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### ABCDE/4 = EDCBA

```Date: 06/30/2003 at 13:22:35
From:
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
error.

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                    ----------
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

- Doctor Jaffee, The Math Forum
http://mathforum.org/dr.math/
```
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