If $g$ is a divisor of $ab,cd$ and $ac+bd$ prove that it is also a divisor of $ac$ and $bd$, where $a,b,c,d$ are integers.
There are several existing solutions of this problem on this site, but I approached this problem in a different way, consider $(ac-bd)^2 = (ac+bd)^2-4abcd$ , since $g^2 \mid \left[ (ac+bd)^2-4abcd \right] $, this implies $ g^2 \mid (ac-bd)^2 $ and hence $ g \mid (ac-bd) $. This further imples that $ g \mid 2ac $ and $ g \mid 2bd $.
Now i am stuck at this point, how do I show from this that $ g \mid ac$ and $ g \mid bd $ ?