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Does there exists set of positive integers?

Source: 2018 China Southeast MO Grade 10 P4

July 30, 2018
number theory

Problem Statement

Does there exist a set A⊆N∗A\subseteq\mathbb{N}^* such that for any positive integer nn, A∩{n,2n,3n,...,15n}A\cap\{n,2n,3n,...,15n\} contains exactly one element? Please prove your conclusion.
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