Question 97
Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding $5000$. Then she fixes $20$ distinct positive integers $a_{1}, a_{2}, \ldots, a_{20}$ such that, for each $k=1,2, \ldots, 20$, the numbers $N$ and $a_{k}$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding $20$, and she tells him back the set $\{a_{k}: k \in S\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?