Posted by **Laura** on Sunday, April 19, 2009 at 2:53pm.

How do you prove that any palindrome, a number that reads the same backwards and forwards, with an even number of digits is divisible by 11?

I know how to show a number is divisible by 11, but how do you prove the above case?

- number theory -
**Reiny**, Sunday, April 19, 2009 at 3:19pm
How do we know a number is divisible by 11?

e.g. is 164587467 divisible by 11?

Add up the odd-number positioned digits

1+4+8+4+7 = 24

6+5+7+6 = 24

if their sum has a difference of zero or a multiple of 11, then the original number divides by 11

e.g. 16437003

sum of even-number positioned digits = 6+3+0+3 = 12

sum of odd-number positioned digits = 1+4+7+0 = 12

now forming the palindrome would obviously make the original odd-number positioned numbers into the even-number positioned digits and vice versa.

so their sum would still have either a difference of zero or a multiple of 11.

BTW, the same property would result if you formed the palindrome of a number with an even number of digits

e.g. 1737032 and 2307371 are palidromes, and both are divisible by 11

the sum of the digits of their respective groupings are 6 and 17

and the difference between these two is 11

## Answer this Question

## Related Questions

- algebra - A palindrome is a number that reads the same forwards as it does ...
- Maths - A palindrome number is one that reads the same both forwards and ...
- math - What are the divisibility short cuts that help you out? Odd numbers can ...
- teaching math - Indicate whether the deductive reasoning used is an example of ...
- teaching math - Indicate whether the deductive reasoning used is an example of ...
- math - Indicate whether the deductive reasoning used is an example of affirming ...
- math - Using the numbers 1 through 9 with no repeats, find a 9 number such that...
- math - 1. a door code is made up of six unique digits the last digit is 8. 2....
- Math - I need to know if the deductive reasoning that is used in this example is...
- COmbinatorics - A DNA chain is a string of As, Ts, Gs, and Cs. How many DNA ...