Does $p^n$ divide $\binom{p^{n+m-1}}{m}$? 
Let $n, m \in \mathbf N$ and $p$ an odd prime number. Then does  $p^n$ divide $\binom{p^{n+m-1}}{m}$ ?

It seems true, but I can not find a clue. Can I have any hint? 

Hint
$$\binom{p^{n+m-1}}{m}=\dfrac{p^{n+m-1}}{m}\binom{p^{n+m-1}-1}{m-1}$$
