Disc. math/logic: division & modulus proofs

[n00neimp0rtnt]

Homework Statement

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)

Homework 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"

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.

 

[CEL]

If a | c, then c = ma. In the same way, d = nb.
The rest of the demonstration is up to you.