Both of these seem trivial and are obviously true, but other than saying "These are obviously true"

Yet my professors seem to get away with doing that all the time...

a.

a | c, so c = ak.

b | c, so c = bl.

gcd(a, b) = 1, so 1 = an + bm.

We want to get c = abz. So you got to play around with what you have. I'll give you one more hint. c = anc + bmc. See if you can get it from there.

Oh, and this isn't really a hint, but more a reminder. You must use everything you are given in a proof. Otherwise, you wouldn't be given it. The only thing I used so far is that 1 = an + bm.

b.

This one is easier than it seems. Let a | n such that a is not 1. Since a does not divide 1, but a divides n, a does not divide n + 1. I proved that fact a few days ago in reply to one of your other questions. Since this applies for all a's that aren't 1...