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number theory

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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

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