I have learned about $X_n = \mathbb{Z} / n\mathbb{Z}$. I understand that a zero divisor is an element $x\neq 0$ in $X_n$ such that $xy = 0$ for some $y\neq 0$. I understand that an element $x$ in $X_n$ is invertible if there is a $y$ in $X$ such that $xy=1$.
My question is if it is true that $X_n$ is the disjoint union of the invertible elements and the zero divisors?
I think this is true since for example in $X_5$ there are no zero divisors and all elements are invertible. In $X_6$ the invertible elements are $1, 5$ and the zero divisors are $2,3, 4$. And $X_6$ is exactly the union of those two sets and the two sets are disjoint.
If I am right that this is correct, I would like a hint on how to prove it. (This is not homework, but I would like to try to proof myself if I can get help started.)
Hint: Take $a \in X_n$ and consider the map $x \mapsto ax$. What does its injectivity or surjectivity tell you about whether $a$ is a unit or a zero divisor?