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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?

See The Solution Submitted by Federico Kereki    
Rating: 3.5000 (4 votes)

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More thoughts | Comment 2 of 4 |

... on probability

For n = 1 to 377 (where I stopped the program), such representation is always found in a "short" time, comparatively speaking in regard to the size of the numbers.

At n=375, for example, 2^n =       76957043352332967211482500195592995713046365762627825523336510555167425334955489475418488779072100860950445293568, which, multiplied by 1258101, gives a number with no zeros:         96819733198613458381733344978575743499579345812327829918735187265966692981332836264513476151439389165262616174283194368.

That is, 2^375 is a 113-digit number, and the number of possibilities for the last 113 digits of its multiples is a cycle of values whose cardinality is a 79-digit number, and we've looked through only about 1 and a quarter million to find what we're looking for.

  Posted by Charlie on 2004-08-27 15:41:52
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