Question 1
Determine all finite nonempty sets $S$ of positive integers satisfying
$$\frac{i+j}{\gcd(i,j)}\in S\qquad\text{for all }i,j\in S,$$
where $\gcd(i,j)$ is the greatest common divisor of $i$ and $j$.
$$\frac{i+j}{\gcd(i,j)}\in S\qquad\text{for all }i,j\in S,$$
where $\gcd(i,j)$ is the greatest common divisor of $i$ and $j$.