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Date: 06/30/2003 at 13:22:35
Subject: Big Number Puzzle

"It is rumoured that there is a five-digit number that, when it is 
quartered, gives an answer which is its digits in reverse order."

For example, if the number were 54,321 (which it isn't) a quarter 
would be 12,345 (which it isn't).

Date: 06/30/2003 at 21:54:15
From: Doctor Jaffee
Subject: Re: Big Number Puzzle

Hi Molly,

This was a very challenging problem, but once I figured out the system 
to solve it, I was able to get the answer with hardly any trial and 

Here is what I did. Let's call the number abcde. I wrote down on a 
paper two equations. First,

      edcba                      edcba
   --------    and second,       x  4
   4) abcde                    ----------

Now, a basic rule of arithmetic is that only numbers whose last two 
digits are evenly divisible by 4 can themselves be divided by 4. 
In other words, 'de' must be divisible by 4. That tells me that 'a' 
can't be 0,1,2,3,4,5,6, or 7. If 'a' were 0 then abcde would only be a 
4-digit number and if 'a' were one of the other numbers 'e' would have 
to be 1. But that can't be because 4 can only be divided into even 
numbers. So 'a' must be 8 or 9, which would make 'e' = 2. Then a 
couldn't be 9 because 9 x 4 = 36 and that would mean e = 6 and we've 
already established that e must be 2.

In the two equations above, replace each of the four 'a's with 8 and 
replace each of the 4 'e's with 2. Then continue working both the 
division problem and the multiplication problem together to figure out 
what b, d, and c must equal.

Give it a try and if you want to check your solution with me or if you 
have difficulties or other questions write back to me and I'll try to 
help you some more. Good luck,

- Doctor Jaffee, The Math Forum 
Associated Topics:
High School Puzzles
Middle School Puzzles

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