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Math Help - Find integer solutions to 1/x + 1/y = 1/14

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    Find integer solutions to 1/x + 1/y = 1/14

    Find all solutions where x and y are integers:

    \frac{1}{x} + \frac{1}{y} = \frac{1}{14}

    this can be rearranged to:

    \frac{xy}{x + y} = 14

    I know how to solve diophantine equations of the form

    ax + by = c

    Obviously the given equation is in a different form. How do I solve?
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  2. #2
    Senior Member abhishekkgp's Avatar
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    Re: Find integer solutions to 1/x + 1/y = 1/14

    Quote Originally Posted by VinceW View Post
    Find all solutions where x and y are integers:

    \frac{1}{x} + \frac{1}{y} = \frac{1}{14}

    this can be rearranged to:

    \frac{xy}{x + y} = 14

    I know how to solve diophantine equations of the form

    ax + by = c

    Obviously the given equation is in a different form. How do I solve?
    \frac{1}{14}= \frac{1}{x}+ \frac{1}{y}< \frac{1}{|x|}+ \frac{1}{|y|}. Let |x|<|y| so \frac{1}{14}<\frac{1}{|x|} + \frac{1}{|y|} < 2 \frac{1}{|x|} which gives |x|< 28. now there are only 56 values of x to be fed into the equation and see which ones give an integral value of y. the computation can be further reduced by using some divisibility by 7 and 2 which are the factors of 14.
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    Senior Member abhishekkgp's Avatar
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    Re: Find integer solutions to 1/x + 1/y = 1/14

    i just found another way. put x=14+a, \, y=14+b, we immediately get ab=14^2.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Find integer solutions to 1/x + 1/y = 1/14

    Quote Originally Posted by VinceW View Post
    Find all solutions where x and y are integers:

    \frac{1}{x} + \frac{1}{y} = \frac{1}{14}

    this can be rearranged to:

    \frac{xy}{x + y} = 14

    I know how to solve diophantine equations of the form

    ax + by = c

    Obviously the given equation is in a different form. How do I solve?


    \frac{1}{x} + \frac{1}{y} = \frac{1}{14}


    \frac{x+y}{xy} =  \frac{1}{14}


    xy = 14(x+y)


    xy-14x-14y+196=196


    (x-14)(y-14) = 196
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