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 50 - Digit Number (Posted on 2003-11-15)
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: solution | Comment 18 of 39 |
(In reply to solution by Charlie)

Your observation about the powers of 10 mod 13 leads to the conclusion that replacing the powers of 10 by their mod 13 equivalents gives 4*0+10*x+1+4*0 as a mod 13 representation of the 50-digit number since 1+10+9+12+3+4=39 is divisble by 13. Now 10*x+1 is divisble by 13 if x=9, and one may readily check by brute force that 91 is the only two-digit number that ends in 1 and is divisible by 13. Hence x=9 uniquely renders the 50-digit number divisble by 13 as desired.
 Posted by Richard on 2003-11-20 20:35:49

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