you don't need induction, you can prove it directly.
n^5 - n = n(n-1)(n+1)(n^2+1)
one of n-1,n, or n+1 is divisible by 3.
one of n,n+1 is divisible by 2. hence n^5 - n is divisible by 6.
now if n = 5k, 5|n. if n = 5k+1, 5|n-1, and if n = 5k+4 5|n+1. so in those 3 cases, it is obvious, 5|n^5 - n.
but if n = 5k+2, n^2+1 = (5k+2)^2 + 1 = 25k^2 + 20k + 5 = 5(5k^2 + 4k + 1).
and if n = 5k+3, n^2+1 = (5k+3)^2 + 1 = 25k^2 + 30k + 10 = 5(5k^2 + 6k + 2).
so in all cases, n^5 - n is divisivble by 2,3 and 5, and since these are all co-prime, their product 30 divides n^5 - n.
my apologies to chisigma, i wrote this up before dinner, and didn't post it until after you replied