Does $a \mid bc$ imply $\frac{a}{(a,b)} \mid c$?

Question

If $a \mid bc$, then does $\frac{a}{(a,b)} \mid c$? I doubt anybody here is industrious enough to show this via a diagram, but who knows.

Answer 1

Set $d=(a,b)$, $a=a'd, b=b'd$. We have $(a',b')=1$. Now, the hypothesis is $a|bc$, or $a'd|b'dc$. Cancelling $d$, we get $a'|b'c$. Since $(a',b')=1$, $a'|c$.

Diagram: