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Math Help - Mupltiple Choice

  1. #1
    Member Rimas's Avatar
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    Mupltiple Choice

    If m and n are integers such that 2m-n=3 what are the possible values of m-2n?

    [a]-3 only
    [b] 0 only
    [c] only multiples of 3
    [d]any integer
    [e] none of these
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  2. #2
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    Quote Originally Posted by Rimas View Post
    If m and n are integers such that 2m-n=3 what are the possible values of m-2n?
    Let x=-n
    Let us solve the Diophantine equation,
    2m+x=3
    Note,
    (m,n)=(1,1) is a trivial solution.

    Then all solutions are, (note that \gcd(2,1)=1)
    m=1-t
    x=1+2t
    That means,
    m-2n=m+2x=1-t+2(1+2t)=1-t+2+4t=3+3t=3(1+t)
    It must be a multiple of 3.
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  3. #3
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    Hello, Rimas!

    Another approach . . .


    If m and n are integers such that 2m-n\,=\,3,
    what are the possible values of m-2n ?

    [a] -3 only . . [b] 0 only . . [c] only multiples of 3
    . . . [d] any integer . . [e] none of these

    We are given: . 2m - n \,-\,3\quad\Rightarrow\quad n\:=\:2m - 3

    Then m - 2n\;=\;m - 2(2m-3)\;=\;m - 4m + 6 \;=\;6 - 3m \;=\;3(2-m)


    Therefore, m-2n must be a multiple of 3 . . . answer choice (c)

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