Help with proving modulus, please.

Question

Prove that (x mod 2)(y mod 2) = (xy) mod 2.

Hi, I would really appreciate if anyone can help me on this question. It is really difficult asking questions during online class. Thank you.

Answer 1

x mod 2 = k then x = 2m + k where m is an integer

y mod 2 = l then y = 2n + l where l is an integer

xy = (2m + k) (2n + l) = 4mn + 2ml + 2kn + kl = 2 (2mn + ml + kn) + kl

xy mod 2 = kl

(x mod 2) (y mod 2) = kl

Proved