In trying to determine the spectrum of a well-known ergodic transformation, I came up with the following useful (for me) result.

Let $p$ be a prime and $a$ a positive integer. Then for $M$ a positive integer, we have $$ {{M} \choose {p^a}} \equiv {\rm floor}\left(\frac M{p^a} \right) \mod p. $$ [The thing on the left is supposed to represent $M$ choose $p^a$, and the right side is supposed to be the floor of $M/p^a$, but on my screen, it looks confusing.]

I have a tedious argument for this (which is probably correct), but why re-invent the binomial wheel? This identity must be known. Can someone provide a reference?