If a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?

If a ≡ b (mod m) and c ≡ d (mod m), which of the following is true?

• A. ac ≡ bd (mod m)
• B. ac ≡ ad (mod m)
• C. a ≡ c (mod m)
• D. b ≡ d (mod m)

Answer: (A)

Congruences multiply: if a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m). To see why, express a = b + km and c = d + lm for some integers k and l. Then ac = (b + km)(d + lm) = bd + m(bl + dk) + m^2 kl, which is bd plus a multiple of m. Therefore ac − bd is divisible by m, so ac ≡ bd (mod m). This shows that pairing the two congruences and multiplying preserves the congruence in the expected way.

Congruences multiply: if a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m). To see why, express a = b + km and c = d + lm for some integers k and l. Then ac = (b + km)(d + lm) = bd + m(bl + dk) + m^2 kl, which is bd plus a multiple of m. Therefore ac − bd is divisible by m, so ac ≡ bd (mod m). This shows that pairing the two congruences and multiplying preserves the congruence in the expected way.