$\frac{(mn)!}{m!(n!)^m}\in\mathbb N^*$? A diligent student gives me (not for the first time!) a problem to solve; it is now:
Prove or disprove $$(m,n)\in \mathbb N^*\Rightarrow \frac{(mn)!}{m!(n!)^m}\in\mathbb N^*$$

We have
$$\eqalign{\frac{((m+1)n)!}{(m+1)!(n!)^{m+1}}
  &=\frac{(mn+n)\cdots(mn+1)}{(m+1)n!}\frac{(mn)!}{m!(n!)^m}\cr
  &=\frac{(mn+n-1)\cdots(mn+1)}{(n-1)!}\frac{(mn)!}{m!(n!)^m}\cr
  &=\binom{mn+n-1}{n-1}\frac{(mn)!}{m!(n!)^m}\ ,\cr}$$
note that the first factor is an integer and use induction on $m$.
