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Math Help - Palindromes

  1. #1
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    Palindromes

    Any help would be appreciated on the following.

    A palindrome is a number that reads same backwards as fowards (for instance, 373 and 521125 are palindromes). Prove that any palindrome with an even number of digits is divisible by 11.

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by lyla View Post
    Any help would be appreciated on the following.

    A palindrome is a number that reads same backwards as fowards (for instance, 373 and 521125 are palindromes). Prove that any palindrome with an even number of digits is divisible by 11.

    Thanks in advance.
    Hi lyla,

    Intuitively, this is obvious because you add and subtract each of the digits;

    Lemma: If k is even, 10^(k) = 99....9 (even # of 9's = k) + 1
    ex. k =2; 10^2 = 100 = 99 + 1
    k = 4; 10^4 = 10000 = 9999 + 1

    Lemma 2: if k is odd, 10^(k) = 10....01 (k-1 0's) - 1
    ex. k = 3; 1000 = 1001 - 1
    Which we know 1001 is div. by 11

    Theorem: n = (a_k*a_(k-1)*...*a_2*a_1*a_0)_10; 11|n iff 11|summation[(-1)^(i)*a_i, from i = 0...k]

    With the summation above, without loss of generality, a_k - a_(k-1) + a_(k-2) - ...

    Proof: n = a_k*10^k + a_(k-1)*10^(k-1) + ... + a_2*10^2 + a_1*(10) + a_0

    Note that a_2*10^2 = 99 + 1 and a_1*10 = (11 - 1), etc.
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    Thus, it should be obvious from this that just adding and subtracting consecutive terms will determine if the number is div. by 11. With palindromes, since you're adding and subtracting the same number of digits, it will = 0, and thus will be divisible by 11.

    Q.E.D.
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  3. #3
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    Quote Originally Posted by lyla View Post
    Any help would be appreciated on the following.

    A palindrome is a number that reads same backwards as fowards (for instance, 373 and 521125 are palindromes). Prove that any palindrome with an even number of digits is divisible by 11.

    Thanks in advance.
    It has been answered.
    Klicken Hier.
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  4. #4
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    Thank you very much for the clarification.
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