Construct a family of sets $A_n$ such that $$|A_n|=\Theta\left((\log n)^2\right)$$ and the elements of $A_n$ are chosen uniformly at random mod $n$.

Say that a set $S$ represents $m\mod{n}$ if there are some $s,t\in S$ such that $$\frac{s}{t}\equiv m\mod{n}$$

**Conjecture:** For any $m$, there is an $A_n$ that represents $m\mod{n}$ almost surely

I formulated this conjecture while working on a somewhat tangentially related problem. My primary interest is to gain a heuristic understanding of whether or not such a result is possible. In addition, I would like to learn more about the kinds of techniques employed to solve problems of this nature.