♣r❡s❡♥t t❤❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ♥❡❡❞❡❞ t♦ ✉♥❞❡rst❛♥❞ t❤✐s ♣❤❡♥♦♠❡♥♦♥✳ ❊♥❞✐♥❣ ✇✐t❤ ❛ ❤✐♥t ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥✳

❑❡②✇♦r❞s

P❡♥❞✉❧✉♠ ✇❛✈❡s✱ ❧❝♠✱ ❧❝♠✇✱ ❝♦♠♠❡♥s✉r❛❜✐❧✐t②✳

(12)

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❙❡❣♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✷ ❈♦r♣♦s ❡♠ ▼✳❈✳❯✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✸ Pê♥❞✉❧♦ ❙✐♠♣❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✷✳✶ Pê♥❞✉❧♦s ❛ss♦❝✐❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❙❡q✉ê♥❝✐❛ ❞❡ ♣♦s✐çõ❡s ❞♦s ♣ê♥❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❊♥❝♦♥tr♦ ❞♦s ♣ê♥❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

(13)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶ ❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛ ✸

✶✳✶ ❙❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ❡ mmcg ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶✳✶✳✶ mmc ❡mdc ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

✶✳✶✳✷ ❙❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✶✳✶✳✸ mmcg ❡mdcg ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✷ ❊♥❝♦♥tr♦ ❞❡ ❝♦r♣♦s ❡ ❛♣❧✐❝❛çõ❡s ❞♦mmc ❡ ❞♦ mmcg ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✶✳✸ Pê♥❞✉❧♦ ❙✐♠♣❧❡s ❡ ♦ ▼✳❍✳❙✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷ ❆♣❧✐❝❛çã♦ ❞♦ mmcg ♥♦ ❢❡♥ô♠❡♥♦ ❞❡ ❜❛t✐♠❡♥t♦ ✷✷

✸ ❙✉❣❡stã♦ ❞❡ ❛❜♦r❞❛❣❡♠ ✷✽

✹ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✸✸

✺ ❆♥❡①♦ ✸✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✷

(14)

■♥tr♦❞✉çã♦

◆❡ss❡ tr❛❜❛❧❤♦✱ tr❛t❛r❡♠♦s ❞❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❣❡♥❡r❛❧✐③❛❞♦ ♥♦ ♠♦✈✐♠❡♥t♦ ❞❡ ♣ê♥❞✉❧♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❖ ♦❜❥❡t✐✈♦ ✐♥✐❝✐❛❧ é ✈❡r✐✜❝❛r s❡ ❞♦✐s ♦✉ ♠❛✐s ♣ê♥❞✉❧♦s ❡♠ ♠♦✈✐♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♣♦ré♠ ❛♥❛❧✐s❛❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡✱ ✈♦❧t❛r✐❛♠ ❛ s❡ ❡♥❝♦♥tr❛r ♥❛ ♠❡s♠❛ ♣♦s✐çã♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ t❛✐s ♣ê♥❞✉❧♦s s❡ ❡♥❝♦♥tr❛ss❡♠✱ q✉❛❧ s❡r✐❛ ♦ ♣❛❞rã♦ ♣❛r❛ t❛✐s ❡♥❝♦♥tr♦s❄

❖s P❛r❛♠êtr♦s ❝✉rr✐❝✉❧❛r❡s ♥❛❝✐♦♥❛✐s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✭P❈◆✰✮ ❞❡t❡r♠✐♥❛ ❞❡♥tr❡ ♦s ❝♦♥t❡ú❞♦s ♣r♦❣r❛♠át✐❝♦s ♣❛r❛ ♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ ♦ ❡♥s✐♥♦ ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱

mmc ❡ ❞❡ ▼♦✈✐♠❡♥t♦ ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ✭M.H.S.✮✳❬✼❪ ❆ss✐♠✱ ❝♦♠♦ ❛♠❜♦s sã♦ ❝♦♥✲

t❡ú❞♦s q✉❡ ❞❡✈❡♠ s❡r tr❛❜❛❧❤❛❞♦s ♥♦ ❡♥s✐♥♦ ❜ás✐❝♦✱ t❡♠♦s ❛q✉✐ ❛ s✉❣❡stã♦ ❞❡ ✉♠❛ ❛❜♦r❞❛❣❡♠ ✐♥t❡r❞✐s❝✐♣❧✐♥❛r q✉❡ ❝♦♥t❡♠♣❧❡ ❛♠❜♦s ❝♦♥t❡ú❞♦s ❞❡ ✉♠❛ ❢♦r♠❛ ♠❛✐s ❛♣❧✐✲ ❝á✈❡❧ ❛♦ ❝♦t✐❞✐❛♥♦ ❞♦ ❛❧✉♥♦✳

❈♦♠ ❛ ✜♥❛❧✐❞❛❞❡ ❞❡ ❡❧✉❝✐❞❛r ♦s ♦❜❥❡t✐✈♦s ❛❝✐♠❛✱ ♠♦♥t❛♠♦s ✉♠❛ s❡q✉ê♥❝✐❛ ❡♠ q✉❡ ♥♦ ❝❛♣ít✉❧♦ ❞❛ ❢✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛✱ ❡①♣❧✐❝❛♠♦s t♦❞♦s ♦s ♣r✐♥❝✐♣❛✐s ❝♦♥❝❡✐t♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦✳ ◆❡ss❡ ❝❛♣ít✉❧♦✱ t❡r❡♠♦s s❡çõ❡s ❡①♣❧✐❝❛♥❞♦ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱

mmc✱ ❡♥tr❡ ♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ❞❡♣♦✐s ❞❡✜♥✐♠♦s s❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ❡ ✐♥❝♦♠❡♥s✉✲

rá✈❡✐s ♣❛r❛ ♣♦❞❡r♠♦s ❣❡♥❡r❛❧✐③❛r ♦ ❝♦♥❝❡✐t♦ ❞♦ mmc✳ ❊♠ s❡❣✉✐❞❛✱ ❡①♣❧✐❝❛♠♦s t♦❞♦s

♦s ❝♦♥❝❡✐t♦s ❢ís✐❝♦s q✉❡ sã♦ ♥❡❝❡ssár✐♦s à ❝♦♠♣r❡❡♥sã♦ ❞♦ tr❛❜❛❧❤♦✱ ❞❡✐①❛♥❞♦ ♥❛ ❜✐❜❧✐♦✲ ❣r❛✜❛ ❛❧❣✉♠❛s r❡❢❡rê♥❝✐❛s ♣❛r❛ ♦ ❧❡✐t♦r ♠❛✐s ❝✉r✐♦s♦ ❛♠♣❧✐❛r s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦ ❡♠ t❛❧ ár❡❛✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡①♣❧✐❝❛♠♦s ❞❡ ❢❛t♦ ❛❧❣✉♥s ❝❛s♦s ♣❛r❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ♣ê♥❞✉✲ ❧♦s✳ ❉❡ss❡ ♠♦❞♦✱ ♠♦str❛♠♦s ♦ ❝❛s♦ ❞❡ ✈ár✐♦s ♣ê♥❞✉❧♦s ❡♠ ♠♦✈✐♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s ♣♦ré♠ ❛♥❛❧✐s❛❞♦s s✐♠✉❧tâ♥❡❛♠❡♥t❡✳ ▼♦str❛♥❞♦ q✉❡ ♣❛r❛ t❛✐s ♣ê♥❞✉❧♦s ✈♦❧t❛r❡♠ ❛ s❡ ❡♥❝♦♥tr❛r ♥❛ ♠❡s♠❛ ♣♦s✐çã♦ é ♥❡❝❡ssár✐❛ ❛ ❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❞❡ s❡✉s ♣❡rí♦❞♦s✳ ❘❡s✲ ♣♦♥❞❡♥❞♦ ❛ss✐♠ ♥♦ss❛ q✉❡stã♦ ❢✉♥❞❛♠❡♥t❛❧✱ ♦s ♣ê♥❞✉❧♦s ✈♦❧t❛♠ ❛ s❡ ❡♥❝♦♥tr❛r❡♠ s❡ ♦s s❡✉s ♣❡rí♦❞♦s ❢♦r❡♠ ❝♦♠❡♥s✉rá✈❡✐s✳ ❙❡♥❞♦ q✉❡ s❡ ❡♥❝♦♥tr❛rã♦ ♥♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱ ❣❡♥❡r❛❧✐❛❞♦ ♦✉ ♥ã♦✱ ❡♥tr❡ s❡✉s ♣❡rí♦❞♦s✳ ❋✐♥❛❧♠❡♥t❡✱ ❞❡✐①❛♠♦s ❝♦♠♦ s✉❣❡stã♦ ❞❡ ❛♣❧✐❝❛çã♦ ❡♠ ❢♦r♠❛ ❞❡ ♣❧❛♥♦s ❞❡ ❛✉❧❛ ❡①♣❧✐❝❛t✐✈♦s ♣❛r❛ q✉❡ ❡ss❛ ❛❜♦r❞❛❣❡♠ ♣♦ss❛

(15)

s❡r ❛♣❧✐❝❛❞❛ ♣❡❧♦s ❧❡✐t♦r❡s✳

(16)

❈❛♣ít✉❧♦ ✶

❋✉♥❞❛♠❡♥t❛çã♦ t❡ór✐❝❛

◆❡st❡ ❝❛♣ít✉❧♦ tr❛t❛r❡♠♦s ❞♦s ❝♦♥❝❡✐t♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ r❡s✲ t❛♥t❡ ❞♦ tr❛❜❛❧❤♦✳ P♦rt❛♥t♦✱ ♦ ♦❜❥❡t✐✈♦ ✐♥✐❝✐❛❧ é ❡st❛❜❡❧❡❝❡r ♦s ♣r✐♥❝í♣✐♦s ♠❛t❡♠át✐❝♦s ❡ ❢ís✐❝♦s ♣❛r❛ ❛ ❡❧✉❝✐❞❛çã♦ ❞❛s ♦♥❞❛s ❞❡ ♣ê♥❞✉❧♦s✳ ◗✉❡ tr❛t❛ ❞❛ ♦❜s❡r✈❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❞❡ ❛❧❣✉♥s ♣ê♥❞✉❧♦s ❡♠ ♠♦✈✐♠❡♥t♦s ✐♥❞❡♣❡♥❞❡♥t❡s✳

✶✳✶ ❙❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s ❡

mmcg

✶✳✶✳✶

mmc

❡

mdc

❉❡♥tr♦ ❞♦ ❝♦♥❥✉♥t♦ ❞♦s ✐♥t❡✐r♦s t❡♠♦s ❞❡✜♥✐❞♦s ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ✭mmc✮ ❡

♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ✭mdc✮✳ ❖ mmc é ♦ ♠❡♥♦r ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s A ❡ B✳ ❉✐r❡♠♦s q✉❡ ✉♠ ♥ú♠❡r♦ M é ♠ú❧t✐♣❧♦ ❞❡ x s❡ ♣✉❞❡r♠♦s ❛❝❤❛r n _∈Z t❛❧ q✉❡ M =n_·x✳ ❆ss✐♠ ❞✐r❡♠♦s q✉❡M é ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ ❞♦✐s ♥ú♠❡r♦s x❡ys❡ ❡①✐st✐r❡♠ m, n _∈ Z t❛✐s q✉❡ M = n _·x = m_·y✳ ❆❧é♠ ❞✐ss♦✱ é ✐♠♣♦rt❛♥t❡ ❞❡✜♥✐r♠♦s ♦ ♠❡♥♦r

♠ú❧t✐♣❧♦ ❝♦♠✉♠ ♦✉ ♦ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠✱mmc✱ ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❉✐r❡♠♦s q✉❡ M é ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x ❡ y✱ ❡s❝r❡✈❡♥❞♦ q✉❡ M =mmc(x, y) ♦✉ q✉❡ M = [x, y]✱ s❡ M s❛t✐s✜③❡r ❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✿

• M s❡r ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡ x ❝♦♠♦ ❞❡ y✱ ♦✉ s❡❥❛✱ M =n_·x ❡ M =m_·y❀

• M s❡r ♣♦s✐t✐✈♦❀

(17)

• s❡ M1 t❛♠❜é♠ é ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x ❡ y✱ ❞❡✈❡♠♦s t❡r q✉❡ M < M1✳

P♦r s❡♠❡❧❤❛♥t❡ ♠♦❞♦✱ ♣♦❞❡♠♦s ❢❛❧❛r q✉❡ D é ✉♠ ❞✐✈✐s♦r ❞❡ x s❡ ❡①✐st❡ u t❛❧ q✉❡ u _∈ Z ❡ x = D _·u✱ ❡s❝r❡✈❡♥❞♦ q✉❡ D _| x✱ D ❞✐✈✐❞❡ x✳ P♦rt❛♥t♦✱ ♣♦❞❡♠♦s ✉s❛r ❛

s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳

❉❡✜♥✐çã♦ ✶✳✷✳ ❉✐③❡♠♦s q✉❡ D é ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ x ❡ y✱ ❡s❝r❡✈❡♥❞♦ q✉❡ D=mdc(x, y) ♦✉ q✉❡ D= (x, y)✱ s❡ D s❛t✐s✜③❡r ❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✿

• D s❡r ❞✐✈✐s♦r t❛♥t♦ ❞❡ x ❝♦♠♦ ❞❡ y✱ ♦✉ s❡❥❛✱ x=D_·u ❡ y=D_·v❀

• s❡ D1 t❛♠❜é♠ é ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ x ❡ y✱ ❞❡✈❡♠♦s t❡r q✉❡ D > D1

❉❡♠♦♥str❛r❡♠♦s ❞✉❛s ♣r♦♣r♦s✐çõ❡s s♦❜r❡ ♦ mmc✱ q✉❡ s❡rã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛

❝♦♠♣r❡❡♥sã♦ ❞♦ r❡st❛♥t❡ ❞❡ss❡ tr❛❜❛❧❤♦✳

Pr♦♣r♦s✐çã♦ ✶✳✶✳ ❖ ♠ó❞✉❧♦ ❞♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ✐♥t❡✐r♦s q✉❛✐sq✉❡r é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❡♥tr❡ ♦mmc ❡ mdc ❞❡ss❡s ♥ú♠❡r♦s✳

|x_·y_|= [x, y]_·(x, y) ✭✶✳✶✮

❉❡♠♦♥str❛çã♦✳ ❚❛❧ ♣r♦♣r✐❡❞❛❞❡ ♣♦❞❡ s❡r ✈❡r✐✜❝❛❞❛ ❢❛❝✐❧♠❡♥t❡ ♣♦✐s t♦♠❛♥❞♦ d ✉♠

❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ x ❡ y ♣♦❞❡r❡♠♦s ❡s❝r❡✈❡r q✉❡✿ m = |x·y|

d ✳ ❖♥❞❡ m s❡rá ✉♠

♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x ❡ y ♣♦✐s m =_| x _{| ·}|y |

d ❡ m =| y | ·

|x_|

d ✳ ❉❡ss❛ ❢♦r♠❛✱ t❡r❡♠♦s

q✉❡ t♦❞♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x ❡ y s❡rá ❞❛ ❢♦r♠❛ ♠❡♥❝✐♦♥❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳ ❙❡♥❞♦

❛ss✐♠✱ t♦♠❛♥❞♦ ♦ ♠❛✐♦rd ♣♦ssí✈❡❧ q✉❡ s❡❥❛ ❞✐✈✐s♦r ❞❡ x ❡ ❞❡ y t❡r❡♠♦s q✉❡ d= (x, y)

❡ ♣♦r ❝♦♥s❡q✉ê♥❝✐❛✱m s❡rá ♦ ♠❡♥♦r ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x❡ y✱ s❡♥❞♦ ❛ss✐♠ m= [x, y]✱

♣♦rt❛♥t♦ t❡r❡♠♦s q✉❡✿

m = _|x_{| ·}|y| d

m = _|y_{| ·}|x| d

|x_·y_| = m_·d

|x_·y_| = [x, y]_·(x, y)

❬✻❪

(18)

Pr♦♣r♦s✐çã♦ ✶✳✷✳ ❖ mdc ❞❡ ❞♦✐s ♥ú♠❡r♦s q✉❡ ♣♦ss✉❡♠ ✉♠ ❢❛t♦r ❝♦♠✉♠✱ é t❛❧ ❢❛t♦r

❝♦♠✉♠ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ mdc ❞♦s ♥ú♠❡r♦s ♦r✐❣✐♥❛✐s ❞✐✈✐❞♦s ♣♦r t❛❧ ❢❛t♦r✱ ❡s❝r❡✈❡♠♦s

❡♥tã♦ q✉❡✿

(a_·x, a_·y) = a_·(x, y) ✭✶✳✷✮

❉❡♠♦♥str❛çã♦✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ mdc t❡♠♦s q✉❡ D é ♦ mdc ❡♥tr❡ x ❡ y s❡ D ❢♦r ♦

♠❛✐♦r ✐♥t❡✐r♦ q✉❡ s❡❥❛ ❞✐✈✐s♦r ❞❡ ❛♠❜♦s ♦s ♥ú♠❡r♦s✳ ❙❡♥❞♦a _∈Z t❛❧ q✉❡ x_a _∈/ Z ❡ y_a _∈/ Zt❡r❡♠♦s q✉❡ _|a_{| ·}D s❡rá ♦ ♠❛✐♦r ❞✐✈✐s♦r ❞❡a_·x ❡ ❞❡ a_·y ❞❡ss❛ ❢♦r♠❛ t❡r❡♠♦s q✉❡✿

(a_·x, a_·y) = _|a _{| ·}D

(a_·x, a_·y) = _|a _{| ·}(x, y)

Pr♦♣r♦s✐çã♦ ✶✳✸✳ ❖mmc ❞❡ ❞♦✐s ♥ú♠❡r♦s q✉❡ ♣♦ss✉❡♠ ✉♠ ❢❛t♦r ❝♦♠✉♠✱ é t❛❧ ❢❛t♦r

❝♦♠✉♠ ♠✉❧t✐♣❧✐❝❛❞♦ ♣❡❧♦ mmc ❞♦s ♥ú♠❡r♦s ♦r✐❣✐♥❛✐s ❞✐✈✐❞♦s ♣♦r t❛❧ ❢❛t♦r✱ ❡s❝r❡✈❡♠♦s

❡♥tã♦ q✉❡✿

[a_·x, a_·y] =a_·[x, y] ✭✶✳✸✮

❉❡♠♦♥str❛çã♦✳ ❖ q✉❡ ♣♦❞❡ s❡r ❡♥t❡♥❞✐❞♦ ♣♦✐s ❞❛❞♦ ✉♠ ✐♥t❡✐r♦ a t❛❧ q✉❡ (a, x) = (a, y) = 1 t❡r❡♠♦s q✉❡ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶ t❡♠♦s q✉❡✿

|a_·x_·a_·y_| = [a_·x, a_·y]_·(a_·x, a_·y)

|a_·x_·a_·y_| = [a_·x, a_·y]_·a_·(x, y)

P♦✐s ❝♦♠♦ (a, x) = (a, y) = 1 t❡♠♦s q✉❡ s❡ d ❞✐✈✐❞❡ x ❡ ❞✐✈✐❞❡ y ✐rá t❛♠❜é♠ ❞✐✈✐❞✐r a_·x ❡ a_·y ❡♥tr❡t❛♥t♦✱ a_·d t❛♠❜é♠ ❞✐✈✐❞✐rá a_·x ❡ a_·y ❡ ❝♦♠♦ ♣♦r ❞❡✜♥✐çã♦ ♦ mdc

❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❡❧❡s t❡♠♦s q✉❡(a_·x, a_·y) =a_·(x, y)

❡ ❛✐♥❞❛ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✶ ❝❤❡❣❛r❡♠♦s q✉❡✿

a2 _{| ·}x_·y_| = [a_·x, a_·y]_·a_·(x, y)

a_·(x, y)_·[x, y] = [a_·x, a_·y]_·(x, y)

a_·[x, y] = [a_·x, a_·y]

(19)

✶✳✶✳✷ ❙❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s

❯♠ s❡❣♠❡♥t♦ é ✉♠❛ ♣❛rt❡ ❞❡ ✉♠❛ r❡t❛✱ q✉❡ ❢♦r❛ ❧✐♠✐t❛❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♦ s❡❣♠❡♥t♦

AB é ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ✐♥✐❝✐❛✲s❡ ❡♠ A ❡ t❡r♠✐♥❛ ❡♠ B✳ P♦❞❡♠♦s ❛ss♦❝✐❛r à

❡ss❡ s❡❣♠❡♥t♦ ✉♠ ✈❛❧♦r ♣❛r❛ s❡✉ ❝♦♠♣r✐♠❡♥t♦✳ P❛r❛ ❞❡✜♥✐r ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ s❡❣♠❡♥t♦AB t❡♠♦s q✉❡ ❛♥❛❧✐③❛r q✉❛♥t❛s ✈❡③❡s ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ u é ♠❡♥♦r ❞♦

q✉❡AB✳

❉❡✜♥✐çã♦ ✶✳✸✳ ❙❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈❡③❡s q✉❡ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ u ❢♦r ♠❡♥♦r ❞♦ q✉❡

