Divisibility of a Repunit

There is a repunit divisible by 2009.

Repunit is an integer writen exclusively with 1's (1, 11, 111, etc.) Among the first 2010 repunits, two have the same remainder when divided by 2009. Their difference which is divisible by 2009 consists of a series of 1's followed by a series of 0's, meaning that the number is the product of a repunit and a power of 10. Since a power of 10 has no common factors with 2009, the repunit must be divisible by 2009.

Reference

  1. I. F. Sharygin, Mathematical Mosaic, Mir, 2002, problem 65.1 (in Russian)

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