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

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

    Find all solutions where x and y are integers:

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

    this can be rearranged to:

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

    I know how to solve diophantine equations of the form

    $\displaystyle ax + by = c$

    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:

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

    this can be rearranged to:

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

    I know how to solve diophantine equations of the form

    $\displaystyle ax + by = c$

    Obviously the given equation is in a different form. How do I solve?
    $\displaystyle \frac{1}{14}= \frac{1}{x}+ \frac{1}{y}< \frac{1}{|x|}+ \frac{1}{|y|}$. Let $\displaystyle |x|<|y|$ so $\displaystyle \frac{1}{14}<\frac{1}{|x|} + \frac{1}{|y|} < 2 \frac{1}{|x|}$ which gives $\displaystyle |x|< 28$. now there are only 56 values of $\displaystyle x$ to be fed into the equation and see which ones give an integral value of $\displaystyle y$. the computation can be further reduced by using some divisibility by $\displaystyle 7$ and $\displaystyle 2$ which are the factors of $\displaystyle 14$.
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  3. #3
    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 $\displaystyle x=14+a, \, y=14+b$, we immediately get $\displaystyle ab=14^2$.
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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:

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

    this can be rearranged to:

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

    I know how to solve diophantine equations of the form

    $\displaystyle ax + by = c$

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


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


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


    $\displaystyle xy = 14(x+y)$


    $\displaystyle xy-14x-14y+196=196$


    $\displaystyle (x-14)(y-14) = 196$
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