KöMaL 2025 B5436

Let $p$ be an odd prime. Let $d_k(n)$ denote the number of positive divisors of $n$ with remainder $k$ modulo $p$ ($0\leq k<p$). Let $A\subseteq\{0,1,\dots,p-1\}=S$ satisfy $\sum_{k\in A}d_k(n)\geq\sum_{j\in S\setminus A}d_j(n)$ for every positive integer $n$. Find the minimum number of elements of $A$.