♦ s❡❣♠❡♥t♦ AB ❢♦r ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦✱ ❞✐r❡♠♦s q✉❡ t❛❧ s❡❣♠❡♥t♦ é ❝♦♠❡♥s✉rá✈❡❧ ❡♠

r❡❧❛çã♦ à ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ ❡st❛❜❡❧❡❝✐❞❛✳ ❆ss✐♠✱ s❡✉ ❝♦♠♣r✐♠❡♥t♦ s❡rá ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ é ♠❡♥♦r ❞♦ q✉❡ ♦ s❡❣♠❡♥t♦ ❛ s❡r ♠❡❞✐❞♦ ❡ t❛❧ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ é ❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✳ ❊♥tr❡t❛♥t♦✱ s❡ ♥ã♦ ❢♦r ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ♥ú✲ ♠❡r♦ ✐♥t❡✐r♦ ❞❡ ✈❡③❡s q✉❡ ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ ❛❝♦♠♦❞❛✲s❡ ❡♠ AB t❛❧ s❡❣♠❡♥t♦ s❡rá

❝❧❛ss✐✜❝❛❞♦ ❝♦♠♦ s❡❣♠❡♥t♦ ✐♥❝♦♠❡♥s✉rá✈❡❧ ❡♠ r❡❧❛çã♦ à ✉♥✐❞❛❞❡ ❡st❛❜❡❧❡❝✐❞❛✳

❋✐❣✉r❛ ✶✳✶✿ ❙❡❣♠❡♥t♦s

P❛r❛ ✉♠ ú♥✐❝♦ s❡❣♠❡♥t♦✱ ♣♦❞❡r❡♠♦s tr♦❝❛r ♥♦ss❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✱ ♦✉ s❡❣♠❡♥t♦ ✉♥✐tár✐♦✱ ❞❡ ❢♦r♠❛ q✉❡ ♦ s❡❣♠❡♥t♦ q✉❡ ❛♥t❡s ❡r❛ ✐♥❝♦♠❡♥s✉rá✈❡❧ ❡♠ r❡❧❛çã♦ ❛♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦us❡rá ❝♦♠❡♥s✉rá✈❡❧ ❡♠ r❡❧❛çã♦ ❛♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦s✳ ❖✉ s❡❥❛✱ ❛ ❝❧❛s✐✜❝❛çã♦

q✉❛♥t♦ ❛ ❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ é r❡❧❛t✐✈❛✳ ◆❛ ❋✐❣✉r❛ ✶✳✶ t❡♠♦s ✉♠ ❡①❡♠♣❧♦ ❞✐ss♦✿ ♦ s❡❣♠❡♥t♦AB é ✐♥❝♦♠❡♥s✉rá✈❡❧ ❡♠ r❡❧❛çã♦ ❛♦ s❡❣♠❡♥t♦u❡♥tr❡t❛♥t♦✱ ❡❧❡ é ❝♦♠❡♥s✉rá✈❡❧

❡♠ r❡❧❛çã♦ ❛♦ s❡❣♠❡♥t♦s✳

❉❡✜♥✐çã♦ ✶✳✹✳ P♦❞❡♠♦s ❛ss✐♠ ❣❡♥❡r❛❧✐③❛r q✉❡ ❞♦✐s s❡❣♠❡♥t♦s AB ❡ CD sã♦ ❝♦♠❡♥✲

s✉rá✈❡✐s s❡ ❡①✐st✐r ✉♠ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ q✉❡ ♣♦ss❛ ♠❡❞✐r✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱ ❛♠❜♦s s❡❣♠❡♥t♦s✳ ❆❣♦r❛✱ s❡ ♥ã♦ ❡①✐st✐r ✉♠ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ q✉❡ ♣♦ss❛ ♠❡❞✐r✱ ❛♦ ♠❡s♠♦ t❡♠♣♦✱AB ❡ CD ❡❧❡s s❡rã♦ ✐♥❝♦♠❡♥s✉rá✈❡✐s✳

❯♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥s❡q✉ê♥❝✐❛ ❞❡ ❞♦✐s s❡❣♠❡♥t♦s s❡r❡♠ ❝♦♠❡♥s✉rá✈❡✐s é q✉❡ ❛ r❛③ã♦ ❡♥tr❡ ❡❧❡s s❡rá ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳ P♦✐s ❛♠❜♦s ♦s s❡❣♠❡♥t♦s s❡rã♦ ❡s❝r✐t♦s ❝♦♠♦ ✉♠

(20)

♣r♦❞✉t♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✐♥t❡✐r♦ ♣♦r ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛✳ ❈♦♠♦ ❡①❡♠♣❧♦ ♣♦❞❡♠♦s t❡r✿ AB =m_·u ❡ CD =n_·u ♣♦rt❛♥t♦ AB

CD = m·u

n·u = m

n✳ ❈♦♠♦ t❛♥t♦ m q✉❛♥t♦n sã♦

♥ú♠❡r♦s ✐♥t❡✐r♦s ❡ ❝♦♥s✐❞❡r❛♥❞♦n ₆= 0 t❡r❡♠♦s q✉❡ m

n s❡rá ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧✳

❊♥tã♦✱ ♦✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❡①♣r❡ss❛r q✉❡ ❞♦✐s s❡❣♠❡♥t♦s ❞❡ ❞✐♠❡♥sõ❡s x ❡ y sã♦

❝♦♠❡♥s✉rá✈❡✐s é ❞✐s❝✉t✐r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❞♦✐s ♥ú♠❡r♦s ✐♥t❡✐r♦sm ❡ n t❛✐s q✉❡✿

m_·y=n_·x ✭✶✳✹✮

❚❛❧ ❝♦♥s❡q✉ê♥❝✐❛ é ✐♠♣❧✐❝❛❞❛ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞♦s s❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s✳ P♦✐s s❡

x ❡ y sã♦ ❝♦♠❡♥s✉rá✈❡✐s✱ ♣♦❞❡♠♦s t♦♠❛r ✉♠ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ u t❛❧ q✉❡ x =m_·u ❡ y=n_·u✳ P♦rt❛♥t♦✱ ❞✐✈✐❞✐♥❞♦ ❛♠❜❛s ❡q✉❛çõ❡s t❡r❡♠♦s q✉❡✿

x

y =

m_·u n_·u x

y =

m n x_·n = y_·m

❊♠ q✉❡ ♣❡❧❛ ❞❡✜♥✐çã♦ t❡♠♦s q✉❡ m ❡ n _∈Z

◆❛ ❆♥t✐❣✉✐❞❛❞❡✱ ♦s ❣r❡❣♦s ❛❝r❡❞✐t❛✈❛♠ q✉❡ ❞♦✐s s❡❣♠❡♥t♦s ❡r❛♠ s❡♠♣r❡ ❝♦♠❡♥✲ s✉rá✈❡✐s ❡♠ r❡❧❛çã♦ ❛ ❛❧❣✉♠❛ ✉♥✐❞❛❞❡✱ ♦✉ s❡❥❛✱ s❡♠♣r❡ ♣♦❞❡rí❛♠♦s tr♦❝❛r ♦ s❡❣♠❡♥t♦ ✉♥✐tár✐♦ ♣❛r❛ q✉❡ t❛✐s s❡❣♠❡♥t♦s ❢♦ss❡♠ ❝♦♠❡♥s✉rá✈❡✐s✳ ❊♥tr❡t❛♥t♦✱ ❡♥tr❡ ✹✺✵ ❡ ✹✵✵ ❛✳❈✳ ♣r♦✈♦✉✲s❡ q✉❡ ♦ ❧❛❞♦ ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❡r❛♠ ✐♥❝♦♠❡♥s✉rá✈❡✐s✳❬✺❪

❙❡ ❢♦ss❡ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠❛ ✉♥✐❞❛❞❡ ❡♠ q✉❡ ♦ ❧❛❞♦ ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ ♠❡s♠♦ q✉❛❞r❛❞♦ ❢♦ss❡ ❝♦♠❡♥s✉rá✈❡✐s✱ t❡rí❛♠♦s q✉❡ t❡r✿

m_·x = n_·x√2

m2_·x2 = n2_·x2_·2

m2 _{= 2}_·_n2 _{✭✶✳✺✮}

❖ q✉❡ é ✉♠ ❛❜s✉r❞♦ ♣♦✐s ❞♦ ❧❛❞♦ ❡sq✉❡r❞♦ ❞❛ ❊q✉❛çã♦ ✭✶✳✺✮ t❡♠♦s q✉❡✱ q✉❛♥❞♦ t❛✐s ♥ú♠❡r♦s ❢♦r❡♠ ❡s❝r✐t♦s ❝♦♠♦ ❢❛t♦r❛çã♦ ❡♠ ♥ú♠❡r♦s ♣r✐♠♦s✱ ♦ ♣r✐♠♦2 ❛♣❛r❡❝❡rá ✉♠❛

(21)

q✉❛♥t✐❞❛❞❡ ♣❛r ❞❡ ✈❡③❡s ♥❛ ❢❛t♦r❛çã♦ ❞♦ ✐♥t❡✐r♦✱ m✱ ❡♥q✉❛♥t♦ q✉❡ ❞♦ ❧❛❞♦ ❞✐r❡✐t♦ ❡❧❡

❛♣❛r❡❝❡rá ✉♠ q✉❛♥t✐❞❛❞❡ í♠♣❛r ❞❡ ✈❡③❡s✳ P♦r ✐ss♦✱ ♦ ❧❛❞♦ ❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ sã♦ ✐♥❝♦♠❡♥s✉rá✈❡✐s✳

✶✳✶✳✸

mmcg

❡

mdcg

❆s ♥♦çõ❡s ❞❡ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠✱mmc✱ ❡ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠✱mdc✱ ❞❡ ❞♦✐s

♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♣♦❞❡♠ s❡r ❡①t❡♥❞✐❞❛s ♣❛r❛ ♣❛r❡s ❞❡ ♥ú♠❡r♦s r❡❛✐s ❝♦♠❡♥s✉rá✈❡✐s✳ ❉❡ss❛ ❢♦r♠❛✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ s❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s✱ ❡❧✉❝✐❞❛❞❛s ♥❛ ❙❡çã♦ ✶✳✶✳✷✱ ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣❧✐❝❛r ❛ ✐❞❡✐❛ ❞❡mmc❡mdc❞❡ ❢♦r♠❛ ❣❡♥❡r❛❧✐③❛❞❛✳

◗✉❛♥❞♦ ❞♦✐s s❡❣♠❡♥t♦s AB ❡ CD ❞❡ ❞✐♠❡♥sõ❡s x ❡ y✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❢♦r❡♠

❝♦♠❡♥s✉rá✈❡✐s✱ ♣♦❞❡r❡♠♦s ❡s❝r❡✈❡r q✉❡ x = m_·u ❡ y = n_·u ❝♦♠ m ❡ n _∈ Z ✳ ❖✉

❛✐♥❞❛✱ ❝♦♠♦ ♥❛ ❡q✉❛çã♦ ✭✶✳✹✮ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡m_·y=n_·x t❛✐s q✉❡ m ❡ n _∈Z✳

