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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(4): solution | Comment 23 of 39 |
(In reply to re(3): solution by Richard)

I don't want to knock you Richard....but you typed this:

we have a=m*(a/m) + a%m=m*q+r. It is quite easy to show that (a+b)%m= (a%m+b%m)%m and (ab)%m= ((a%m)*(b%m))%m, which means we can remainder first before adding or multiplying without affecting the remainder of the result. It is this that makes casting out nines work as a checksum. Since (10^n)%9=1, 176923%9 =(1+7+6+9+2+3)%9=1. The same idea can be applied to digit problems using any value for m, although m=9 is the nicest for decimals, and m=11 is second nicest (since (10^n)%11=(-1)^n). that over. That does not make *any* sense at all!! You're telling me, when you typed it, you thought it would make sense to a normal 17 year old!? E.g. the first equation... a=m*(a/m)...I can understand that, as the 'm's cancel and so you are left with 'a', but've mysteriously added on a%m...and that is equal to m*q + r. What the heck is 'q' and 'r'? said "It is this that makes casting out nines work as a checksum". It is *what*? Here is an is that that makes eggs lay placebo generating coefficients of friction. It makes no sense!!! PLUS....what is "casting" out nines??? What is a "checksum"! If you're going to explain yourself, Richard, please do it properly!
  Posted by Kirk on 2003-11-22 18:49:19

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