Question 127
For two positive integers $m$ and $n$, let $\operatorname{gcd}(m, n)$ denote their greatest common divisor.
Let $S$ be an infinite set of positive integers such that there are four pairwise distinct numbers $v, w, x, y \in S$ with $\operatorname{gcd}(v, w) \neq \operatorname{gcd}(x, y)$.
Prove that there are three pairwise distinct numbers $a, b, c \in S$ such that $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, c) \neq \operatorname{gcd}(b, c)$.
Let $S$ be an infinite set of positive integers such that there are four pairwise distinct numbers $v, w, x, y \in S$ with $\operatorname{gcd}(v, w) \neq \operatorname{gcd}(x, y)$.
Prove that there are three pairwise distinct numbers $a, b, c \in S$ such that $\operatorname{gcd}(a, b)=\operatorname{gcd}(a, c) \neq \operatorname{gcd}(b, c)$.