KöMaL 2021 B5187

A subset of the set $S=\{1,2,\dots,n\}$ is called primitive, if it does not contain two elements such that one is a divisor of the other. Show that if it is not possible to add a further element of $S$ to a particular primitive set $A\subseteq S$ and keep it primitive, then either $A=\{1\}$ or the size of $A$ is greater than or equal to the number of primes up to $n$.