A graph $G$ is called $\textit{divisibility graph}$ if the vertices can be assigned distinct positive integers such that between two vertices assigned $u, v$ there is an edge iff $\frac{u} {v}$ or $\frac{v} {u}$ is a positive integer. Show that for any positive integer $n$ and $0 \leq e \leq \frac{n(n-1)}{2}$, there is a $\textit{divisibility graph}$ with $n$ vertices and $e$ edges.
HIDE: Remark on source of 10.3 It appears to be Kvant 2022 Issue 10 M2719, so it will not be posted; the same problem was also used as 9.4.