4. What is the maximum number of numbers that can be taken from $\{1,2, \cdots, 100\}$ such that the sum and product of any two different numbers are not multiples of 100?
4.45 items.

Example: $1,2, \cdots, 49$ excluding $20, 25, 30, 40$.
Divide $\{1,2, \cdots, 100\}$ into the following 45 groups:
$$
\begin{array}{l}
\{10,20,30, \cdots, 100\}, \\
\{4,96,25,75\}, \\
\{i, 100-i\} \text { (other numbers), }
\end{array}
$$

Then at most one number can be taken from each group. Therefore, at most 45 numbers can be taken.