elementary number theory - 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. $gcd(a_i,a_j)=1$.

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

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