Show that n is any odd integer greater than 1, then n^ - n is divisible by 8

^=5

You mean n^5 - n is divisible by 8? Then read the following:
n^5 - n = n(n^4 - 1) = n(n-1)(n+1)(n^2 + 1)
Now n is odd so n-1 and n+1 are both even as well as n^2 + 1 is even. So there are three two'2 in the factorisaton. Hence its divisible by 8.
To make it more lucid, take n = 2k-1 and substitute, you will get it.