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Math Help - Linear Diophantine Equation question

  1. #1
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    Linear Diophantine Equation question

    hi everyone .

    i need a little help in a direction of how to start this question...

    "For what values of c does 2x + 5y = c have exactly one strictly positive integer solution?"


    im not very good with applying constraints to things . ive been told that a little bit of enumeration will be needed in near the end to get the answer though.

    thank you for your time!
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by melissaanderson View Post
    hi everyone .

    i need a little help in a direction of how to start this question...

    "For what values of c does 2x + 5y = c have exactly one strictly positive integer solution?"


    im not very good with applying constraints to things . ive been told that a little bit of enumeration will be needed in near the end to get the answer though.

    thank you for your time!
    Ok. Start with telling us why a LDE would only have one positive solution?
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  3. #3
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    Thm: Given any c\geq 0, there exists exactly one solution (x,y) such that y=0 or y=1. Proof: Given c, it is either even or odd. If c is even then (\frac{c}2,0) is a solution. If c is odd, then c-5 is even, so (\frac{c-5}2,1) is a solution. Call this initial solution (x_0,y_0). (Note that if c is odd but c-5<0, no positive solution exists, not even this initial one.)

    Cor: Now for any n=1,2,..., (x_0-5n,y_0+2n) is also a solution. To be a strictly positive solution, x_0-5n\geq 0, so x_0\geq5 for a second solution to exist.

    Thus, for a second solution to NOT exist, y_0=0 or y_0=1, and x_0<5. So, consider the function c(x,y)=2x+5y on the domain (x,y)=(0 \to 4, 0 \to 1). These resulting values of c are the ones for which only one positive (x,y) solution exists.
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