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Thread: Find all possible values of a divisor

  1. #1
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    Find all possible values of a divisor

    $\displaystyle m=6k+1$.
    $\displaystyle n=3k-1$.
    $\displaystyle k$ is fixed odd integer.
    $\displaystyle a|m$ and $\displaystyle a|n$.
    Find all possible values of $\displaystyle a$.

    I know that $\displaystyle m$ is odd and $\displaystyle n$ is even. I know
    $\displaystyle m=...,-17,-5,7,19,31,..$
    $\displaystyle n=...,-10,-4,2,8,14,20,...$

    I can find that $\displaystyle a=(6k+1)/q=(3k-1)/r$ (where q and r are any integers). I can even solve for k=[q(3k-1)-r]/[6r]. I guess I can say that when k is an integer in that format, then a divides m and n (or something along those lines). But that's all I've got.

    Frankly, I think the answer is $\displaystyle a=\pm1$, but cannot prove it.

    BTW, this is a take home quiz, so don't give answers. A hint of where to go would be nice, though.

    Edit: My friend suggests Euclidean Algorithm on the definition of $\displaystyle m$ and $\displaystyle n$ (I don't have time to try it now, so I can't give feedback).
    Last edited by MSUMathStdnt; Sep 15th 2011 at 05:08 PM.
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