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)
| 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"
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.