Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.
Source: 1993 Bulgaria NMO, Round 4, p5
Tags: number theory, periodic, remainder
Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.