Question 119
Let $\mathcal{S}$ be an infinite set of nonzero natural numbers containing four pairwise distinct integers $a, b, c, d$ such that $\operatorname{gcd}(a, b) \neq \operatorname{gcd}(c, d)$. Prove that $\mathcal{S}$ contains three pairwise distinct integers $x, y, z$ such that $\operatorname{gcd}(x, y)=\operatorname{gcd}(y, z) \neq \operatorname{gcd}(z, x)$.