I'm a physicist taking elementary number theory so while I've had a class discussing proofs in the past, my experience with rigorous mathematical proofs is very limited. I've got a problem and I'm not sure how many things I need to show to adequately complete it. The question is:
Prove that if $d$ is a common divisor of $a$ and $b$, then $d = gcd(a,b)$ if and only if $gcd(a/d,b/d) = 1$.
The math itself is very simple. But what I actually need to show is less clear to me. I know iff statements need to be shown both ways ($A\Rightarrow B$ and $B\Rightarrow A$ since it's essentially an equality statement), so would I need to show both directions for that and THEN assume 'if $d$ is a common divisor, then...'? Or should I start with that assumption and do something extra later on?
Much appreciation for the help.