Show that if $a$ is a divisor of $bc$ and $(a, b) = 1$, then $a$ is a divisor of $c$.

Question

Suppose I set $ax+by=1$ such that $x, y$ are integers and $a$ and $b$ are elements.

The greatest common divisor of $a, b$ is $1$.

If $a=bc$, then $bcx+by=1$.

But then what do I do?

Answer 1

We have $$ax+by=1\Rightarrow acx+bcy=c$$

But $a|bc\Rightarrow bc=ak$, $k\in\Bbb{Z}$

Thus $$acx+aky=c\\a(cx+ky)=c\\am=c\\a|c$$