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Modulo power identity

Published on Saturday, 28th November 2015, 07:00 pm; Solved by 212;
Difficulty rating: 60%

Problem 536

Let S(n) be the sum of all positive integers m not exceeding n having the following property:
a m+4a (mod m) for all integers a.

The values of m ≤ 100 that satisfy this property are 1, 2, 3, 5 and 21, thus S(100) = 1+2+3+5+21 = 32.
You are given S(106) = 22868117.

Find S(1012).