I'm looking for a quick proof that $SL_2(\mathbb Z/n\mathbb Z)\cong \oplus_{p\mid n}SL_2(\mathbb Z/p^{e_p}\mathbb Z)$ for some nonnegative integers $e_p$. I've argued as follows, but I'm hoping for a 'slicker' (one line?) proof as I feel like this should just follow easily from some property I'm forgetting about. Anyway, here is my argument.
We know there are $e_p$ such that $\mathbb Z/n\mathbb Z\cong \oplus_{p\mid n} \mathbb Z/p^{e_p}\mathbb Z$ from group theory, therefore if $A\in SL_2(\mathbb Z/n\mathbb Z)$ then $A$ has a unique decomposition as $A=\sum_{p\mid n} A_p$, for some matrices $A_p$ with entries in $\mathbb Z/p^{e_p}\mathbb Z$. The fact that each $A_p$ has determinant $1$ follows from the decomposition of the terms of $A$ and the fact that $\det A=1$.