If integers a, b, and m are greater than zero and a is congruent to b modulo m, then it follows that a^n is congruent to b^n modulo m for all positive integers n. The congruence is denoted as a ≡ b (mod m), and the proof can be approached using mathematical induction on n. The inductive step requires showing that if the statement holds for n, it also holds for n+1. Additionally, expanding (b + km)^n using the binomial theorem can help in deriving the conclusion. This discussion emphasizes the importance of understanding congruences and their properties in modular arithmetic.