1. The problem statement, all variables and given/known data Show that if a, b, c, and d are integers such that a | c and b | d, then ab | cd. Let m be a positive integer. Show that a mod m = b mod m if a ≡ b(mod m) 2. Relevant equations | means "divides," so a | b means "a divides b" or "b can be divided by a" mod gets the remainder; a mod m means "the remainder after m is divided by a" ≡ means "is congruent to" 3. The attempt at a solution For the proof of the first one, I can easily substitute real values: a = 4 b = 3 c = 16 d = 9 and from that I would get (4)(3) | (16)(9) 12 | 144 which is obviously 12, for which the statement holds true; however, since this is a universal proof and not an existential one, that statement is far from enough to prove it. For the proof of the second statement, I am unsure about how to treat a congruency in a proof like this. Proofs are probably my weakest point in this course, so thanks in advance for any help.