Let's say we have some positive integer $n$ and a set $S$ that is the set of all highly composite numbers greater than or equal to $n$. Is it possible to prove that all members of $S$ share a factor other than $1$? Since it's fairly trivial to prove that all HCN are even other than $1$, is it possible to prove that all members of S have a greatest common factor greater than $2$? If so, is there some method that can be utilized to find the GCF of $S$ for any arbitrarily large $n$? Additionally, can it be proven that there exists a positive integer $n$ such that $\text{GCF}(S)\geq m$ for an arbitrarily large positive integer $m$?
Empirically, I have found possible correlations, such as the observation that if $n=6$, it appears that $\text{GCF}(S)=6$. It also appears that if $n=840$, $\text{GCF}(S)=420$.