Problem 455: Powers With Trailing Digits

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Let f(n) be the largest positive integer x less than 109 such that the last 9 digits of nx form the number x (including leading zeros), or zero if no such integer exists.

For example:

f(4) = 411728896 (4411728896 = ...490411728896) f(10) = 0 f(157) = 743757 (157743757 = ...567000743757) Σf(n), 2 ≤ n ≤ 103 = 442530011399 Find Σf(n), 2 ≤ n ≤ 106.