❈♦♠♦ x ❡ y sã♦ ♠❡❞✐❞❛s ❞❡ s❡❣♠❡♥t♦s ❝♦♠❡♥s✉rá✈❡✐s t❡r❡♠♦s q✉❡ ❛ r❛③ã♦ ❡♥tr❡

❡❧❡s s❡rá ✉♠ ♥ú♠❡r♦ r❛❝✐♦♥❛❧ ❡ ♣♦❞❡r❡♠♦s ❡s❝r❡✈❡r q✉❡✿

x

y =

m_·u n_·u x

y =

m n

❚♦♠❛♥❞♦ u ❞❡ ❢♦r♠❛ q✉❡ ❡❧❡ s❡❥❛ ♦ ♠❛✐♦r s❡❣♠❡♥t♦ ✉♥✐tár✐♦ q✉❡ ❢❛ç❛ AB ❡ CD

s❡r❡♠ ❝♦♠❡♥s✉rá✈❡✐s t❡r❡♠♦s q✉❡(m, n) = 1❡ ❛ss✐♠ ❛ ❢r❛çã♦ m

n é ❛ ❢r❛çã♦ ✐rr❡❞✉tí✈❡❧ ❞❡

|x_|

|y_| ✳ ❖ q✉❡ ♥♦s ♣❡r♠✐t❡ ❡s❝r❡✈❡r t❛✐s ❣r❛♥❞❡③❛s ❝♦♠♦ ♥❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛✳ ❆❧é♠ ❞✐ss♦

♣♦❞❡♠♦s✱ ❢❛❝✐❧♠❡♥t❡✱ ✈❡r✐✜❝❛r q✉❡ M =m_{· |}y _|= n_{· |}x _| é ♦ ♠❡♥♦r ♠ú❧t✐♣❧♦ ❝♦♠✉♠

❡♥tr❡x❡y♣♦✐s(m, n) = 1♣♦rq✉❡ t♦♠❛♠♦s ❛ ♠❛✐♦r ✉♥✐❞❛❞❡ ❞❡ ♠❡❞✐❞❛ q✉❡ ♣❡r♠✐t✐❛ ❛

❝♦♠❡♥s✉r❛❜✐❧✐❞❛❞❡ ❡♥tr❡ ♦s s❡❣♠❡♥t♦s✳ P♦✐s s❡ (m, n)₆= 1 s❡r✐❛ ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠

u1 = (m, n)·ut❛❧ q✉❡ ♦s s❡❣♠❡♥t♦s AB ❡ CD ❛✐♥❞❛ ❢♦ss❡♠ ❝♦♠❡♥s✉rá✈❡✐s✱ ❡♥tr❡t❛♥t♦

s❡ (m, n) ₆= 1 t❡r❡♠♦s q✉❡ u1 > u ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦ ♣♦✐s ❡s❝♦❧❤❡♠♦s u ♦ ♠❛✐♦r

s❡❣♠❡♥t♦ ♣♦ssí✈❡❧ ♣❛r❛ q✉❡AB ❡CD ❢♦ss❡♠ ❝♦♠❡♥s✉rá✈❡✐s✳ ❆ss✐♠ ♣♦❞❡✲s❡ ❛ss♦❝✐❛r ❛

t❛✐s ♥ú♠❡r♦s ✉♠ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❣❡♥❡r❛❧✐③❛❞♦(mmcg)r❡s♣❡✐t❛♥❞♦ ❛ s❡❣✉✐♥t❡

❞❡✜♥✐çã♦✿

❉❡✜♥✐çã♦ ✶✳✺✳ M =_| y _{| ·}m =_| x _{| ·}n s❡rá ♦ mmcg ❡♥tr❡ x ❡ y✱ ❡s❝r❡✈❡♥❞♦ q✉❡ M =mmcg(x, y)✱ s❡ M s❛t✐s✜③❡r ❛s ❝♦♥❞✐çõ❡s ❛ s❡❣✉✐r✿

(22)

• M s❡r ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡ x ❝♦♠♦ ❞❡ y❀

• M s❡r ♣♦s✐t✐✈♦❀

• s❡ M1 t❛♠❜é♠ é ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❞❡ x ❡ y ❞❡✈❡♠♦s t❡r q✉❡ M < M1✳

mmcg(x, y) = m_·y=n_·x ✭✶✳✻✮

❉❡✜♥✐çã♦ ✶✳✻✳ D s❡rá ♦ ♠á①✐♠♦ ❞✐✈✐s♦r ❝♦♠✉♠ ❣❡♥❡r❛❧✐③❛❞♦ ✭mdcg✮ ❡♥tr❡ x ❡ y s❡ D=u ♦♥❞❡ u é ♦ ♠❛✐♦r s❡❣♠❡♥t♦ q✉❡ ❝♦♥s❡❣✉❡ ❞✐✈✐❞✐r ❡♠ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✐♥t❡✐r❛ ❞❡

✈❡③❡s t❛♥t♦ AB ❝♦♠♦ CD s❡♥❞♦ ❛ss✐♠ u= |x|

m =

|y_|

n ✳ ▲♦❣♦ D s❡rá ♦ mdcg ❡♥tr❡ x

❡ y s❡ ❡❧❡ s❛t✐s✜③❡r ❛s ❝♦♥❞✐çõ❡s✿

• D s❡r ❞✐✈✐s♦r t❛♥t♦ ❞❡ x ❝♦♠♦ ❞❡ y✱ ♦✉ s❡❥❛✱ x=D_·u ❡ y=D_·v❀

• s❡ D1 t❛♠❜é♠ é ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ x ❡ y ❞❡✈❡♠♦s t❡r q✉❡ D > D1

mdcg(x, y) = x

m =

y

n ✭✶✳✼✮

Pr♦♣r♦s✐çã♦ ✶✳✹✳ ❈♦♥s✐❞❡r❛♥❞♦ x ❡ y _∈ Q t❛✐s q✉❡ s✉❛s ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s s❡rã♦✱

r❡s♣❡❝t✐✈❛♠❡♥t❡a/b ❡ c/d♣♦❞❡r❡♠♦s ❝❛❧❝✉❧❛r ♦ mmcg❡♥tr❡ x❡ y❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

mmcg(x, y) = [a, c]

(b, d) ✭✶✳✽✮

❉❡♠♦♥str❛çã♦✳ ❈♦♠♦x= a

b ❡y= c

d✱ t❡r❡♠♦s q✉❡✿

x

y =

a/b c/d =

a_·d b_·c

❉✐✈✐❞✐♥❞♦ t❛♥t♦ ♦ ♥✉♠❡r❛❞♦r ❝♦♠♦ ♦ ❞❡♥♦♠✐♥❛❞♦r ♣♦r (a, c)_·(b, d) t❡r❡♠♦s q✉❡✿

x

y =

a_·d

(a, c)_·(b, d)

c_·b

(a, c)_·(b, d) =

a

(a, c)·

d

(b, d)

c

(a, c)·

b

(b, d)

(23)

❆ss✐♠✱ t❛♥t♦ ♦ ♥✉♠❡r❛❞♦r ❝♦♠♦ ♦ ❞❡♥♦♠✐♥❛❞♦r s❡rã♦ ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s ❡ ♣♦r ❡ss❛ r❛③ã♦✱ ❝♦♠♦ ❢❡✐t♦ ♥❛ ❊q✉❛çã♦ ✭1.6✮✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ♦ mmcg ❡♥tr❡ x ❡ y✱

r❡♣r❡s❡♥t❛❞♦ ♣♦rM✱ s❡rá ✐❣✉❛❧ ❛✿

M = y_· a

(a, c)

d

(b, d)

M = x_· c

(a, c)

b

(b, d)

M = a

b · c

(a, c)

b

(b, d)

M = a_· c

(a, c) 1 (b, d)

❈♦♠♦ ♦ ♠ó❞✉❧♦ ❞♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ♥ú♠❡r♦s é s❡♠♣r❡ ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ mmc

♣❡❧♦mdc❡♥tr❡ ❡❧❡s✱ q✉❡ ❢♦✐ ❞❡♠♦♥str❛❞♦ ♥❛ Pr♦♣♦s✐çã♦ ✶✳✶ t❡♠♦s q✉❡✿ [a, c] = a·c (a, c)

❡ ❝♦♠ ✐ss♦ t❡r❡♠♦s q✉❡✿

M =mmcg(x, y) = [a, c] (b, d)

Pr♦♣r♦s✐çã♦ ✶✳✺✳ ❈♦♥s✐❞❡r❛♥❞♦ x ❡ y _∈ Q t❛✐s q✉❡ s✉❛s ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s s❡rã♦✱

r❡s♣❡❝t✐✈❛♠❡♥t❡ a/b ❡ c/d ♣♦❞❡r❡♠♦s ❝❛❧❝✉❧❛r ♦ mdcg ❡♥tr❡ x ❡ y ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

mdcg(x, y) = (a, c)

[b, d] ✭✶✳✾✮

❉❡♠♦♥str❛çã♦✳ P❡❧♦ q✉❡ ✜③❡♠♦s ♥❛ ❞❡♠♦♥str❛çã♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❞✐③❡r q✉❡✿

x_· c

(a, c)

b

(b, d) = y·

a

(a, c)

d

(b, d)

y =

x_· c

(a, c)

b

(b, d)

a

(a, c)

d

(b, d)

❉✐ss♦ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✿ a x

(a,c) ·

d

(b,d)

❞✐✈✐❞❡ t❛♥t♦ x ❝♦♠♦ y✱ ❛❧é♠ ❞✐ss♦ q✉❡ t❛❧

(24)

✈❛❧♦r é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❞❡ x ❡ y s❡♥❞♦ ❛ss✐♠✱ r❡♣r❡s❡♥t❛♥❞♦ ♦ mdcg(x, y) ♣♦r

D✱ t❡r❡♠♦s✿

D = _a x

(a,c) ·

d

(b,d)

D = _aa/b

(a,c) ·

d

(b,d)

D = a·(a, c)·(b, d)

a_·b_·d

D = (a, c)·(b, d)

b_·d

▼❛s ❝♦♠♦b_·d= (b, d)_·[b, d] ♣♦❞❡r❡♠♦s ❡s❝r❡✈❡r q✉❡✿

mdcg(x, y) = (a, c) [b, d]

❚❛✐s ❝♦♥❝❡✐t♦s ❢♦r❛♠ ❜❛s❡❛❞♦s ♥♦ tr❛❜❛❧❤♦ ❞❡ ❈②❞❛r❛ ❘✐♣♦❧❧ ❡t ❛❧❬✺❪✳ ❊ss❛s ❣❡♥❡r❛✲ ❧✐③❛çõ❡s q✉❛♥❞♦ ❛♣❧✐❝❛❞❛s ❛ ♥ú♠❡r♦s ✐♥t❡✐r♦s ♥ã♦ ♣❡r❞❡♠ s✐❣♥✜❝❛❞♦ ♣♦✐s ✈♦❧t❛♠ ♣❛r❛ ❛s ❞❡✜♥✐çõ❡s ✉s✉❛✐s ❞♦mmc❡ ❞♦mdc❞♦s ♥ú♠❡r♦s ✐♥t❡✐r♦s✳ P♦✐s✱ s✉♣♦♥❤❛ ❞♦✐s ✐♥t❡✐r♦s x ❡ y ❡♥tã♦ t❡r❡♠♦s q✉❡ ♣❛r❛ ❡s❝r❡✈ê✲❧♦s ❝♦♠♦ ❢r❛çõ❡s✱ ❜❛st❛ t♦♠❛r♠♦s a = x✱ c =y

