Problem

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Tags: number theory



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\[ \frac{1}{d_i}\prod_{j=1}^n d_j \in S_i.\]