Almost the most basic theorem of number theory is used: the necessary and sufficient condition of integer (a, b)= 1 is that X exists, and Y is an integer so that ax+by= 1.

If a|bc, BC = am and (a, b)= 1, then x and y exist such that ax+by= 1.

Both sides multiplied by C have ACX+BCY = C.

Use am instead of b, that is, ACX+ Amy = C.

That is, a(cx+my)=c, so a | c.