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set S_{k} contains infinitely many multiples of m

Source: 4th QEDMO 2007 p11

November 9, 2020
Setsnumber theorymultiple

Problem Statement

Let S1,S_{1}, S2,S_{2}, ...,..., SnS_{n} be finitely many subsets of N\mathbb{N} such that S1∪S2∪...∪Sn=N.S_{1}\cup S_{2}\cup...\cup S_{n}=\mathbb{N}. Prove that there exists some k∈{1,2,...,n}k\in\left\{ 1,2,...,n\right\} such that for each positive integer m,m, the set SkS_{k} contains infinitely many multiples of m.m.
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