Suppose $a\mid d$, $b\mid d$, $a=u\cdot\gcd(a, b)$ and $b=v\cdot\gcd(a, b)$. Show that there is an integer $r$ such that $d=ruv\cdot \gcd(a, b)$
Here, $a, b, u, v$ and $d$ are also integers. I know that $\gcd(a, b) = ma + nb$ (from the extended Euclidean algorithm). So I started with the equations:
1) $a=u(ma + nb)$
2) $b=v(ma + nb)$
But unfortunately I can no longer proceed. How to approach the proof?