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Thread: prove that then n is divisible by 11

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    prove that then n is divisible by 11

    Prove that is n is a positive integer such that the alternately adding and subtracting its digits in decimal representation yeilds divisible by 11 number, then n is divisible by 11.
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    Quote Originally Posted by mandy123 View Post
    Prove that is n is a positive integer such that the alternately adding and subtracting its digits in decimal representation yeilds divisible by 11 number, then n is divisible by 11.
    $\displaystyle n=a_0+a_1~10+a_2~10^2+...+a_k~10^k$

    Now $\displaystyle 10^r \equiv (-1)^r \mod 11$

    so:

    $\displaystyle n \equiv a_0 - a_1 + a_2 +\ ...\ +(-1)^ka_k \mod 11 $

    That is $\displaystyle n$ is congurent to the number obtained by alternately adding and subtracting its digits in decimal representation, but $\displaystyle n$ is congruent to $\displaystyle 0$ modulo $\displaystyle 11$, and therefore so is alternately adding and subtracting its digits in decimal representation.

    RonL
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