2.3. Let $S \cong\{1,2, \cdots, 2002\}$, for any $a, b \in S(a, b$ can be the same $)$, there is always $a b \notin S$, find the maximum value of $|S|$.


Parse $|S|_{\max }=1958$.
First, $1 \notin S$. Otherwise, let $a=b=1$, then $a, b, a b \in S$, which contradicts the problem statement!
Second, if $a \in\{2,3,4,5,6\}$ and $a \in S$, then among the following 45 pairs of numbers, each pair's two numbers cannot both belong to $S$.
$$
\begin{array}{c}
(1, a),(2,2 a), \cdots,(a-1,(a-1) a), \\
(a+1,(a+1) a),(a+2,(a+2) a), \cdots, \\
(2 a-1,(2 a-1) a), \\
\cdots \cdots \\
(44 a+1,(44 a+1) a),(44 a+2,(44 a+2) a), \cdots, \\
(45 a-1,(45 a-1) a) .
\end{array}
$$

Since $(45 a-1) a \leqslant(45 \times 6-1) \times 62002$, i.e., $a b \notin S$, at this point $|S|=1958$. In summary, the maximum value of $|S|$ is 1958.