Lifting the Exponent

(Lemma from MAA official solution, 2020 AIME I Problems/Problem 12)

Denote $v_p(n)$ the highest power of prime $p$ that divides $n$ . Let $p$ be an odd prime, and let $a$ and $b$ be integers that are not multiples of $p$ such that $p \mid (a-b)$ . Let $n$ be a positive integer. Then $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ .