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 2 of 4: Rule out odd elements
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k odd and maximal  ⟹  k+2gcd⁡(k,2)=k+2>kk\text{ odd and maximal}\implies\frac{k+2}{\gcd(k,2)}=k+2>k
Detailed analysis

Suppose an odd element exists and let kk be the largest odd element of SS. Since 2∈S2\in S and gcd⁡(k,2)=1\gcd(k,2)=1, the closure condition gives (k+2)/gcd⁡(k,2)=k+2∈S(k+2)/\gcd(k,2)=k+2\in S, contradicting maximality. Hence every element of SS is even.