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?