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Math Help - Congruences and Diophantine equations

  1. #1
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    Congruences and Diophantine equations

    How do I use congruences to solve Diophantine equations?
    For example 12x+25y=331.
    I know this can be expressed as 12x=331(mod 25) but I don't know how to solve without just going through the numbers 1-25 to see which one works.

    also how can I find the number of solutions for x^3=x^2(mod 50)?
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  2. #2
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    Quote Originally Posted by Willie_Trombone View Post
    How do I use congruences to solve Diophantine equations?
    For example 12x+25y=331.
    I know this can be expressed as 12x=331(mod 25) but I don't know how to solve without just going through the numbers 1-25 to see which one works.

    also how can I find the number of solutions for x^3=x^2(mod 50)?
    Notice, 12(-2) + 25(1) = 1 \implies 12(-662) + 25( 331) = 331.
    Therefore, all solutions are given by x=-662 + 25t \text{ and }y=331 - 12t, t\in \mathbb{Z}.

    also how can I find the number of solutions for x^3=x^2(mod 50)?
    This can be written as x^2(x-1)\equiv 0(\bmod 50).
    Therefore, x^2(x-1)\equiv 0 (\bmod 2) \text{ and }x^2(x-1)\equiv 0(\bmod 25).
    Can you solve?
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  3. #3
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    haha, sorry I'm still not sure what to do at that point in the second problem. I just really can't seem to get the hang of this material.
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