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### Reversing the Digits

```Date: 05/29/2003 at 19:00:19
From: Sarah
Subject: Reversing digits

Certain pairs of numbers that have two digits have the same product
when you reverse their digits. For example:

12 x 42 = 21 x 24
504       504

24 x 63 = 42 x 36
1512      1512   ...etc

My teacher says there are 11 other possibilities. Is there a pattern
to follow to find them, or should I randomly check every number until
it works, as I have been doing up until now?
```

```
Date: 05/29/2003 at 23:21:45
From: Doctor Ian
Subject: Re: Reversing digits

Hi Sarah,

I don't know if there's a formula, but maybe we can be a little more

Suppose our numbers look like 'ab' and 'cd'.  Then we can represent
them as

10a + b

and

10c + d

Do you see why?  Using a representation like this is often the key to
solving this kind of problem.

We're going to multiply the numbers to get

(10a + b)(10c + d)

and we're going to multiply them with the digits switched,

(10b + a)(10d + c)

and the products are supposed to be equal:

(10a + b)(10c + d) = (10b + a)(10d + c)

If we expand this, we get

100ac + 10bc + 10ad + bd = 100bd + 10ad + 10bc + ac

Now, that looks like a mess, but if we look closely, we can see that
10bc and 10ad appear on both sides of the equation.  So they aren't
really contributing anything.  Let's get rid of them:

100ac + bd = 100bd + ac

This is looking more interesting. Let's subtract bd from both sides,
and ac from both sides:

100ac - ac = 100bd - bd

99ac = 99bd

ac = bd

So what we're _really_ looking for are pairs of single-digit factors
that give the same product. For example,

a c   b d
24 = 3*8 = 6*4

So we would expect

10*3 + 6 = 36

and

10*8 + 4 = 84

to be one of the pairs we're looking for.  Does it work?

36 * 84 = 63 * 48

3024 = 3024

Excellent!  Can you take it from here?

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