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### ABCD x 4 = DCBA

```
Date: 08/22/97 at 05:55:57
From: Denise
Subject: Puzzle

ABCD x 4 = DCBA. What are the values of A,B,C and D? The values are
between 0-9 and each is different.

We have been trying for days on paper and have absolutely no idea. Can

Thanks so much for your time.
```

```
Date: 08/22/97 at 15:05:26
From: Doctor Wilkinson
Subject: Re: Puzzle

First of all, A cannot be greater than 2, because 4 x 3 = 12, and
we would have another digit to the left of the D in the product.
A cannot be 0, because we don't write numbers with a 0 on the left.
And A cannot be 1, because 4 times anything is even, so we can't get
a 1 as the last digit of the product.  So A must be 2, and we have

2BCD
4
----
DCB2

Now looking at the leftmost digit of the product, we see that D must
be 8 or 9, and it can't be 9, because 4 X 9 is 36, and we wouldn't get
the 2 in the product.  So we have

2BC8
4
----
8CB2

Now since there's no carry into the leftmost position of the product
(otherwise we would have a 9 and not an 8), B must be 0, 1, or 2, and
it's not 2, because we've already used 2. It's not 0 either, but I'm
going to let you figure out why. Assuming you've figured that out, we
have to have B = 1, and you should be able to finish it off from
there.  (There are only a few possibilities at this point).

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
Elementary Puzzles
Middle School Puzzles

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