Permutation and Modulus

Given an integer $N$, find a permutation $P = P_1, P_2, \ldots, P_N$ of length $N$ that satisfies the following two conditions: - $P_i \neq i$ $(1 \leq i \leq N)$; - Consider the set which contains $P_i$ $\%$ $i$ for all $1 \leq i \leq N$. This set should have the **minimum** number of **distinct** integers possible, over all permutations $P$. Note that a permutation of length $N$ is an array con