Prove that `n^2-n` divisible by 2 for every positive integer n.

Prove that n^(2)-n is divisible by 2 for every positive integer n.

Prove that n + n is divisible by 2 for any positive integer n.

Prove that x^n-y^n is divisible by x - y for all positive integers n.

If p be a natural number, then prove that, p^(n+1)+(p+1)^(2n-1) is divisible by (p^(2)+p+1) for every positive integer n.

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

If p is a natural number, then prove that p^(n+1) + (p+1)^(2n-1) is divisible by p^(2) + p +1 for every positive integer n.

The expression 31^(n) + 17^(n) is divisible by, for every odd positive integer n :

Show that n^2-1 is divisible by 8, if n is an odd positive integer.

Using binomial theorem, prove that 5^(4n)+52n-1 is divisible by 676 for all positive integers n.