❡b =d = 1✱ ❡♥tã♦ t❛♥t♦ ♦mmcg q✉❛♥t♦ ♦mdcg ✈♦❧t❛♠ ♣❛r❛ ❛s ❞❡✜♥✐çõ❡s ✐♥✐❝✐❛✐s ❞❡ mmc ❡mdc✳

mmcg(x, y) = mmc(a, c)

mdc(b, d)

mmcg(x, y) = mmc(a, c)

mmcg(x, y) = mmc(x, y)

e

mdcg(x, y) = mdc(a, c)

mmc(b, d)

mdcg(x, y) = mdc(a, c)

mdcg(x, y) = mdc(x, y)

(25)

❆❧é♠ ❞✐ss♦ ✈❛❧❡ ❛s ♣r♦♣♦s✐çõ❡s s❡♠❡❧❤❛♥t❡s ❛s Pr♦♣♦s✐çõ❡s ✶✳✶ ❡ ✶✳✸

Pr♦♣r♦s✐çã♦ ✶✳✻✳ ❖ ♠ó❞✉❧♦ ❞♦ ♣r♦❞✉t♦ ❞♦s ♥ú♠❡r♦s é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❡♥tr❡ ♦mmcg

❡ mdcg ❞❡ss❡s ♥ú♠❡r♦s✳

|x_·y_|=mmcg(x, y)_·mdcg(x, y) ✭✶✳✶✵✮

❉❡♠♦♥str❛çã♦✳ ❈♦♥s✐❞❡r❛♥❞♦ x ❡ y ❝♦♠♦ r❛❝✐♦♥❛✐s t❛✐s q✉❡ s✉❛s ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s

s❡rã♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ a/b ❡ c/d t❡r❡♠♦s q✉❡✿

|x_·y_| = mmcg(x, y)_·mdcg(x, y)

|x_·y_| = mmc(a, c)

mdc(b, d) ·

mdc(a, c)

mmc(b, d)

|x_·y_| = a·c

b_·d

Pr♦♣r♦s✐çã♦ ✶✳✼✳ ❙❡ ♦s ♥ú♠❡r♦s x ❡ y ❢♦r❡♠ t❛✐s q✉❡ mmcg(x, y) =a t❡r❡♠♦s q✉❡ ♦ mmcg(x, y) s❡rá ♦ ♣r♦❞✉t♦ ❞❡ a ♣❡❧♦ mmcg(x1, y1) ♦♥❞❡ x1 = x_a ❡ y1 = y_a✳

mmcg(a_·x, a_·y) = a_·mmcg(x, y) ✭✶✳✶✶✮

❉❡♠♦♥str❛çã♦✳ P♦rq✉❡ ♣♦❞❡r❡♠♦s ❡s❝r❡✈❡r q✉❡ ❞❛❞♦ ✉♠ r❡❛❧ a t❛❧ q✉❡ mdcg(a, x) =

mdcg(a, y) = 1 t❡r❡♠♦s✱ ❞❛ Pr♦♣♦s✐çã♦ ✶✳✻✱ q✉❡✿

|a_·x_·a_·y_| = mmcg(a_·x, a_·y)_·mdcg(a_·x, a_·y)

|a_·x_·a_·y_| = mmcg(a_·x, a_·y)_·a_·mdcg(x, y)

P♦✐s ❝♦♠♦ mdcg(a, x) = mdcg(a, y) = 1 t❡♠♦s q✉❡ s❡ d ❞✐✈✐❞❡ x ❡ ❞✐✈✐❞❡ y ✐rá

t❛♠❜é♠ ❞✐✈✐❞✐r a_·x ❡ a_·y ❡♥tr❡t❛♥t♦✱ a_·d t❛♠❜é♠ ❞✐✈✐❞✐rá a_·x ❡ a_·y ❡ ❝♦♠♦ ♣♦r

❞❡✜♥✐çã♦ ♦ mdcg ❡♥tr❡ ❞♦✐s ♥ú♠❡r♦s é ♦ ♠❛✐♦r ❞✐✈✐s♦r ❝♦♠✉♠ ❡♥tr❡ ❡❧❡s t❡♠♦s q✉❡ mdcg(a_·x, a_·y) =a_·mdcg(x, y) ❡ ❡♥tã♦ ❝❤❡❣❛r❡♠♦s q✉❡✿

(26)

a2 _{| ·}x_·y_| = mmcg(a_·x, a_·y)_·a_·mdcg(x, y)

a_·mdcg(x, y)_·mmcg(x, y) = mmcg(a_·x, a_·y)_·mdcg(x, y)

a_·mmcg(x, y) = mmcg(a_·x, a_·y)

❈♦♠♦ ❛ ❣❡♥❡r❛❧✐③❛çã♦ t❛♥t♦ ❞♦ mmc q✉❛♥t♦ ❞♦ mdc ♥ã♦ ♣❡r❞❡ ❛ ❡ssê♥❝✐❛ ❞❡ t❛✐s

❝♦♥❝❡✐t♦s✱ ♣♦❞❡♠♦s ❥✉st✐✜❝❛r t❛❧ ❢❡✐t♦ ❡ ❛❣♦r❛ ❛♣❧✐❝á✲❧♦s ❛♦s ❝❛s♦s q✉❡ ♦✉tr♦r❛ ♥ã♦ ♣♦❞í❛♠♦s✳

❊①❡♠♣❧♦ ✶✳✶✳ ◗✉❛❧ ♦ mmcg ❡ ♦ mdcg ❡♥tr❡ ♦s ♥ú♠❡r♦s 2_·√3 ❡ 7_·√3❄

◆❡ss❡ ❝❛s♦✱ ♣♦❞❡♠♦s s✐♠♣❧✐✜❝❛r ♥♦ss♦s ❝á❧❝✉❧♦s ❛♣❡♥❛s ❝♦❧♦❝❛♥❞♦ ♦ ❢❛t♦r ❝♦♠✉♠ ❡♠ ❡✈✐❞ê♥❝✐❛ ❡ ❛ss✐♠ ❝❛❧❝✉❧❛r♠♦s ♦mmcg ❞❛ ♣❛rt❡ ♥ã♦ ❝♦♠✉♠✱ ❞❡ss❡ ♠♦❞♦ t❡r❡♠♦s

q✉❡✿

mmcg(2_·√3,7_·√3) = √3_·mmcg(2,7)

mmcg(2_·√3,7_·√3) = √3_·14

mmcg(2_·√3,7_·√3) = 14_·√3

P❛r❛ ♦mdcg ❜❛st❛ ♣❡❣❛r♠♦s ♦ ♠❛✐♦r ♥ú♠❡r♦ q✉❡ ❞✐✈✐❞❡ ❛♠❜♦s ♦s ♥ú♠❡r♦s✱ ❞❡ss❛

❢♦r♠❛ t❡r❡♠♦s q✉❡❀

mdcg(2_·√3,7_·√3) =√3

❊①❡♠♣❧♦ ✶✳✷✳ ◗✉❛❧ ♦ mmcg ❡ ♦ mdcg ❡♥tr❡ ♦s ♥ú♠❡r♦s 5

14 ❡ 9 10❄

P❛r❛ r❡s♦❧✈❡r t❛❧ ❡①❡♠♣❧♦ ❞❡✈❡✲s❡ ✉s❛r ❛ ♣r♦♣♦s✐çã♦ ✶✳✽✱ ❝♦♠♦ ❡ss❡s ♥ú♠❡r♦s ❥á ❡stã♦ ♥❛ ❢♦r♠❛ ❞❡ ❢r❛❝õ❡s ✐rr❡❞✉tí✈❡✐s✱ t❡r❡♠♦s q✉❡ a = 5✱b = 14✱c = 9 ❡ d = 10 ❡

(27)

❛ss✐♠✿

mmcg(x, y) = [a, c] (b, d)

mmcg 5 14, 9 10

= [5,9] (14,10)

mmcg 5 14, 9 10 = 45 2

P❛r❛ ♦ mdcg t❡r❡♠♦s q✉❡✿

mdcg(x, y) = (a, c) [b, d]

mdcg 5 14, 9 10

= (5,9) [14,10]

mdcg 5 14, 9 10 = 1 70

❊①❡♠♣❧♦ ✶✳✸✳ ◗✉❛❧ ♦ mmcg ❡ ♦ mdcg ❡♥tr❡ ♦s ♥ú♠❡r♦s 2_·√5 ❡ 7_·√3❄

◆❡ss❡ ❝❛s♦✱ ❝♦♠♦ ❡ss❡s ♥ú♠❡r♦s ♥ã♦ sã♦ ❝♦♠❡♥s✉rá✈❡✐s✱ ♥ã♦ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ♦

mmcg ❡ ♦ mdcg ❡♥tr❡ ❡❧❡s✳

✶✳✷ ❊♥❝♦♥tr♦ ❞❡ ❝♦r♣♦s ❡ ❛♣❧✐❝❛çõ❡s ❞♦

mmc

❡ ❞♦

mmcg

❊①✐st❡♠ ✐♥ú♠❡r❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❝♦♠ ♦ ✉s♦ ❞♦ ♠í♥✐♠♦ ♠ú❧✲ t✐♣❧♦ ❝♦♠✉♠✱ ❡ ❛❣♦r❛ ❛♣r❡s❡♥t❛r❡♠♦s✱ t❛♠❜é♠✱ ❛❧❣✉♠❛s ❛♣❧✐❝❛çõ❡s ♣❛r❛ ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❣❡♥❡r❛❧✐③❛❞♦ ♦♥❞❡ ❛ ❞❡✜♥✐çã♦ ✐♥✐❝✐❛❧ ♥ã♦ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞❛ ❞✐r❡t❛✲ ♠❡♥t❡✳

❊①❡♠♣❧♦ ✶✳✹✳ ❈♦♥s✐❞❡r❡ q✉❡ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ A ❝♦♠♣❛r❡ç❛ ♥♦ ❝♦❧é❣✐♦ ❞❡ ✸

❡♠ ✸ ❞✐❛s ❡♥q✉❛♥t♦ ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ B ❝♦♠♣❛r❡ç❛ ❞❡ ✺ ❡♠ ✺ ❞✐❛s✱ ✐♥❞❡♣❡♥✲

