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Problem of the Week 837

Casting Out Zeros

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One may perform the following two operations on a natural number:

  1. Multiply it by any natural number;
  2. Delete zeros in its decimal representation.
For any natural number n, can one perform a sequence of these operations that will transform n to a one-digit number?

Source: 1991 Leningrad Mathematical Olympiads


This seems to be a hard one. One week after the original posting (May 2, 1997), Tom Halverson had received no solutions (either via email or by Macalester students). Here is a way to turn 11 into a single digit number.

  1. multiply by 82 to get 902.
  2. delete the 0
  3. multiply by 9 to get 828
  4. multiply by 25 to get 20700
  5. delete the 0s to get 27
  6. multiply by by 4 to get 108
  7. delete the 0 to get 18
  8. multiply by 5 to get 90
  9. delete the 0

© Copyright 1997 Stan Wagon. Reproduced with permission.

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2 October 1998