For a positive integer $n$, let $S_n$ be the set of positive integers that do not exceed $n$ and are coprime to $n$. Define $f(n)$ as the smallest positive integer that allows $S_n$ to be partitioned into $f(n)$ disjoint subsets, each forming an arithmetic progression. Prove that there exist infinitely many pairs $(a, b)$ satisfying $a, b > 2025$, $a \mid b$, and $f(a) \nmid f(b)$.