Question 84
Let $n$ be a positive integer, coprime to $2$ and $5$. Show that there is a positive integer $k \geq n$ with the property: for any base $10$ positive integer $\overline{a_1a_2...a_k}$, divisible by $k$, with all digits different from $0$, the numbers $\overline{a_2...a_k a_1}$, $\overline{a_3...a_k a_1 a_2}$, ..., $\overline{a_k a_1...a_{k-1}}$ are also divisible by $k$.