Let n be a natural number and let a, b ∈ Z. Use the definition of congruence, Lemma and induction to show that if a ≡ b (mod n) then
$\displaystyle a^k$ ≡ $\displaystyle b^k$ (mod n), for all integers k ≥ 0
You ask lots of questions, some of which are very long and which look like homework, and you show no self work at all....show what you've done and where you're stuck.