Prove that if d s a positive integer, d/a and d/b, then gcd(a,b)=d iff
gcd(a/d,b/d)=1
The following addendum to kalagota's answer is probably obvious, but just in case: $\displaystyle ax + by = d$ implies that $\displaystyle \gcd(a,b)|d$, so that $\displaystyle \gcd(a,b)=d$ since conversely $\displaystyle d|a$ and $\displaystyle d|b$ imply $\displaystyle d|\gcd(a,b)$.