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50 - Digit Number (Posted on 2003-11-15) Difficulty: 3 of 5
A number of 50 digits has all its digits equal to 1 except the 26th digit. If the number is divisible by 13, then find the digit in the 26th place.

See The Solution Submitted by Ravi Raja    
Rating: 3.3333 (6 votes)

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re(3): a little more straightforward | Comment 9 of 39 |
(In reply to re(2): a little more straightforward by Richard)

First of all, you mistakenly placed the x in the 25th digit, not the 26th. Second of all, I found a mistake in Eric's solution.

My earlier reasoning was that if you took the first and last 24 ones, you'd be left with 1X,000,000,000,000,000,000,000,000 not 1X. Therefore, 10^24 should be divisible by 13. I checked this on my calculator, not thinking about prime factorization or the fact that the calculator only has 14 significant digits. You are right, it is not divisible by 13.

So I looked again at the "straightforward" solution and found that you cannot cancel any 1's except the last 6. If you cancel out the others, you'd be assuming that 1,000,000 is divisible by 13, when it's really not. Coincidentally, his solution gets the same answer anyway.
  Posted by Tristan on 2003-11-19 19:06:25

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