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Math Help - A multiple of 17

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    Question A multiple of 17

    Let x and y be integers such that 2x+3y is a multiple of 17. Show that 9x+5y must also be a multiple of 17.

    Thank you very much.
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  2. #2
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    Quote Originally Posted by Jenny20 View Post
    Let x and y be integers such that 2x+3y is a multiple of 17. Show that 9x+5y must also be a multiple of 17.

    Thank you very much.
    We have,
    2x+3y=17k
    Let us find all solutions for a given k>0.

    This is a linear diophantine equations with two variables.
    We see that,
    2(-1)+3(1)=1
    Thus,
    2(-17k)+3(17k)=17k
    Thus, all solutions are,
    x=-17k+3t
    y=17k-2t
    For some integer t.
    Thus,
    9x+5y is,
    9(-17k+3t)+5(17k-2t)=-153k+27t+85k-10t
    Combine,
    -68k-17t=17(-4-t)
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