An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}\_{>0}$, then

$$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$

This assertion was made in a way (i.e. without proof or citation) that suggests the author considers it pretty obvious, but it isn't so to me. If there's a simple proof of it, I'd like to see it.

(Note: $q$ need not be coprime to $p$.)

(Searching online I found a result called [Lucas' Theorem](http://en.wikipedia.org/wiki/Lucas%27_theorem), which looks like it may include the equivalence above as a special case, but I'm not sure yet. In any case, I did not find the proof of Lucas' theorem particularly easy, but, if the equivalence above is indeed a special case of Lucas' theorem, I hope there may be an easier proof for it.)