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 4 of 4: Verify the unique set
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S={2}S=\{2\}
Detailed analysis

The preceding contradictions show that no element other than 22 can occur. The set S={2}S=\{2\} satisfies the condition because (2+2)/gcd⁡(2,2)=2(2+2)/\gcd(2,2)=2, so it is the unique solution.