OIM 2020 Problem 4

Show that there exists a set $\mathcal{C}$ of $2020$ distinct, positive integers that satisfies simultaneously the following properties: $\bullet$ When one computes the greatest common divisor of each pair of elements of $\mathcal{C}$ , one gets a list of numbers that are all distinct. $\bullet$ When one computes the least common multiple of each pair of elements of $\mathcal{C}$ , one gets a list of numbers that are all distinct.