Edit: See here
Basic Number Theory: LCM/GCD Proof
(Make sure to read posts #1 through #4 to get a full treatment of part (2).)
Although the proof for part (1) is lengthy for my taste, since
And since (a,b) | b, we know is an integer, hence a | m, and by symmetry b | m also.
Edit 2: I think aman_cc's proof for part (2) below is nicer than the one on the other forum.