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No zeroes allowed (Posted on 2004-08-27) Difficulty: 3 of 5
Is there any integer multiple of N=2^2004 that includes no zeroes in its decimal representation?

  Submitted by Federico Kereki    
Rating: 3.5000 (4 votes)
Solution: (Hide)
Yes, there are such numbers. Start with N, analyzing its digits from right to left. Each time you find a zero (say, at the Mth position) add N times 10^M; this will wipe out the zero (since N doesn't end in zero) without affecting the digits to the right of the Mth place.

Keep on doing this until the rightmost 2004 digits are non-zero. At that time, you'll have a multiple of N that can be expressed as X.10^2004+Y, and Y has no zeroes in it. As the sum is a multiple of N, and X*10^2004 is also a multiple of N, it follows that Y must be a multiple of N, so Y is a number as we wanted.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: SolutionCharlie2004-08-27 21:40:55
SolutionSolutionNick Hobson2004-08-27 17:03:45
More thoughtsCharlie2004-08-27 15:41:52
Some ThoughtsFirst thoughtsCharlie2004-08-27 15:13:48
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