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Thread: weird problem

  1. #1
    mms
    mms is offline
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    weird problem

    Hi, i didnt know where to put this problem

    Show that every positive integer has a multiple made of only sevens and zeros.

    thanks for your help.
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  2. #2
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    So we have an integer $\displaystyle n$ and we want an integer $\displaystyle m$ such that $\displaystyle nm = s$, where $\displaystyle s$ consists of only 7's and 0's.

    This is a constructive proof, but I'm sure there are other ways of doing it.

    Firstly take a number of the form $\displaystyle x = 1111\ldots 111$, constisting only of 1's. Then calculate $\displaystyle r = x \mod{n}$

    If $\displaystyle r=0$ then we can simply multiply r by 7 and we are done.

    If not then we find 2 other numbers of the form $\displaystyle y_1=111\ldots 111$ > $\displaystyle y_2 = 11\ldots 111$ such that $\displaystyle y_1 \equiv y_2 \mod{n} $. Then $\displaystyle y_1 - y_2 $ is divisible by n and consists only of 1's and 0's, we simply multiply this by 7 and we are done.

    The justification for the existence of such $\displaystyle y_i$ is that if we take $\displaystyle n$ elements from the set $\displaystyle \left\{1,11,111,1111,\ldots \right\}$ , then there must be at least 2 with the same remainder mod n.

    Pomp.
    Last edited by pomp; Jul 13th 2009 at 09:44 AM. Reason: typos. lots of them.
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