Binomial Coefficient involving Power of Prime/Proof 2

Lemma
Let $p$ be a prime number, and let $k \in \Z$.

Then:
 * $\dbinom {p^n k} {p^n} \equiv k \pmod p$

where $\dbinom {p^n k} {p^n}$ is a binomial coefficient.

Proof
By invoking Lucas' Theorem repeatedly, we have