Problem

Source: Iranian TST 2018, third exam day 2, problem 4

Tags: number theory, Iran, Iranian TST



We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. Proposed by Morteza Saghafian