Linear equation over $\mathbb{Z}/n\mathbb{Z}$

Question

For given $a,b\in \mathbb{Z}/n\mathbb{Z}$ is there a criterion which allows one to determine whether there exists $x\in \mathbb{Z}/n\mathbb{Z}$ with $ax=b$?

Answer 1

There is a solution if and only if the greatest common divisor of $a$ and $n$ is a factor of $b$.

Answer 2

Yes. You want to see whether there exist $x$ and $y$ such that $$ ax+ny=b $$ (using for $a$ and $b$ representatives of the given elements in $\mathbb{Z}/n\mathbb{Z}$).

What does Bézout's theorem say?

Hint: consider the greatest common divisor of $a$ and $n$.