Proof: Palindromes Divisible by 11

• Math100
In summary, a palindrome with an even number of digits is divisible by 11. This is because the decimal expansion of the palindrome can be split into two parts, each with the same digits in reverse order. This results in a difference of 0 when the digits are subtracted from each other, making the sum of these differences 0. Therefore, 11 must divide the original palindrome.
Math100
Homework Statement
A palindrome is a number that reads the same backward as forward (for instance, ## 373 ## and ## 521125 ## are palindromes). Prove that any palindrome with an even number of digits is divisible by ## 11 ##.
Relevant Equations
None.
Proof:

Suppose ## N ## is a palindrome with an even number of digits.
Let ## N=a_{m}10^{m}+\dotsb +a_{2}10^{2}+a_{1}10+a_{0} ##, where ## 0\leq a_{k}\leq 9 ##, be the
decimal expansion of a positive integer ## N ##, and let ## T=a_{0}-a_{1}+a_{2}-\dotsb +(-1)^{m}a_{m} ##.
Note that ## m ## is odd.
Then ## N=a_{0}10^{m}+\dotsb +a_{m-1}10+a_{m} ##.
This means ## a_{i}=a_{m-i} ## for ## 0\leq i\leq m ##.
Since ## m ## is odd, it follows that ## T=(a_{0}-a_{m})+(a_{2}-a_{m-1})+\dotsb +(a_{m-1}-a_{1})\implies 0+0+\dotsb +0=0 ##.
Thus ## 11\mid T\implies 11\mid N ##.
Therefore, any palindrome with an even number of digits is divisible by ## 11 ##.

Delta2 and fresh_42
Looks ok, except that it is better to write at the end
... ## T=(a_{0}-a_{m})+(a_{2}-a_{m-1})+\dotsb +(a_{m-1}-a_{1})= 0+0+\dotsb +0=0 ##.

Delta2 and Math100

1. What is a palindrome?

A palindrome is a word, phrase, or sequence that reads the same backward as forward, such as "racecar" or "madam".

2. How do you determine if a number is divisible by 11?

To determine if a number is divisible by 11, you can use the divisibility rule for 11, which states that if the difference between the sum of the digits in the odd places and the sum of the digits in the even places is either 0 or a multiple of 11, then the number is divisible by 11.

3. What is the proof that palindromes are divisible by 11?

The proof for palindromes being divisible by 11 is based on the divisibility rule for 11. Since palindromes have the same digits in the same order when read backward and forward, the difference between the sums of the digits in the odd and even places is always 0, making them divisible by 11.

4. Are there any exceptions to the proof for palindromes divisible by 11?

Yes, there are a few exceptions to the proof for palindromes divisible by 11. These exceptions occur when the number of digits in the palindrome is even and the sum of the digits in the odd and even places is not 0, but rather a multiple of 11. For example, the number 121 is a palindrome and is divisible by 11, but the number 122 is not divisible by 11.

5. Can this proof be applied to numbers other than palindromes?

Yes, the proof for palindromes divisible by 11 can be applied to any number that has the same digits in the same order when read backward and forward. This includes numbers that are not palindromes but have repeating patterns, such as 121212 or 1234321.

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