Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$ [duplicate]
Solution 1:
Hint $\ $ By the $ $ GCD Distributive Law, $ $ and $\ d\mid a,b\iff d\mid (a,b)\ \,$ [gcd Universal Property]
$$\begin{eqnarray} d\,\left(\dfrac{a}d,\dfrac{b}d\right) &\!\!=& (a,b)\\[.4em] \Rightarrow\ \left(\dfrac{a}d,\dfrac{b}d\right) &\!\!=& (a,b)\,/\,d\\[.4em] \!{\rm Thus}\ \ \ 1 = \left(\dfrac{a}d,\dfrac{b}d\right) &\!\!\iff\!& (a,b)\!=\!d\end{eqnarray}$$