Question 160
A set $S$ of positive integers is called magic if for any two distinct members of $S$, $i$ and $j$,
$$
\frac{i+j}{\operatorname{GCD}(i, j)}
$$
is also a member of $S$. The $\operatorname{GCD}$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $\operatorname{GCD}(36,80)=4$.
Find and describe all finite magic sets.
$$
\frac{i+j}{\operatorname{GCD}(i, j)}
$$
is also a member of $S$. The $\operatorname{GCD}$, or greatest common divisor, of two positive integers is the largest integer that divides evenly into both of them; for example, $\operatorname{GCD}(36,80)=4$.
Find and describe all finite magic sets.