Four-Digit Palindromes

Date: 10/21/98 at 16:32:10
From: Jessica Evans
Subject: Palindrome

Hi, 

This is a question from my Thinking Mathmatically class. I have figured 
out all of the possiblities of Palindromes but have no idea why it 
works. Please help.

A number like 12321 is a called a palindrome because it reads the same 
backward and forward. A friend of mine claims that all palindromes with 
four digits are exactly divisible by eleven. Are they?

Thank you.

Date: 10/22/98 at 11:01:53
From: Doctor Nick
Subject: Re: Palindrome

Hi Jessica -

Yes, four digit palindromes are always divisible by 11. 

Suppose a and b are digits, and we consider the palindrome that looks 
like abba.

Now, abba = 1000*a + 100*b + 10*b + a.

We can rearrange this to 1000*a + a + 100*b + 10*b = 1001*a + 110*b.

Now, 1001 = 7 * 11 * 13, and 110 = 2 * 5 * 11, so:

   abba = 1001*a + 110*b = 11 * (7*13*a + 2*5*b)

So abba is a multiple of 11.

Here's a specific example:

 3223 = 3000+200+20+3 = 3003+220 = 11*273+11*20
      = 11*293

In fact, more is true: every palindrome with an even number of digits 
is divisible by 11. You can show this the same way as for four digits, 
but you have to be a bit more general. I'll let you think about it, 
and if you're interested in the details, write back and I'll write 
them up.  

What it all comes down to is this:

   10^k + 10^j

is always divisible by 11 if k+j is an odd number.

have fun,

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