Problem

Source: Mexico National Olympiad 2020 P1

Tags: combinatorics, number theory, greatest common divisor, vector



A set of five different positive integers is called virtual if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors. Proposed by Víctor Almendra