ELMO 2021 Problem 4

The set of positive integers is partitioned into $n$ disjoint infinite arithmetic progressions $S_1, S_2, \ldots, S_n$ with common differences $d_1, d_2, \ldots, d_n$ , respectively. Prove that there exists exactly one index $1\leq i \leq n$ such that$$\frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.$$