❞❡♥❞❡t❡♠❡♥t❡ ❞♦ ❞✐❛ ❞❛ s❡♠❛♥❛ q✉❡ ❝❛✐❛ s❡✉ ♣❧❛♥tã♦✳ ❙❡ ❛♠❜♦s ♣r♦❢❡ss♦r❡s ❡st✐✈❡r❛♠ ♣r❡s❡♥t❡s ♥♦ ❝♦❧é❣✐♦ ♥♦ ❞✐❛ ✸✵ ❞❡ ♥♦✈❡♠❜r♦✱ q✉❛❧ s❡rá ♦ ❞✐❛ ❞♦ ♠ês ❞❡ ❞❡③❡♠❜r♦✱ ❞♦ ♠❡s♠♦ ❛♥♦✱ q✉❡ t❡r❡♠♦s ♦ ♣ró①✐♠♦ ❡♥❝♦♥tr♦ ❡♥tr❡ ♦s ♣r♦❢❡ss♦r❡s❄

(28)

❈♦♠♦ ♦ ♣r♦❢❡ss♦r A ❝♦♠♣❛r❡❝❡ ♥♦ ❝♦❧é❣✐♦ ❞❡ ✸ ❡♠ três ❞✐❛s✱ ♣♦❞❡♠♦s ♣❡♥s❛r q✉❡

♥❡ss❡ ♣r✐♠❡✐r♦ ♠ês ❞❡ ♦❜s❡r✈❛çã♦ ❡❧❡ ✐rá ❛♦ ❝♦❧é❣✐♦ ♥♦s ❞✐❛s ♠ú❧t✐♣❧♦s ❞❡ ✸✳ ❆♥❛❧♦✲ ❣❛♠❡♥t❡✱ ♦ ♣r♦❢❡ss♦r B ❡st❛rá ♣r❡s❡♥t❡ ♥♦s ❞✐❛s ♠ú❧t✐♣❧♦s ❞❡ ✺✳ ❆ss✐♠ ♣❛r❛ q✉❡ ♦s

❞♦✐s ♣r♦❢❡ss♦r❡s ❡st❡❥❛♠ ♥♦ ❝♦❧é❣✐♦✱ ♦ ❞✐❛ ❞❡✈❡ s❡r ♠ú❧t✐♣❧♦ t❛♥t♦ ❞❡ ✸ ❝♦♠♦ ❞❡ ✺✳ P♦r ❡ss❛ r❛③ã♦✱ ♣❛r❛ r❡s♣♦♥❞❡r ❛ ❡ss❡ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛ ❜❛st❛ ♣❡❣❛r♠♦s ♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ❡♥tr❡ ♦s ♣❡rí♦❞♦s✱ q✉❡ é ♦ t❡♠♣♦ q✉❡ t❛❧ s✐t✉❛çã♦ ❣❛st❛ ♣❛r❛ ✈♦❧t❛r ❛ ♦❝♦rr❡r✳

▲♦❣♦ ♥♦ ❝❛s♦ ❞❡ss❡ ❡①❡♠♣❧♦ ❜❛st❛ ♣❡❣❛r♠♦s[3,5]s❡♥❞♦ ♦ ♣❡rí♦❞♦ q✉❡ ❡❧❡s s❡ ❡♥❝♦♥✲

tr❛rã♦✱ ❡♥tã♦✱ ❝♦♠♦[3,5] = 15✱ t❡♠♦s q✉❡ ❞❡ ✶✺ ❡♠ ✶✺ ❞✐❛s ❡❧❡s s❡ ❡♥❝♦♥tr❛♠✱ ♣♦rt❛♥t♦

♦ ♣r✐♠❡✐r♦ ❞✐❛ ❞♦ ♠ês ❞❡ ❞❡③❡♠❜r♦ ❞♦ r❡❢❡r✐❞♦ ❛♥♦ ❡♠ q✉❡ ❛♠❜♦s ♣r♦❢❡ss♦r❡s ❡st❛rã♦ ♣r❡s❡♥t❡s s❡rá ♦ ❞✐❛ ✶✺✳

❚❛❧ ❡①❡♠♣❧♦ é ✉♠ ♠♦❞❡❧♦ ❝❧áss✐❝♦ ❞❡ ✈ár✐♦s ♣r♦❜❧❡♠❛s✳ ❊♥tr❡t❛♥t♦ ❡①✐st❡ ❝❛s♦s ♦♥❞❡ t❛❧ r❡s♦❧✉çã♦ ✜❝❛ ✉♠ ♣♦✉❝♦ ❧✐♠✐t❛❞❛ ♣♦r ❝♦♥t❛ ❞❡ s✉❛ ❞❡✜♥✐çã♦✳ ❈♦♠♦ ❡①❡♠♣❧♦ t❡♠♦s ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ♣♦❞❡♠♦s t✐r❛r ❞❡ ✉♠❛ ❜r✐♥❝❛❞❡✐r❛ ❝♦♠ ❞♦✐s t✐♠❡s ❞❡ ❢✉t❡❜♦❧ ❜r❛s✐❧❡✐r♦✳

❊①❡♠♣❧♦ ✶✳✺✳ ❙❛❜❡♥❞♦ q✉❡ ♦ ❱❛s❝♦ ❞❛ ●❛♠❛ ❢♦✐ r❡❜❛✐①❛❞♦ ❞❡ ❞✐✈✐sã♦ ✸ ✈❡③❡s ♥♦s ú❧t✐♠♦s ✽ ❛♥♦s ❡ ♦ ❇♦t❛❢♦❣♦ ❢♦✐ r❡❜❛✐①❛❞♦ ✷ ✈❡③❡s ❛ ❝❛❞❛ ✶✷ ❛♥♦s✳ ❙❡ ❝♦♥s✐❞❡r❛r♠♦s ♦s ❡✈❡♥t♦s ❛❝✐♠❛ ❝♦♠♦ r❡❣r❛✱ ♦✉ s❡❥❛✱ ❝❛❞❛ t✐♠❡ s❡r r❡❜❛✐①❛❞♦ é ❛❧❣♦ ♣❡r✐ó❞✐❝♦✱ ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❤✐♣♦t❡t✐❝❛♠❡♥t❡ q✉❡ ♥♦ ❛♥♦ ❞❡ ✷✵✵✵ ❛♠❜♦s ❝❧✉❜❡s ❢♦r❛♠ r❡❜❛✐①❛❞♦s✳ ◗✉❛❧ s❡rá ♦ ❛♥♦ q✉❡ ❛♠❜♦s ♦s t✐♠❡s s❡rã♦ r❡❜❛✐①❛❞♦s ❥✉♥t♦s❄

◆❡ss❡ ❝❛s♦✱ t❡♠♦s q✉❡ ♦ ♣❡rí♦❞♦ ♣❛r❛ ♦ r❡❜❛✐①❛♠❡♥t♦ ❞♦ ❇♦t❛❢♦❣♦ é ✐❣✉❛❧ ❛ ✻ ❛♥♦s✱ ♣♦✐s ♦ ♠❡s♠♦ s♦❢r❡ ✷ ✈❡③❡s ♥♦s ú❧t✐♠♦s ✶✷ ❛♥♦s✳ ❊♥tr❡t❛♥t♦✱ ♦ ♣❡rí♦❞♦ ♣❛r❛ ♦ r❡❜❛✐①❛♠❡♥t♦ ❞♦ ❱❛s❝♦ é ❞❡ 8

3 ❛♥♦s ❡ ❝♦♠♦ ❡ss❡ ♥ú♠❡r♦ ♥ã♦ ♣❡rt❡♥❝❡ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s

♥ú♠❡r♦s ✐♥t❡✐r♦s✱ ♥ã♦ ♣♦❞❡r❡♠♦s ✉t✐❧✐③❛r ♦mmc✳ ❆ss✐♠✱ t❛❧ ❡①❡♠♣❧♦ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦

♣❡❧❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ mmc✳ ❈♦♠♦ ❛♠❜♦s ♣❡rí♦❞♦s sã♦ ♥ú♠❡r♦s r❛❝✐♦♥❛✐s✱ ♣♦❞❡♠♦s

✉t✐❧✐③❛r ❛ ❡q✉❛çã♦ ✭✶✳✽✮ ♣❛r❛ r❡s♦❧✈ê✲❧♦✳ ❈♦♠♦ ❛♠❜♦s ♣❡rí♦❞♦s ❥á ❡stã♦ ♥❛ ❢♦r♠❛ ❞❡ ❢r❛çõ❡s ✐rr❡❞✉tí✈❡✐s✱ ❜❛st❛ t♦♠❛r a = 6✱ b = 1✱ c = 8 ❡ d = 3 ❡ ❞❡ss❛ ❢♦r♠❛ t❡r❡♠♦s

q✉❡✿

(29)

x = mmcg

6,8

3

x = [6,8] (1,3)

x = 24

1

x = 24

P♦rt❛♥t♦✱ ❛♠❜♦s ♦s ❝❧✉❜❡s r❡❜❛✐①❛rã♦ ❡♠ ✷✵✷✹✱ ❝♦♥s✐❞❡r❛♥❞♦ q✉❡ ♦ ❢❡♥ô♠❡♥♦ ❞❡ r❡❜❛✐①❛♠❡♥t♦ s❡❥❛ ♣❡rí♦❞✐❝♦✳ ❆❧é♠ ❞✐ss♦ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛♠❜♦s ❝❧✉❜❡s r❡❜❛✐①❛rã♦ ❛ ❝❛❞❛ ✷✹ ❛♥♦s✳

❖✉tr❛ ❛♣❧✐❝❛çã♦ ❞♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ ✭mmc✮ é ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ❞♦✐s

❝♦r♣♦s ❡♠ ♠♦✈✐♠❡♥t♦ ❝✐r❝✉❧❛r ✉♥✐❢♦r♠❡ ✭M.C.U.✮ ✈♦❧t❡♠ ❛ s❡ ❡♥❝♦♥tr❛r❡♠ ♥♦ ♠❡s♠♦

♣♦♥t♦✳ ❙✉♣♦♥❤❛ ❞♦✐s ❝♦r♣♦s✱ ♦ ❝♦r♣♦A q✉❡ ❣✐r❛ ♥♦ s❡♥t✐❞♦ ❤♦rár✐♦ ❡ ❣❛st❛ ✉♠ t❡♠♣♦ TA ♣❛r❛ ❞❛r ✉♠❛ ✈♦❧t❛ ❡♠ t♦r♥♦ ❞♦ ♣❡r❝✉rs♦ r❡❣r❡ss❛♥❞♦ à ♦r✐❣❡♠ O✳ ❊♥q✉❛♥t♦ ✉♠

