stuck on congruence proof

Hey guys (and gals),

I was wondering if you could help me out. I am stuck on this proof I know a bunch of the basic steps but can't put them together properly. If I could get your help it would be great the question is:

Prove that for every $\displaystyle a,m \epsilon Z$ such that $\displaystyle gcd(a,m)=1$, with $\displaystyle m>1$ there is a $\displaystyle b \epsilon Z$ such that $\displaystyle ab \equiv 1 (\bmod m)$. Show therefore, that we can "divide" by a by multiplying the congruence by the corresponding b:

$\displaystyle ax \equiv c (\bmod m) \Leftrightarrow x \equiv bc (\bmod m) $