Divisibility proof problem I need assistance with the following proof. 
Let a,b,c,m be integers, with m $\geq$ 1. Let d = (a,m). Prove that m divides ab-ac if and only if $\frac md $ divides b-c.
Alright, I know that since d = (a,m) there exists an r and t such that $ar + mt = d$
I figure since we're trying to prove m divides ab-ac iff $\frac md$ divides $b-c$, we look to $\frac md$ dividing $b-c$
Only I get stuck when trying to work with $\frac md$ dividing $b-c$ algebraically. 
I've tried relating d = (a,m) to  $\frac md$ but I'm still stuck. 

Hint: This comes down to the following: We have $m|xy$, if and only if
$$\left.\frac{m}{\gcd(m,x)}\right|y$$
In words: every factor of $m$ that is not in $x$ must be in $y$ for $xy$ to be divisible by $m$.
