Prove that n^13 - n is divisible by 2, 3, 5, 7 and 13 for any integer n.
n^13-n=n(n^12-1).
We will rely on Fermat's Little Theorem.
If p|n there is nothing to prove.
We will assume that p doth not divide n.
That is, prove 2,3,5,7,13 divide n^12-1
First, the case p=2 is trivial.
If p=3 then,
n^2=1(mod 3)
Raise both to the power of 6,
n^12=1(mod 3)
If p=5 then,
n^4=1(mod 3)
Raise both to the power of 3,
n^12=1(mod 3)
If p=7 then,
n^6=1(mod 7)
Raise both to the power of 2,
n^12=1(mod 7)
If p=13 then,
n^12=1(mod 13)
Q.E.D.