Problem

Source: 2024 Vietnam National Olympiad - Problem 6

Tags: number theory



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