MathLabs

Problem 1

Determine all finite nonempty sets SS of positive integers satisfying i+jgcd⁡(i,j)∈S\frac{i+j}{\gcd(i,j)}\in S for all i,j∈Si,j\in S.
Step 1 of 4: Force 2 into the set
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i=j=k∈S  ⟹  k+kgcd⁡(k,k)=2∈Si=j=k\in S\implies\frac{k+k}{\gcd(k,k)}=2\in S
Detailed analysis

For any k∈Sk\in S, applying the condition with i=j=ki=j=k gives (k+k)/gcd⁡(k,k)=2(k+k)/\gcd(k,k)=2. Thus 2∈S2\in S.