It's been bugging me all afternoon - is there an easy proof that if $\gcd(a,b) = 1$ there is a prime number $p$ of the form $p = an+b$? I'm aware of Dirichlet's theorem that there are in fact infinitely many primes of this form but am hoping there is an elementary proof of this weaker form.
The fact that I have been unable to find any proof of this online suggests to me that it's in fact very easy and I am simply missing something obvious.