Idempotents

If we calculate $a^{2} mod 6$ for $0 \leq a \leq 5$ we get: $0, 1, 4, 3, 4, 1$ .

The largest value of $a$ such that $a^{2} \equiv a mod 6$ is $4$ .
Let's call $M (n)$ the largest value of $a < n$ such that $a^{2} \equiv a (\mod n)$ .
So $M (6) = 4$ .

Find $\sum M (n)$ for $1 \leq n \leq 10^{7}$ .

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.