Let $A$ be a fixed matrix in $M_2\mathbb{Z}$ with determinant $n \neq 0$.

**Question 1** How many ways can I write $A = XY$ for $X, Y \in M_2\mathbb{Z}$?

The answer to this question is pretty clearly infinite, since for any $\gamma \in GL_2\mathbb{Z}$ and any such pair $(X, Y)$, then $(X \gamma^{-1}, \gamma Y)$ is another such factorization. So let's get rid of that.

**Question 2** Consider the set $S_A = \{(X, Y) \in (M_2\mathbb{Z})^2 \mid A = XY\}$. The group $G = GL_2\mathbb{Z}$ acts naturally on this set via
$$
\gamma \cdot (X, Y) = (X \gamma^{-1}, \gamma Y)
$$
What is the cardinality of $S_A/G$?

Unfortunately, this depends on our choice of $A$ (fixing $n$). If we compare $$ A_1 = \begin{pmatrix}4 & 0 \\ 0 & 1 \end{pmatrix} \qquad \qquad A_2 = \begin{pmatrix}2 & 0 \\ 0 & 2 \end{pmatrix} $$ then in the first case, we find $|S_{A_1}/G| = 3$ while in the second, $|S_{A_2}/G| = 4$. This is due to the fact that the second matrix is not primitive. We could exclude such matrices, or we could define an equivalence relation on factorizations via $$ (X, kY) \sim (kX, Y) $$ which is compatible with the action of $G$. For a fixed $A$, define $T_A = S_A/_\sim$.

**The Real Question** Does the cardinality of $T_A/G$ depend on $A$? If not, what is it?

By computation, it seems that this does not depend on $A$ and moreover, $$ |T_A/G| = \sum_{d \mid n} 1 = \sigma_0(n) $$

This seems like it should be a pretty obvious question about arithmetic/algebraic groups, but it's not really my area of expertise.