MathDB

Let

n_1, n_2, \cdots , n_{51}

be distinct natural numbers each of which has exactly

2023

positive integer factors. For instance,

2^{2022}

has exactly

2023

positive integer factors

1,2, 2^{2}, 2^{3}, \cdots 2^{2021}, 2^{2022}

. Assume that no prime larger than

11

divides any of the

n_{i}

's. Show that there must be some perfect cube among the

n_{i}

's.

4