Ponder This Challenge - December 1999 - Zero-less multiply of 2**1999

Ponder This Challenge:

Millennial Puzzle

The puzzlemaster misstated the December puzzle the first time around. Let's try again.

Consider the integer N=2^1999 (2 raised to the 1999 power). Is there a positive integer multiple of N whose decimal representation does not contain the digit 0? How would you construct such an integer or prove that it doesn't exist?

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Solution

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  Yes. Work from right to left. Start with N, and notice that its rightmost digit is not zero. Find the rightmost zero; say it's in the Kth position. Then add (10^K)*N (that is, N shifted over K positions) to change that digit without affecting any to the right of it.

  Now find the next zero, and again add (10^J)*N for some appropriate J. With each iteration, the rightmost zero moves further to the left. Continue until the rightmost zero is at least 1999 places to the left (or there are none). Now our integer is (10^1999)*A + B, where B has no zeros.

  Subtracting off (10^1999)*A = (2^1999)*(5^1999)*A, we are left with B, which is a multiple of (2^1999) with no zeros in its decimal representation.

Solvers

• Correct submissions for the original puzzle: (None)
• Kunal Shroff (12.6.1999 @ 4:57 PM EST)
• Krishna Alamuru (12.6.1999 @ 3:33 AM EST)
• Sridhar Srinivasan (12.6.1999 @ 1:24 PM EST)
• Joseph DeVincentis (12.6.1999 @ 10:48 AM EST)
• Jimmy Hu (12.6.1999 @ 1:57 PM EST)
• Philippe Fondanaiche (12.6.1999 @ 6:14 PM EST)
• John Feiler (12.7.1999 @ 2:39 AM EST)
• Philip Nanni (12.7.1999 @ 2:52 AM EST)
• Ionel Santa (12.7.1999 @ 5:35 AM EST)
• Ilan Algor (12.7.1999 @ 6:61 AM EST)
• Robert Galejs (12.7.1999 @ 9:10 AM EST)
• Correct submissions for the revised puzzle: (None)
• Joseph DeVincentis (12.7.1999 @ 10:26 AM EST)
• Jimmy Hu (12.7.1999 @ 1:32 PM EST)
• Alamuru Krishal (12.8.1999 @ 3:26 PM EST)
• Stanislav Jordanov (12.9.1999 @ 9:25 AM EST)
• Dean Flatt (12.7.1999 @ 7:06 PM EST)
• James Chan (12.7.1999 @ 9:17 PM EST)
• Sridhar Srinivasan (12.7.1999 @ 11:51 AM EST)
• Dave Biggar (12.12.1999 @ 9:32 AM EST)
• Ionel Santa (12.13.1999 @ 3:22 AM EST)
• Mihai Cioc (12.17.1999 @ 5:41 PM EST)
• Amos Guler (12.21.1999 @ 6:22 AM EST)
• Ming-wei Wang (12.22.1999 @ 9:19 PM EST)
• Antoni & Tomek Tomaszewicz (12.24.1999 @ 5:12 PM EST)
• Brian Carnes (12.27.1999 @ 5:27 PM EST)