## Repunit Divisibility

#### Problem

A number consisting entirely of ones is called a repunit. We shall define $R(k)$ to be a repunit of length $k$; for example, $R(6) = 111111$.

Given that $n$ is a positive integer and $GCD(n, 10) = 1$, prove that there always exists a value, $k$, for which $R(k)$ is divisible by $n$.

Problem ID: 293 (03 Oct 2006) Difficulty: 4 Star