By Fermat's little theorem, we know that if n and 5 are coprime, then , that is to say is divisible by 5 and hence n^5-n is too.
If n and 5 aren't coprime, that is n is a multiple of 5, then n^5-n is obviously divisible by 5.
You can work with algebra :
you can see that n(n+1)(n-1)(n-2)(n+2) is the product of 5 consecutive integers. Therefore, there is one of the factors that is divisible by 5. Hence it is divisible by 5.
5n(n+1)(n-1) is obviously divisible by 5.