Proving that every reduced residue class contain at least one prime I don't know if I expressed this clearly, but I want to know if the following is true and also some help proving it in case it is.
$\forall a,b \in \mathbb{N} , \gcd{(a,b)} = 1 \Rightarrow \exists  p \equiv b \pmod{a}$
Where $p$ is a prime and $b < a$.
I know the converse is true, because a prime can only be congruent to reduced residue classes. 
If it is too trivial, some hints can suffice, if it is not, pointing to good material would be greatly appreciated.
Thanks in advance.

This is a consequence of Dirichlet's theorem. I don't know a way to prove your theorem without it.
