Suppose gcd(a,b) = 1.

That is, b=1k and a = 1m for some integers k,m.

Suppose b|c that is, c = bx for some integers x.

Then c = (1k)x where 1k is an integer.

I think now you need to show that k = 1 because a = 1m (and so 1 is the largest divisor of a).

That's all I've got...it might not be complete or correct. Can someone double check it?