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Thread: Show that Z=E mod 9, where E is the sum of digits of r (in dec. repres. of r)

  1. #1
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    Show that Z=E mod 9, where E is the sum of digits of r (in dec. repres. of r)

    Hi,

    I have the following problem and I don't know how to go about this... I would really appreciate if you could give me a hand. The problem says:

    "Let $\displaystyle r \in \mathbb{N}$. Show that

    $\displaystyle r \equiv \Sigma mod 9$,

    where $\displaystyle \Sigma$ is the sum of digits of $\displaystyle r$ (in decimal representation of r)."
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  2. #2
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    Quote Originally Posted by jmgilbert View Post
    Hi,

    I have the following problem and I don't know how to go about this... I would really appreciate if you could give me a hand. The problem says:

    "Let $\displaystyle r \in \mathbb{N}$. Show that

    $\displaystyle r \equiv \Sigma mod 9$,

    where $\displaystyle \Sigma$ is the sum of digits of $\displaystyle r$ (in decimal representation of r)."


    If $\displaystyle n=A_k\times 10^k +A_{k-1}\times 10^{k-1}+\ldots+A_1\times 10+A_0$ , and

    since $\displaystyle 10^m=1\!\!\pmod 9\,,\,\,\forall\,m\in\mathnn{N}$ , we get

    $\displaystyle n=\sum\limits^k_{i=0}A_i\times 10^i=\sum\limits^k_{i=0}A_i\!\!\pmod 9$

    Tonio
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