# Math Help - Binomial coefficient

1. ## Binomial coefficient

Let $p\geq3$ be a prime number, and let m and $k be positive integers. Show that $\dbinom{p^m}{k}$ is divisible by p.

2. ## Re: Binomial coefficient

Originally Posted by alexmahone
Let $p\geq3$ be a prime number, and let m and $k be positive integers. Show that $\dbinom{p^m}{k}$ is divisible by p.
We have the simple identity $\binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot\frac{p^m}{k}$ for $0. Hence, (a) $k\cdot\binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot p^m$.

Let $p^j$ be the largest power of $p$ that divides $k$. So $\frac{k}{p^j}\cdot\binom{p^m}{k}=\binom{p^m-1}{k-1}\cdot p^{m-j}$, by (a).

From $p^j\leq k it follows that $m-j>0$ and so $p$ divides $\frac{k}{p^j}\cdot\binom{p^m}{k}$, where $p$ does not divide $\frac{k}{p^j}$. Euclid's Lemma implies then that $p$ must divide $\binom{p^m}{k}$.

By the way, the argument works for $p=2$ also.