number theory - Why is $nchoose k$ periodic modulo $p$ with period $p^e$?

Given some integer $k$, define the sequence $a_n={n\choose k}$. Claim: $a_n$ is periodic modulo a prime $p$ with the period being the least power $p^e$ of $p$ such that $k

In other words, $a_{n+p^e}\equiv a_{n} (\text{mod } p)$. But the period $p^e$ is smaller than I'd have expected (it is obvious that a period satisfying $k! < p^e$ would work). So how can I prove that it works?