2023 Putnam B5

Determine which positive integers $n$ have the following property:

For all integers $m$ that are relatively prime to $n$, there exists a permutation $\pi: \{1,2,\dots,n\} \to \{1,2,\dots,n\}$ such that $\pi(\pi(k)) \equiv mk \pmod{n}$ for all $k \in \{1,2,\dots,n\}$.

确定哪些正整数 $n$ 具有以下属性：

对于与 $n$ 互质的所有整数 $m$，存在一个排列 $\pi: \{1,2,\dots,n\} \ 到 \{1,2,\dots,n\}$，使得 $\pi(\pi(k)) \equiv mk \pmod{n}$ 对于 \{1,2,\dots,n\}$中的所有$k \。