Pigeonholes and Repunits

• Published in: The American mathematical monthly Vol. 121; no. 6; pp. 529 - 533
• Main Author: Wu, Chai Wah
• Format: Journal Article
• Language: English
• Published: Washington Taylor & Francis 01.06.2014; Mathematical Association of America; Taylor & Francis Ltd
• Subjects: Decimals; Integers; Mathematical analysis; Mathematical problems; Mathematical theorems; Numbers; Zero

It is well known that any integer k has a multiple consisting of only the digits 1 and 0. As an extension of this result, we study integers of the form 111 ⋯ 000 or 111 ⋯ 111 that are a multiple of k. We show that if k > 2 and k is not a power of 3, then the multiple can be chosen to have at most...