dirichlet and gcds Let (a,b)=1 and c>0. Prove that there is an integer x such that (ax+b, c)=1.
Right now, I have the following approach: 
Let's assume that for every x, (ax+b,c)$\neq$ 1. Then $\exists$ d, where d/c that also d/ax+b=> ax$\equiv$-b (mod d). I'm not sure how to continue from here on.

The critical thing is that $d|ax+b$ for all $x$.  So take two different $x$'s and get $d|a$, then $d|b$ and you are done.
