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Thread: Diophantine Equation? Or something?

  1. #1
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    Diophantine Equation? Or something?



    Lost. Sorry.
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  2. #2
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    $\displaystyle 4x^2 - y^2 = (2x + y)(2x - y) = 480$.

    Now list all the pairs of numbers that can multiply to 480:

    $\displaystyle (480, 1); (240, 2); (160, 3); (120, 4); (96, 5); (80, 6); $

    $\displaystyle (60, 8); (48, 10); (40, 12); (32, 15); (30, 16); (24, 20)$

    The pairs with one odd number and one even number will result in y not being an integer, so you can throw those out. Also, the pairs such that the sum of the numbers is not divisible by 4 will result in x not being an integer, so you can throw those out.

    1. $\displaystyle 2x + y = 120$

    $\displaystyle 2x - y = 4$

    $\displaystyle 4x = 124$

    $\displaystyle x = 31, y = 58$ is a solution.

    2. $\displaystyle 2x + y = 60$

    $\displaystyle 2x - y = 8$

    $\displaystyle 4x = 68$

    $\displaystyle x = 17, y = 26$ is a solution.

    3. $\displaystyle 2x + y = 40$

    $\displaystyle 2x - y = 12$

    $\displaystyle 4x = 52$

    $\displaystyle x = 13, y = 14$ is a solution.

    4. $\displaystyle 2x + y = 24$

    $\displaystyle 2x - y = 20$

    $\displaystyle 4x = 44$

    $\displaystyle x = 11, y = 2$ is a solution.

    Hence the solutions are $\displaystyle (31, 58); (17, 26); (13, 14); (11, 2)$.
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  3. #3
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    Very detailed. Thank you muchly.
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