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. 

$ m\mid a(b\!-\!c)\overset{{\rm cancel}\ d\,}\iff\! \frac{m}d\mid \frac{a}d(b\!-\!c)\overset{\rm Euclid}\iff \frac{m}d\mid b\!-\!c,\ $  since $\,\left(\frac{m}d,\frac{a}d\right) = 1,\ $ by $\ (m,a)=d.$
