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Source: 2024 Vietnam National Olympiad - Problem 6

January 6, 2024
number theory

Problem Statement

For each positive integer nn, let τ(n)\tau (n) be the number of positive divisors of nn.
a) Find all positive integers nn such that τ(n)+2023=n\tau(n)+2023=n. b) Prove that there exist infinitely many positive integers kk such that there are exactly two positive integers nn satisfying τ(kn)+2023=n\tau(kn)+2023=n.
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