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Idempotents

Problem 407

Published on Sunday, 23rd December 2012, 10:00 am; Solved by 765

If we calculate a2 mod 6 for 0 ≤ a ≤ 5 we get: 0,1,4,3,4,1.

The largest value of a such that a2 ≡ a mod 6 is 4.
Let's call M(n) the largest value of a < n such that a2 ≡ a (mod n).
So M(6) = 4.

Find ∑M(n) for 1 ≤ n ≤ 107.