HMMT 2021 A10

Let $S$ be a set of positive integers satisfying the following two conditions:

• For each positive integer $n$, at least one of $n, 2 n, \ldots, 100 n$ is in $S$.
• If $a_{1}, a_{2}, b_{1}, b_{2}$ are positive integers such that $\operatorname{gcd}\left(a_{1} a_{2}, b_{1} b_{2}\right)=1$ and $a_{1} b_{1}, a_{2} b_{2} \in S$, then $a_{2} b_{1}, a_{1} b_{2} \in S$.

Suppose that $S$ has natural density $r$. Compute the minimum possible value of $\left\lfloor 10^{5} r\right\rfloor$. Note: $S$ has natural density $r$ if $\frac{1}{n}|S \cap\{1, \ldots, n\}|$ approaches $r$ as $n$ approaches $\infty$.