. Suppose gcd(a, m) = d and that m > 1. Consider the congruence ax ≡ b (mod m). Should there be a solution for every choice of b?

Question

If yes, prove your claim; if not, give a counter example

X is not told so I assume it can be any orbitrary number. b is also abitrary, with that being said, isn't true due to those 2 factors?

Answer 1

No, it has not. For instance, the congruence $\;8x\equiv 5\pmod{12}$ has no solution, because it would imply that $5$ is divisible by $4$.

What can be said is this:

The congruence $\;ax\equiv b\pmod m$ has a solution if and only if $b\equiv 0\mod d=\gcd(a,m)$.

In this case, the congruence is equivalent to $$\frac ad\, x\equiv\frac bd\mod\frac md. $$