Groups and the Group Axioms · Hard problem

On $S=\{1,2,\ldots,10\}$ consider $a*b=\operatorname{lcm}(a,b)$. It is not a binary operation on $S$, because some outputs leave $S$. For how many of the $100$ ordered pairs $(a,b)\in S\times S$ does $a*b$ stay in $S$?