Let 𝑎, 𝑏 ∈ ℤ and 𝑚 ∈ ℕ. Prove that if 𝑎 ≡ 𝑏(mod 𝑚), then 𝑎 3 ≡ 𝑏 3(mod 𝑚).

Answer 1

If a ≡ b (mod m), then a - b is a multiple of m. Since a³ - b³ = (a - b)(a² + ab + b²) and a - b is a multiple of m, then a³ - b³ is a multiple of m. Hence a³ ≡ b³ (mod m).