Suppose $R$ is an integral domain and $r,s \in R-\{0\}$.
If $g=\gcd(r,s)$ and $d$ is any common divisor of $r,s$, show that $g'=\gcd(r',s')$ where $r'd=r, s'd=s, g'd=g$.
I can show that $g'$ is a common divisor of $r',s'$ but can't seem to show that any common divisor of $r',s'$ divides $g'$.
What I got:
$g'dk_1=gk_1 =r=r'd \rightarrow g'd=r'$ since $R$ is an integral domain. Similarly,
$g'dk_2=gk_2 =s=s'd \rightarrow g'd=s'$
Any ideas on how to proceed?