EGMO 2013 Problem 3

Let $\boldsymbol{n}$ be a positive integer.

(a) Prove that there exists a set $S$ of $6 n$ pairwise different positive integers, such that the least common multiple of any two elements of $S$ is no larger than $32 n^{2}$. (b) Prove that every set $T$ of $6 n$ pairwise different positive integers contains two elements the least common multiple of which is larger than $9 \boldsymbol{n}^{2}$.