4. Given the set $S=\{1,2, \cdots, 2005\}, A \subseteq S, A$ such that the sum of any two numbers in $A$ is not divisible by 117, find the maximum value of $|A|$.

 untranslated text remains the same as requested. However, if you need any further assistance or a different translation, feel free to let me know!
4. $F_{i}=\{x ; x \equiv i(\bmod 117)\}$, then $S=\bigcup_{i=0}^{16} F_{i}$ and $\left|F_{0}\right|=17,\left|F_{1}\right|=\left|F_{2}\right|$ $=\cdots=\left|F_{16}\right|=18$.
$\left|F_{17}\right|=\left|F_{18}\right|=\cdots=\left|F_{116}\right|=17, F_{0}$ can only have one number in $A$, if $F_{i} \cap A \neq \varnothing$, there must be $F_{117-i} \cap A=\varnothing, A$ can have at most $1+\left|F_{1}\right|+\left|F_{2}\right|+\cdots$ $+\left|F_{58}\right|=1+18 \times 16+17 \times 42=1003$ elements.