OlympiadBench / 1606 / Xenia and Sergey play the following game. Xenia thinks of a positive integer N…

Xenia and Sergey play the following game. Xenia thinks of a positive integer N not exceeding 5000. Then she fixes 20 distinct positive integers a₁,a₂,…,a₂₀ such that, for each k=1,2,…,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∈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?

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 $\left\{a_{k}: k \in S\right\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?