Question 192
For every pair of positive integers $n, m$ with $n \lt m$, denote by $s(n, m)$ the number of positive integers in the range $[n, m]$ that are coprime with $m$. Find all positive integers $m \ge 2$ that satisfy the following conditions:
i) $\frac{s(n, m)}{m-n} \ge \frac{s(1, m)}{m}$ for all $n = 1, 2, \dots, m-1$;
ii) $2022^m + 1$ is divisible by $m^2$.
i) $\frac{s(n, m)}{m-n} \ge \frac{s(1, m)}{m}$ for all $n = 1, 2, \dots, m-1$;
ii) $2022^m + 1$ is divisible by $m^2$.