The pigeonhole principle - repunits

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MetadadosDescriçãoIdioma
Autor(es): dc.contributorUniversidade Estadual Paulista (UNESP)-
Autor(es): dc.creatorPegg Jr, Ed-
Data de aceite: dc.date.accessioned2019-08-21T17:25:47Z-
Data de disponibilização: dc.date.available2019-08-21T17:25:47Z-
Data de envio: dc.date.issued2011-05-26-
Data de envio: dc.date.issued2011-05-26-
Data de envio: dc.date.issued2011-05-26-
Fonte completa do material: dc.identifierhttp://acervodigital.unesp.br/handle/123456789/4270-
Fonte completa do material: dc.identifierhttp://objetoseducacionais2.mec.gov.br/handle/mec/9139-
Fonte: dc.identifier.urihttp://educapes.capes.gov.br/handle/capes/446644-
Descrição: dc.descriptionKnowledge about algorithms, historical mathematics, integers, number theory, rational numbers, representations of numbers and theorem proving-
Descrição: dc.descriptionIn 1834, Johann Dirichlet noted that if there are five objects in four drawers then there is a drawer with two or more objects. The Schubfachprinzip, or drawer principle, got renamed as the pigeonhole principle, and became a powerful tool in mathematical proofs. Pick a number that ends with 1, 3, 7, or 9. Will it evenly divide a number consisting entirely of ones (a repunit)? Answer: yes. Proof: Suppose 239 was chosen. Take the remainder of 239 dividing 10, 100, 1000, ..., 10^239. The chosen number will not divide evenly into any of those 239 powers of 10, so there are 238 possible remainders, 1 to 238. By the pigeonhole principle, two remainders must be the same, for some 10^a and 10^b. As it turns out, 10^3 and 10^10 both give remainder 44. Subtracting, 9,999,999,000 is the result, which yields 1,111,111 when divided by 9000. When b>a, 10^b-10^a always returns an all-1 number multiplied by 9 and some power of 10, finishing the proof. The reciprocal of the chosen number has a repeating decimal of similar length. Consider: 1,111,111/239 = 4649. 4649 x 9 = 41 841. 1/239 = .00418410041841(...)-
Descrição: dc.descriptionComponente Curricular::Ensino Fundamental::Séries Finais::Matemática-
Publicador: dc.publisherWolfram Demonstrations Project-
Relação: dc.relation238ThePigeonholePrincipleRepunits.nbp-
Direitos: dc.rightsDemonstration freeware using Mathematica Player-
Palavras-chave: dc.subjectEducação Básica::Ensino Fundamental Final::Matemática::Números e operações-
Palavras-chave: dc.subjectAlgorithm-
Palavras-chave: dc.subjectHistorical mathematics-
Palavras-chave: dc.subjectNumber theory-
Palavras-chave: dc.subjectRational number-
Palavras-chave: dc.subjectRepresentation of number-
Palavras-chave: dc.subjectTheorem proving-
Palavras-chave: dc.subjectPrincípio matemático-
Palavras-chave: dc.subjectProva matemática-
Palavras-chave: dc.subjectTeoria da prova-
Título: dc.titleThe pigeonhole principle - repunits-
Aparece nas coleções:Repositório Institucional - Acervo Digital Unesp

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