CMO 2003 Problem 2

Determine the maximal size of the set $S$ such that: i) all elements of $S$ are natural numbers not exceeding $100$ ; ii) for any two elements $a,b$ in $S$ , there exists $c$ in $S$ such that $(a,c)=(b,c)=1$ ; iii) for any two elements $a,b$ in $S$ , there exists $d$ in $S$ such that $(a,d)>1,(b,d)>1$ .

Yao Jiangang