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Math Help - Find all solutions

  1. #1
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    Find all solutions

    Find all six solutions to 12x = 24 mod 30 (= is congruence not equality...)

    Book says to use this theorem:

    Suppose
    d divides a, c, m. Then every solution of

    (
    a/d)x =(c/d) mod (m/d)

    corresponds to
    d distinct solutions (modulo m) of

    ax = c mod m:


    Not sure how to show this because it seems to me that there would be infinite solutions. How do I do this?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by jzellt View Post
    Find all six solutions to 12x = 24 mod 30 (= is congruence not equality...)

    Book says to use this theorem:

    Suppose
    d divides a, c, m. Then every solution of

    (
    a/d)x =(c/d) mod (m/d)

    corresponds to
    d distinct solutions (modulo m) of

    ax = c mod m:


    Not sure how to show this because it seems to me that there would be infinite solutions. How do I do this?
    Of course there are infinite solutions in the integers, but it is asking for solutions in \mathbb{Z}_{30}
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  3. #3
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    and how do I find those solutions in Z30
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  4. #4
    Super Member Bacterius's Avatar
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    The theorem the book asks you to use is the Linear Congruence Theorem which is used to solve linear congruences as yours
    Note that as Drexel said, the values of x "wrap around" (x = 30 is equivalent to x = 0, x = 31 is equivalent to x = 1, ...). But the LCT basically provides a general formula that gives all solutions to your congruence, so you don't have to worry about whether there are infinite solutions, since the formula gives them all
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