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
Lucas' Theorem states that for $n,\,k,\,p\in\Z$ and $p$ be a prime number, with
 * $n=a_rp^r+\cdots+a_1p+a_0$
 * $k=b_rp^r+\cdots+b_1p+b_0$

then:
 * $\displaystyle \dbinom n k \equiv \prod_{j \mathop = 0}^r \binom {a_j}{b_j} \pmod p$

Therefore: