factoring zero in modulo n Let $m,n\in \mathbb{N}$. How many different classes $\overline{y}\in\mathbb{Z}_n$ are there, so that
$$\overline{m}\cdot \overline{y}=\overline{0}$$
Each element is either invertible or a factor of the zero. I know that there are $n-\phi(n)$ zero factors in $\mathbb{Z}_n$ ($\phi$ is the Euler's totient function). I have some trouble finding out how many different ways is there to factor a random element $\overline{m}$.

Hint: If you know the prime factorization of $m$, then you have that the set of $y$ is generated by the element that has the complement prime factorization so that when you multiply $y$ and $m$ you get all prime factors of $n$ (with their multiplicity).
