Question 120
For every positive integer $M \ge 2$, find the smallest real number $C_M$ such that for any integers $a_1, a_2, \dots, a_{2023}$, there always exists some integer $1 \le k \lt M$ such that
$$
\left\{ \frac{ka_1}{M} \right\} + \left\{ \frac{ka_2}{M} \right\} + \dots + \left\{ \frac{ka_{2023}}{M} \right\} \le C_M.
$$
Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer.
$$
\left\{ \frac{ka_1}{M} \right\} + \left\{ \frac{ka_2}{M} \right\} + \dots + \left\{ \frac{ka_{2023}}{M} \right\} \le C_M.
$$
Here, $\{x\}$ is the unique number in the interval $[0, 1)$ such that $x - \{x\}$ is an integer.