❝♦r♣♦ B✱ q✉❡ ❣✐r❛ ♥♦ s❡♥t✐❞♦ ❛♥t✐✲❤♦rár✐♦✱ ❣❛st❛ ✉♠ t❡♠♣♦ TB ♣❛r❛ ❝♦♠♣❧❡t❛r ✉♠❛

✈♦❧t❛✳

❋✐❣✉r❛ ✶✳✷✿ ❈♦r♣♦s ❡♠ ▼✳❈✳❯✳

❊ss❡s ❝♦r♣♦s s❡ ❡♥❝♦♥tr❛rã♦ ❛❧❣✉♠❛s ✈❡③❡s ❡♠ ♣♦♥t♦s ❞✐❢❡r❡♥t❡s ❞❡ss❡ ♣❡r❝✉rs♦✱ ♠❛s s♦♠❡♥t❡ s❡ ❡♥❝♦♥tr❛rã♦ ♥♦ ♠❡s♠♦ ♣♦♥t♦ ❡♠ ✉♠ t❡♠♣♦ ✐❣✉❛❧ ❛♦ mmc ❡♥tr❡ TA ❡ TB✳

P♦✐s ❡♠ t❛❧ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦✱ ♦ ❝♦r♣♦ A t❡rá ❞❛❞♦ ✉♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✈♦❧t❛s NA

❡q✉❛♥t♦ q✉❡ ♦ ❝♦r♣♦ B t❡rá ❞❛❞♦ NB ✈♦❧t❛s✱ ✈♦❧t❛♥❞♦ ♦s ❞♦✐s ❛ s❡ ❡♥❝♦♥tr❛r❡♠ ♥♦

♣♦♥t♦O✳

P♦r ❞❡✜♥✐çã♦✱ ♣♦❞❡rí❛♠♦s r❡s♦❧✈❡r ❡ss❡ ♣r♦❜❧❡♠❛ s♦♠❡♥t❡ s❡ ♦s ❝♦r♣♦s ♣♦ss✉íss❡♠ ♣❡rí♦❞♦s ✐♥t❡✐r♦s ❞❡ t❡♠♣♦s✳ ❊♥tr❡t❛♥t♦✱ ❝♦♠ ❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❞♦ mmc✱

(30)

♣♦❞❡♠♦s ❛❣♦r❛ r❡s♦❧✈❡r t❛❧ ♣r♦❜❧❡♠❛ ♣❛r❛ q✉❛❧q✉❡r ♣❛r ❞❡ r❡❛✐s q✉❡ s❡❥❛♠ ❝♦♠❡♥s✉✲ rá✈❡✐s✳ ❉❡ ❢♦r♠❛ q✉❡ ♦ t❡♠♣♦ q✉❡ ♦s ❝♦r♣♦s ❣❛st❛rã♦ ♣❛r❛ s❡ ❡♥❝♦♥tr❛r❡♠ ♥❛ ♠❡s♠❛ ♣♦s✐çã♦ s❡rá ♦mmcg ❡♥tr❡ ♦s t❡♠♣♦s ❣❛st♦s ♣❛r❛ q✉❡ ❝❛❞❛ ✉♠ ❞❡❧❡s ❝♦♠♣❧❡t❡♠ ✉♠❛

✈♦❧t❛✳

✶✳✸ Pê♥❞✉❧♦ ❙✐♠♣❧❡s ❡ ♦ ▼✳❍✳❙✳

◆❛ ♥❛t✉r❡③❛✱ ❡①✐st❡♠ ✐♥ú♠❡r♦s ♠♦✈✐♠❡♥t♦s ♦s❝✐❧❛tór✐♦s ❡ ♣❡r✐ó❞✐❝♦s✳ ❈♦♠♦ ❡①❡♠♣❧♦ ❞❡ss❡s ♠♦✈✐♠❡♥t♦s t❡♠♦s✿ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✉♠❛ ♠♦❧❛✱ ❞❛ ❝♦r❞❛ ❞❡ ✉♠ ✈✐♦❧ã♦✱ ❞❡ ✉♠❛ ❝r✐❛♥ç❛ s❡♥t❛❞❛ ❡♠ ✉♠ ❜❛❧❛♥ç♦✱ ❞❡ ✉♠❛ ❡s❝♦✈❛ ❞❡ ❞❡♥t❡s ❡❧étr✐❝❛ ❡t❝✳ ❆❧é♠ ❞❡ss❡s ♠♦✈✐♠❡♥t♦s✱ ❛ ❛✉❞✐ç❛♦✱ ✈✐sã♦ ❡ ❢❛❧❛ sã♦ r❡s✉❧t❛♥t❡s ❞❡ ❢❡♥ô♠❡♥♦s ♦s❝✐❧❛tór✐♦s✳❬✸❪

▼♦✈✐♠❡♥t♦ ♦s❝✐❧❛tór✐♦ é ❛q✉❡❧❡ ❡♠ q✉❡ ♦ ❝♦r♣♦ ❜❛❧❛♥ç❛ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ♣♦♥t♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ ❆t✐♥❣✐♥❞♦ ❛ss✐♠ ♣♦♥t♦s ❞❡ ❛♠♣❧✐t✉❞❡s ♠á①✐♠❛s ♣❛r❛ ♦ s❡♥t✐❞♦ ❝♦♥s✐❞❡r❛❞♦ ♣♦s✐t✐✈♦ ❡ ♣❛r❛ ♦ ♥❡❣❛t✐✈♦✳ ❙❡ t❛❧ ♠♦✈✐♠❡♥t♦ ❣❛st❛r ♦ ♠❡s♠♦ t❡♠♣♦ ❡♠ t♦❞❛s ✐❞❛s ❡ ✈✐♥❞❛s✱ s❡rá t❛♠❜é♠ ✉♠ ♠♦✈✐♠❡♥t♦ ♣❡rí♦❞✐❝♦✳ ❉❡♥tr❡ t❛✐s ♠♦✈✐♠❡♥t♦s✱ ♦ ♠❛✐s s✐♠♣❧❡s é ♦ ▼♦✈✐♠❡♥t♦ ❍❛r♠ô♥✐❝♦ ❙✐♠♣❧❡s ✭M.H.S.✮✳

❇❛s✐❝❛♠❡♥t❡✱ ✉♠ ❝♦r♣♦ ❡st❛rá ❡♠ M.H.S. s❡ ♦ ♠❡s♠♦ ❡st✐✈❡r ❡♠ ✉♠ ♠♦✈✐♠❡♥t♦

♦s❝✐❧❛tór✐♦ ❡ ♣❡r✐ó❞✐❝♦✳ ❉♦✐s ❢❛t♦r❡s sã♦ ✐♥❞✐s♣❡♥sá✈❡✐s ♣❛r❛ t❛❧ ♦s❝✐❧❛çã♦✿ ❛ ♣r❡s❡♥ç❛ ❞❡ ✉♠❛ ❢♦rç❛ r❡st❛✉r❛❞♦r❛ ❡ ❛ ✐♥ér❝✐❛ ❞♦ ❝♦r♣♦✳ ❆ ❢♦rç❛ r❡st❛✉r❛❞♦r❛ ❛❣❡ ♥♦ ✐♥t✉✐t♦ ❞❡ ❢❛③❡r ♦ ❝♦r♣♦ ✈♦❧t❛r à ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✳ P♦r ❡ss❛ r❛③ã♦ r❡♣r❡s❡♥t❛r❡♠♦s t❛❧ ❢♦rç❛ ❝♦♠♦✿

~

F =₋K_·~x ✭✶✳✶✷✮

❊♠ q✉❡ _F~ _{r❡♣r❡s❡♥t❛ ♦ ✈❡t♦r ❢♦rç❛ r❡st❛✉r❛❞♦✉r❛ ❞♦ ♠♦✈✐♠❡♥t♦✱} _~x _{r❡♣r❡s❡♥t❛ ♦}

✈❡t♦r ♣♦s✐çã♦ ❡♠ r❡❧❛çã♦ à ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✱K é ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦rçã♦ ❡♥tr❡ ❛

❢♦rç❛ ❡ ❛ ❡❧♦♥❣❛çã♦✱ ♦✉ ♣♦s✐çã♦✱ ❡ ♦ s✐♥❛❧ ♥❡❣❛t✐✈♦ r❡♣r❡s❡♥t❛ q✉❡ t❛❧ ❢♦rç❛ s❡♠♣r❡ t❡rá ♦ s❡♥t✐❞♦ ♦♣♦st♦ ❛♦ ❞❛ ❞❡❢♦r♠❛çã♦ s♦❢r✐❞❛ ♣❡❧♦ ❝♦r♣♦✱ ♣♦r ✐ss♦ ❞♦ ♥♦♠❡ r❡st❛✉r❛❞♦r❛✳ ❏á ❛ ✐♥ér❝✐❛ é ❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♦s ❝♦r♣♦s ♣♦ss✉❡♠ ❞❡ ❝♦♥t✐♥✉❛r s❡✉ ❡st❛❞♦✱ ♦✉ ❡♠ r❡♣♦✉s♦ ♦✉ ❡♠ ♠♦✈✐♠❡♥t♦ r❡t✐❧í♥❡♦ ✉♥✐❢♦r♠❡✳ P♦r ❡ss❛ r❛③ã♦✱ ♦s ❝♦r♣♦s ♥ã♦ ♣❛r❛♠ ♥♦ ✐♥st❛♥t❡ ❡ ♥❛ ♣♦s✐çã♦ ♦♥❞❡ ❛ ❢♦rç❛ é ♥✉❧❛✳ ◆❡ss❡ tr❛❜❛❧❤♦✱ tr❛t❛r❡♠♦s ❞❡ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦M.H.S. ♦ Pê♥❞✉❧♦ ❙✐♠♣❧❡s✳ ◗✉❛♥❞♦ ✉♠ ❝♦r♣♦ ❞❡ ♠❛ss❛ m✱ ♣r❡s♦ ❛ ✉♠

✜♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ L ♦s❝✐❧❛ ❡♠ t♦r♥♦ ❞❡ ✉♠❛ ♣♦s✐çã♦ O✱ ❞❡s❝r❡✈❡♥❞♦ ♦ ❛r❝♦ \BOC

t❡r❡♠♦s ✉♠ Pê♥❞✉❧♦ ❙✐♠♣❧❡s✳

◆❡ss❡ ❝❛s♦✱ ❛ ❢♦rç❛ r❡st❛✉r❛❞♦r❛ ❞♦ M.H.S. s❡rá ❛ ❝♦♠♣♦♥❡♥t❡ ❞❛ ❢♦rç❛ P❡s♦ ♥❛

(31)

❞✐r❡çã♦ t❛♥❣❡♥❝✐❛❧✱ _P~_x_{✳ ❚❛❧ ❝♦♠♣♦♥❡♥t❡✱ ❞❡♣❡♥❞❡ ❞❛ ♣♦s✐çã♦ ❞♦ ❝♦r♣♦✱ ♦✉ s❡❥❛✱ ♥ã♦ é}

❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞♦ ♠♦✈✐♠❡♥t♦ ❡ ♦ P❡s♦ é ❛ ❢♦rç❛ ❣r❛✈✐t❛❝✐♦♥❛❧ q✉❡ ❛ t❡rr❛ ❡①❡r❝❡ ❡♠ ✉♠ ❝♦r♣♦ ♣ró①✐♠♦ à ❡❧❛✳ ❚❛❧ ❢♦rç❛ ♣♦❞❡ s❡r ❝❛❧❝✉❧❛❞❛ ❝♦♠♦ ♦ ♣r♦❞✉t♦ ❡♥tr❡ ❛ ♠❛ss❛ ❞♦ ❝♦r♣♦ ❡ ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ♦✉ ❝❛♠♣♦ ❣r❛✈✐t❛❝✐♦♥❛❧✳

❋✐❣✉r❛ ✶✳✸✿ Pê♥❞✉❧♦ ❙✐♠♣❧❡s

❉❛ ❋✐❣✉r❛ ✶✳✸ t❡♠♦s q✉❡ ♦ ♠ó❞✉❧♦ ❞❡ _P~_x _{s❡rá ❞❛❞♦ ♣♦r✿}

Px =P · senβ ✭✶✳✶✸✮

❙❡ ❝♦♥s✐❞❡r❛r♠♦s β ✉♠ â♥❣✉❧♦ ♣❡q✉❡♥♦✱ ♠❡♥♦r ❞♦ q✉❡ 10◦

✱ ♣♦❞❡♠♦s ✉t✐❧✐③❛r ❛ ❛♣r♦✲ ①✐♠❛çã♦ ♣❛r❛ t❛✐s â♥❣✉❧♦s❬✷❪✱ ♦♥❞❡✿

senβ _≈ tgβ _≈β ✭✶✳✶✹✮

♣❛r❛ β ❡♠ r❛❞✐❛♥♦s✳ P♦✐s t❡♠♦s ❞♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧ q✉❡✿

lim

x→0

senx

x = 1 ✭✶✳✶✺✮

❊♥tã♦ ♣❛r❛ â♥❣✉❧♦s ❛té 10◦ t❡r❡♠♦s q✉❡

β = 0,1745rad ❡ sen10◦

= 0,1736 ❡ ❞❡ss❛

(32)

❢♦r♠❛✱ ♣❛r❛ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ♣❛r❛ três ❝❛s❛s ❞❡❝✐♠❛✐s t❡♠♦s q✉❡ senβ _≈β✳ P♦r ❡ss❛

r❛③ã♦✱ t❡♠♦s q✉❡ ♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞♦s Pê♥❞✉❧♦s s❡rá ♣ró①✐♠♦ ❛♦ ❡s♣❡r❛❞♦ t❡♦r✐❝❛✲ ♠❡♥t❡ s❡ ♦ â♥❣✉❧♦ ❞❡ ❛❜❛♥❞♦♥♦ ❢♦r ♠❡♥♦r ♦✉ ✐❣✉❛❧ à10◦✳

❉❡ss❡ ♠♦❞♦✱ ❝♦♠♦ β= x

L t❡♠♦s q✉❡✿

Px=P · senβ =P ·

x

L =

P

L ·x ✭✶✳✶✻✮

❆ss✐♠ ❛ ♣r✐♠❡✐r❛ ❝❛r❛❝t❡ríst✐❝❛ ♣❛r❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦ Pê♥❞✉❧♦ s❡r ✉♠M.H.S é r❡✲

♣r❡s❡♥t❛❞♦ ♣❡❧❛ ❊q✉❛çã♦ ✭✶✳✶✷✮✱ ♣♦✐s ❛ ❢♦rç❛ r❡st❛✉r❛❞♦r❛ é ♣r♦♣♦r❝✐♦♥❛❧ à ❞❡❢♦r♠❛çã♦ ❡ ❛✐♥❞❛✱ ❝♦♠♦ r❡♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛✱ ♣♦ss✉✐ s❡♥t✐❞♦ ♦♣♦st♦ ❛ ♠❡s♠❛✳ ❉❛ ❙❡❣✉♥❞❛ ❧❡✐ ❞❡ ◆❡✇t♦♥✱ ♦✉ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❛ ❉✐♥â♠✐❝❛✱ t❡♠♦s q✉❡✿

~

FR=m·~a ✭✶✳✶✼✮

❖♥❞❡ _{F r}~ _{é ❛ ❢♦rç❛ r❡s✉❧t❛♥t❡✱ ♦✉ s❡❥❛✱ ❛ s♦♠❛ ✈❡t♦r✐❛❧ ❞❡ t♦❞❛s ❛s ❢♦rç❛s q✉❡ ❛❣❡♠}

♥♦ ❝♦r♣♦✱~a é ♦ ✈❡t♦r ❛❝❡❧❡r❛çã♦ ❞♦ ❝♦r♣♦ ❞❛❞♦ ♣❡❧❛ s❡❣✉♥❞❛ ❞❡r✐✈❛❞❛ ❞❛ ♣♦s✐çã♦ ❡♠

r❡❧❛çã♦ ❛♦ t❡♠♣♦ ❡ m é ❛ ♠❛ss❛ ♦✉ ❛ ♠❡❞✐❞❛ ❞❛ ✐♥ér❝✐❛ ❞♦ ❝♦r♣♦✳ ❉❛ ❞❡✜♥✐çã♦ ❞❡

❛❝❡❧❡r❛çã♦ t❡♠♦s q✉❡✿

~a= ∂

2_~x

∂t2 ✭✶✳✶✽✮

❯t✐❧✐③❛♥❞♦ t❛✐s ❡q✉❛çõ❡s t❡♠♦s q✉❡✿

~

FR=m·~a = m·

∂2_~x

∂t2

~

FR=P~x = −K·~x

−K_·~x = ₋P

L ·~x

❉❡ss❡ ♠♦❞♦✱ ♣❛r❛ ♠❡❧❤♦r ❡✈✐❞❡♥❝✐❛r♠♦s ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦r❝✐♦♥❛❧✐❞❛❞❡ ❡♥tr❡ ❛ ❢♦rç❛ ❡ ❛ ❡❧♦♥❣❛çã♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r q✉❡✿

− P_L ·~x =m_·∂

2_~x

∂t2

′

✭✶✳✶✾✮

(33)

❊♠ q✉❡ ❛ ❝♦♥st❛♥t❡ ❞❡ ♣r♦♣♦rçã♦ ❡♥tr❡_F~ _❡_~x _é_K ₌ P

L✳ ❆ s♦❧✉çã♦ ♣❛r❛ ❛ ❡q✉❛çã♦

❞✐❢❡r❡♥❝✐❛❧ ♦r❞✐♥ár✐❛ ❛❝✐♠❛ é ❞❛ ❢♦r♠❛✿

x=A_·cos(ω_·t+φ0) ✭✶✳✷✵✮

❊♠ q✉❡ A r❡♣r❡s❡♥t❛ ❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♠♦✈✐♠❡♥t♦✱ ❞✐stâ♥❝✐❛ ❡♠ q✉❡ ♦ ❝♦r♣♦ ♦s❝✐❧❛

❡♠ r❡❧❛çã♦ ❛ ♣♦s✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ O❀ φ0 r❡♣r❡s❡♥t❛ ❛ ❢❛s❡ ✐♥❝✐❛❧✱ q✉❡ ❡stá ❧✐❣❛❞❛ à

♣♦s✐çã♦ ♦♥❞❡ ♦ ❝♦r♣♦ ❝♦♠❡ç❛ ❛ ♦s❝✐❧❛r ❡ω r❡♣r❡s❡♥t❛ ❛ ♣✉❧s❛çã♦✳ ❖♥❞❡ ❡ss❛ ♣✉❧s❛çã♦

é t❛❧ q✉❡✿

−P_L ·~x = m_·∂

2_~x

∂t2

−P

L ·(A·cos(ω·t+φ0)) = m· ∂2_x

∂t2(A·cos(ω·t+φ0))

❈♦♠♦ P =m_·g✱ ♦♥❞❡g é ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❧♦❝❛❧✱ t❡♠♦s✿

−m_L·g ·(A_·cos(ω_·t+φ0)) = m·

∂2_x

∂t2(A·cos(ω·t+φ0))

−m_L·g ·(A_·cos(ω_·t+φ0)) = m·(−ω2 ·A·cos(ω·t+φ0))

❉✐✈✐❞✐♥❞♦ ❛♠❜♦s ♦s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ ♣♦r✿ m_·(₋A_·cos(ω_·t+φ0)) t❡r❡♠♦s q✉❡✿

g

L =ω

2

P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡ ❛♥❣✉❧❛r✱ q✉❡ é ❛ t❛①❛ ❞❡ ✈❛r✐❛çã♦ t❡♠♣♦r❛❧ ❞❛ ♣♦s✐✲ çã♦ ❛♥✉❧❛r✱ ❝❤❡❣❛♠♦s q✉❡ ω = 2·π

T ♦♥❞❡ T é ♦ ♣❡rí♦❞♦ ❞♦ ♣ê♥❞✉❧♦✱ ♦✉ s❡❥❛✱ t❡♠♣♦

♥❡❝❡ssár✐♦ ♣❛r❛ q✉❡ ♦ ❝♦r♣♦ ❝♦♠♣❧❡t❡ ✉♠❛ ♦s❝✐❧❛çã♦ ❝♦♠♣❧❡t❛✳

g

L =ω

2 ₌

2_·π T

2

P♦rt❛♥t♦✱ ❝♦♥❝❧✉í♠♦s q✉❡ ♦ ♣❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ s✐♠♣❧❡s é ❞❛❞♦ ♣❡❧❛ ❡q✉❛çã♦ ❡ ❞❡✜♥✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✳

❉❡✜♥✐çã♦ ✶✳✼✳ P❡rí♦❞♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ é ♦ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ❣❛st♦ ♣❛r❛ q✉❡ ♦ ♠❡s♠♦ ♣♦ss❛ ❝♦♠♣❧❡t❛r ✉♠❛ ♦s❝✐❧❛çã♦✱ ♦✉ s❡❥❛✱ s❛✐r ❞❡ ✉♠ ♣♦♥t♦ ♣❛ss❛r ♣♦r t♦❞❛s ❛s ♣♦s✐çõ❡s

(34)

♣♦ssí✈❡✐s ❡ r❡t♦r♥❛r ❛♦ ♠❡s♠♦ ♣♦♥t♦✳ ❚❛❧ ♣❡rí♦❞♦ ❞❡♣❡♥❞❡rá✱ ❡♠ ✉♠ ❝❛s♦ ♦♥❞❡ ❞❡s♣r❡✲ ③❛r♠♦s ❛s ❢♦rç❛s ❞✐ss✐♣❛t✐✈❛s✱ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ♣ê♥❞✉❧♦ ❡ ❞❛ ❛❝❡❧❡r❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ q✉❡ ❢♦r s✉❜♠❡t✐❞♦✳

T = 2_·π_·

s

L

g ✭✶✳✷✶✮

❬✾❪