What is the largest positive integer that always divides n^5-n^3 for n a natural number.

First we note we can factorise the n^3 out of the expression, giving n^3(n^2-1). Secondly, we can see that the second term is a difference of two squares, allowing is to factorise the total to n^3(n-1)(n+1). We see that this is always divisible by 3, as n^5-n^3 is divisible by both n-1, n and n+1 and with 3 consecutive integers, always one of the is divisible by 3. Secondly, we see n^5-n^3 is divisible by 2, as either n or n+1 is even. But it's divisible by 2 multiple times. If n is even, then n^3 is divisible by 8 and hence so is n^5-n^3. If n is odd, then n-1 and n+1 is even. As these are two consecutive even numbers, one of the two is divisible by 4. Hence (n-1)(n+1) is divisible by 8, as one is divisible by at least 2 and the other by at least 4. 

All in all we've found that n^5-n^3 is always divisible by 8 and by 3. As these are coprime, n^5-n^3 is always divisible by 8*3=24. But is this the largest positive integer? To check this, we can sub in n=2, giving n^5-n^3=24. The largest positivee integer that divides 24 is of course 24. So there can't be a larger positive integer that always divides n^5-n^3, so our answer is 24.