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Problem 455

Published on Saturday, 18th January 2014, 10:00 pm; Solved by 282

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.