OMM 2020 Problem 5

A four-element set $\{a, b, c, d\}$ of positive integers is called good if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$ , and it divides $4\times 8=32$ .

Find the greatest possible value of $n$ , such that any four-element set with elements less than or equal to $n$ is good.

Proposed by Victor and Isaías de la Fuente