Let $m,n$ be positive integers. Prove that $$m!n!(3m+n)!(3n+m)!|(5m)!(5n)!$$
My idea is to prove the following inequality $$\lfloor 5x\rfloor+\lfloor 5y\rfloor\ge \lfloor x\rfloor+\lfloor y\rfloor +\lfloor 3x+y\rfloor+\lfloor x+3y\rfloor,0\le x,y<1$$I was stuck at this part. Could anyone have a nice approach to this inequality?