Proof question about division

Question

How to prove that if n|a-b, then n|a^k-b^k, similarly if n|a-b, then n|(a-b)^k

Answer 1

Since a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+...+ab^{k-2}+b^{k-1}) you have that if n divides a-b then n divides a^{k}-b^{k} for any positive integer k.

Also, since (a-b)^{k}=(a-b)(a-b)^{k-1} you have that if n divides a-b then n divides (a-b)^{k} for any positive integer k.