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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?

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    At first glance, it seems thinking of the numbers in base $p$ will help. [Kummer's Theorem](http://planetmath.org/encyclopedia/KummersTheorem.html) will probably help as well.2011-11-15

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