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2023 factors and perfect cube

Source: Indian Statistical Institute (ISI) UGB 2023 P4

May 14, 2023
combinatoricsnumber theory

Problem Statement

Let n1,n2,⋯ ,n51n_1, n_2, \cdots , n_{51} be distinct natural numbers each of which has exactly 20232023 positive integer factors. For instance, 220222^{2022} has exactly 20232023 positive integer factors 1,2,22,23,⋯22021,220221,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}. Assume that no prime larger than 1111 divides any of the nin_{i}'s. Show that there must be some perfect cube among the nin_{i}'s.
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