Pretty Sets

Round 19

Statement

Suppose a set $S$ satisfies the following conditions:
  •    $\text{(1)}$ Every element in $S$ is a positive integer and not greater than $100$
  •    $\text{(2)}$ For any two different elements $a$ and $b$ in S, there is an element $c$ in $S$ such that the greatest common divisor of $a$ and $c$ is equal to $1$ , and the greatest common divisor of $b$ and $c$ is also $1$; and
  •    $\text{(3)}$ For any two different elements $a$ and $b$ in $S$, there is an element $d$, which is different from $a$ and $b$, such that the greatest common divisor of $a$ and $d$, and that of $b$ and $d$ are greater than $1$.
Find the maximum possible number of elements in $S.$
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