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
clever about finding the answers. 

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/