Let $n$ be a positive integer. Determine the maximum number of positive integers less than or equal to $n^2$ that can be colored red in such a way that if $a$ and $b$ are red, with $a \neq b$, then $a \cdot b$ is not red.
Source: 2024 Argentina L3 P3
Tags: number theory
Let $n$ be a positive integer. Determine the maximum number of positive integers less than or equal to $n^2$ that can be colored red in such a way that if $a$ and $b$ are red, with $a \neq b$, then $a \cdot b$ is not red.