The least common multiple of random sets of positive integers

We study the typical behavior of the least common multiple of the elements of a random subset $A\subset \{1,\dots ,n\}$. For example we prove that $\text{lcm}\{a:a\in A\}=2^{n(1+o(1))}$ for almost all subsets $A\subset \{1,\dots ,n\}$.