A note on Pascal's triangle and division by eleven

Authors

DOI:

https://doi.org/10.22481/intermaths.v3i2.11094

Keywords:

Pascal's triangle, Divisibility, Palindromes, Stifel's relationship, Binomial theorem

Abstract

Divisibility is an old topic that to this day intrigues and fascinates researchers and scholars. Several rules are well-known in particular the divisibility by eleven, since, for example, a palindrome with an even number of digits is divisible by eleven. In current times, divisibility has its applications, for example, in cryptography. Here, in this paper, we will show that applying two somewhat intuitive procedures to the lines of Pascal's triangle shall always yield numbers divisible by eleven. Illustrative examples are presented.

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Author Biographies

Leonardo Ribeiro de Castro Carvalho, Geekie S.A., Brasil

Bacharel em Engenharia da Compução, graduado com distinção no Instituto Tecnológico da Aeronáutica, ITA. Premiado em diversas competições internacionais de matemática, dentre elas, IMO, IMC
e Olimpíada IberoAmericana de Matemática. Atualmente é Diretor de Engenharia da empresa Nilo Saúde. E-mail: lrccarvalho@gmail.com

Edmundo Capelas de Oliveira, Departamento de Matemática Aplicada, Imecc - Unicamp, Brasil

Doutor em Física pela Universidade Estadual de Campinas (Unicamp). Fez pós-doutorado junto à Università di Perugia, Itália. Atualmente é Professor Titular (aposentado) junto ao Departamento de Matemática Aplicada do Instituto de Matemática e Estatística e Computação Científica da Unicamp. Tem experiência área de Física, com ênfase em Métodos Matemáticos da Física, atuando principalmente nos temas: cálculo integrodiferencial fracionário, funções especiais, funções analíticas e equações diferenciais. É autor de vários livros-textos. E-mail: capelas@unicamp.br

References

M. Gardner, “Mathematical Games: Tests that show whether a large number can be divided by a number from 2 to 12”, Scientific American, vol. 207, pp. 232 - 246, 1962.

B. Richmond and T. Richmond, A discret transition to advanced mathematics, Pure and Applied Undergraduate Text, American Mathematical Society, vol. 3, 2009.

F. J. Muller, “More on Pascal’s triangle and power of 11”, The Mathematics Teacher, vol. 58, 425 - 428, 1965.

L. Low, “Even more of Pascal’s triangle and power of 11”, The Mathematics Teacher, vol. 59, 461 - 463, 1966.

R. Sivaraman, “Pascal triangle and Pythagorean triples”, International Journal of Engineering Technologies and Management Research, vol. 8, 75 - 80, 2021. https://doi.org/10.29121/ijetmr.v8.i8.2021.1020

R. Merris, Combinatorics, John Wiley & Sons (1996). The pdf of Chapter 1 is made avaliable on the Internet (Accessed on May 24, 2022).

W. D. Wallis and J. C. George, Introduction to Combinatorics, (Discrete Mathematics and its Applications), 2ª ed. Boca Raton: CCR Press – Taylor & Francisc Groups, 2017.

A. W. F. Edwards, The arithmetical triangle, in R. Wilson, and J. J. Watkins (eds), Combinatorics: Ancient and Modern, Oxford University Press, Oxford (2013).

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Published

2022-12-31

How to Cite

Carvalho, L. R. de C., & Capelas de Oliveira, E. (2022). A note on Pascal’s triangle and division by eleven. Intermaths, 3(2), 29-37. https://doi.org/10.22481/intermaths.v3i2.11094

Issue

Vol. 3 No. 2 (2022)

Section

Artigos

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