<s>[INST] How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number? 
How to prove that  $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$?

If $p$ is a prime, then the number of times $p$ divides $N!$ is $e_p(N)=\sum_{k=1}^\infty\left\lfloor\frac{N}{p^k}\right\rfloor$. So I need
$$e_p(5m)+e_p(5n)\geq e_p(m)+e_p(n)+e_p(3m+n)+e_p(3n+m).$$
What to do next?  Thanks in advance.
 [/INST] Let $p$ be a prime number. You should show that 
$$\left\lfloor \frac{5m}{p^k}\right\rfloor+\left\lfloor \frac{5n}{p^k}\right\rfloor \geq \left\lfloor \frac{m}{p^k}\right\rfloor+\left\lfloor \frac{n}{p^k}\right\rfloor+\left\lfloor \frac{3m+n}{p^k}\right\rfloor+\left\lfloor \frac{3n+m}{p^k}\right\rfloor$$
for $k \geq 1.$ By Hermite's identity you will get 
$$\left\lfloor \frac{5m}{p^k}\right\rfloor=\left\lfloor \frac{m}{p^k}\right\rfloor+\left\lfloor \frac{m}{p^k}+\frac{1}{5}\right\rfloor+\left\lfloor \frac{m}{p^k}+\frac{2}{5}\right\rfloor+\left\lfloor \frac{m}{p^k}+\frac{3}{5}\right\rfloor+\left\lfloor \frac{m}{p^k}+\frac{4}{5}\right\rfloor$$ 
Then you should show that 
$$\sum_{i=1}^{4} \left\lfloor \frac{m}{p^k}+\frac{i}{5}\right\rfloor+\sum_{j=1}^{4}\left\lfloor \frac{n}{p^k}+\frac{j}{5}\right\rfloor \geq \left\lfloor \frac{3m+n}{p^k}\right\rfloor+\left\lfloor \frac{3n+m}{p^k}\right\rfloor$$
which can be done by considering different cases for the values $\frac{m}{p^k}, \frac{n}{p^k}$ i.e. each can be in the following intervals $(0,\frac15), [\frac15, \frac25), [\frac25, \frac35), [\frac35, \frac45), [\frac45, 1),$ since $\lfloor x+n \rfloor=\lfloor x\rfloor+n,$ for and real number $x$ and an integer $n,$ you can just ignore the case when $\frac{m}{p^k}, \frac{n}{p^k}$ are bigger that $1.$
Earlier thought: The expression can be rewritten as 
$$\frac{\binom{5m}{3m}(3m)!\binom{5n}{n}n!}{(3m+n)!} \frac {\binom{4n}{3n}(3n)!\binom{2m}{m}m!}{(3n+m)!}$$
I was hoping to find a combinatorial interpretation for this expression but I haven't found yet! 
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