Question 123
Suppose a set $S$ satisfies the following conditions:
(1) every element in $S$ is a positive integer and not greater than $100$;
(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
(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 number of elements in $S$.
(1) every element in $S$ is a positive integer and not greater than $100$;
(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
(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 number of elements in $S$.