Pigeonholes and Repunits

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...

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Bibliographic Details
Published inThe American mathematical monthly Vol. 121; no. 6; pp. 529 - 533
Main Author Wu, Chai Wah
Format Journal Article
LanguageEnglish
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
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Summary: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 k − 1 digits.
ISSN:0002-9890
1930-0972
DOI:10.4169/amer.math.monthly.121.06.529