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Prove that there are no such 2021 numbers

Source: 2021 Peru Cono Sur TST P6

July 11, 2023
number theory

Problem Statement

Prove that there are no positive integers a1,a2,…,a2021a_1, a_2, \ldots , a_{2021} (not necessarily distinct) such that for k=1,2,3,…,2021k = 1, 2, 3, \ldots , 2021 the number of elements in the set Ak={j∈N:1≤j≤2021 and aj∣k}A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \} be exactly aka_k.
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