## Idempotents

### Problem 407

Published on Sunday, 23rd December 2012, 10:00 am; Solved by 1005; Difficulty rating: 20%
If we calculate `a`^{2} mod 6 for 0 ≤ `a` ≤ 5 we get: 0,1,4,3,4,1.

The largest value of `a` such that `a`^{2} ≡ `a` mod 6 is 4.

Let's call M(`n`) the largest value of `a` < `n` such that `a`^{2} ≡ `a` (mod `n`).

So M(6) = 4.

Find ∑M(`n`) for 1 ≤ `n` ≤ 10^{7}.