# number theory

posted by .

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 -

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

## Similar Questions

2. ### math

Indicate whether the deductive reasoning used is an example of affirming the hypothesis or denying the conclusion. 8 If a number is divisible by 3, then the sum of the digits of that number is divisible by 3. The sum of the digits …
3. ### teaching math

Indicate whether the deductive reasoning used is an example of affirming the hypothesis or denying the conclusion. If a number is divisible by 3, then the sum of the digits of that number is divisible by 3. The sum of the digits of …
4. ### teaching math

Indicate whether the deductive reasoning used is an example of affirming the hypothesis or denying the conclusion. If a number is divisible by 3, then the sum of the digits of that number is divisible by 3. The sum of the digits of …
5. ### Math

The question is this: You know that a number is divisible by 6 if it is divisible by both 3 and 2. So why isn't a number divisible by 8 if it is divisible by both 4 and 2?
6. ### Math

I need to know if the deductive reasoning that is used in this example is affirming the hypothesis or denying the conclusion. If a number is divisible by 3, then the sum of the digits of that number is divisible by 3. A number is divisible …
7. ### math

Using the numbers 1 through 9 with no repeats, find a 9 number such that: the first digit is divisible by 1, the first two digits are divisible by 2, the first 3 digits are divisible by 3, and so on until we get to a 9 digit number …
8. ### algebra

A palindrome is a number that reads the same forwards as it does backwards. For example, 98789 is a 5-digit palindrome, and 7337 is a 4-digit palindrome. How many 3-digit palindromes are there?