Given a sequence $a_1,a_2,\ldots ,a_n$, if $\gcd(a_1,a_2,\ldots ,a_n) = 1$, then there exists one pair $a_i,a_j$ st. $\gcd(a_i,a_j)=1$.

Question

Anyone can help prove the following claim using elementary proof (no advanced number theory stuff)?

Given a sequence $a_1,a_2,\ldots,a_n$, if $\gcd(a_1,a_2,\ldots,a_n) = 1$, then there exists at least one pair $a_i,a_j$ for some $i,j\in\{1,2,\ldots,n\}$ with $i\neq j$ such that $\gcd(a_i,a_j)=1$.

Thank you!

Answer 1

A counterexample is $(6,10,15)$

Answer 2

More generally, this is also untrue:

Let $S$ be a finite subset of $\mathbb{Z}$. If the gcd of all elements of $S$ is $1$, then there exists a proper nonempty subset $T$ of $S$ such that the gcd of all elements of $T$ is $1$.

Nevertheless, this is true:

Let $S$ be an infinite subset of $\mathbb{Z}$ such that the gcd of all elements of $S$ is $1$. Then, there exists a finite nonempty subset $T$ of $S$ such that the gcd of all elements of $T$ is $1$. However, the minimum cardinality of such a subset $T$ can be arbitrarily large.