0%

Problem 455


Problem 455


Powers With Trailing Digits

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.


尾数与指数相同的幂

对于任意整数n,记f(n)是小于109且满足nx的最后9位数字恰好等于x的最大的数。如果这样的数不存在,则f(n)等于0。

例如:

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

求Σf(n), 2 ≤ n ≤ 106