Problem 1

Determine all finite nonempty sets

S

of positive integers satisfying

\frac{i+j}{\gcd(i,j)}\in S

for all

i,j\in S

Step 3 of 4: Rule out a second element above 2

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\ell>2\text{ second-smallest}\implies\frac{\ell+2}{\gcd(\ell,2)}=\frac{\ell}{2}+1<\ell

Detailed analysis

If there were an element besides $2$ , let $\ell>2$ be the second-smallest element. It is even, so closure with $i=\ell,j=2$ gives $(\ell+2)/\gcd(\ell,2)=\ell/2+1$ . For $\ell>2$ this lies strictly between $2$ and $\ell$ , contradicting the choice of $\ell$ .