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Math Help - Congruence Equations

  1. #1
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    Congruence Equations

    Could someone help me find the solutions to the following:

    (x^2)+x=0 in Z5.
    (x^2)+x=0 in Z6.

    Thanks. MK
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  2. #2
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    Hello, MK!

    These are of the same type . . . I'll explain the first one.


    x + x .= .0 .(mod 5)

    We can apply standard algebraic methods.

    Factor: .x(x + 1) .= .0 .(mod 5)

    Then: .x .= .0 .(mod 5) . . x .= .5m, for some integer m.

    And: .x+1 .= .0 .(mod 5) . . x .= .5n - 1, for some integer n.

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  3. #3
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    Quote Originally Posted by Soroban View Post
    Hello, MK!

    These are of the same type . . . I'll explain the first one.



    We can apply standard algebraic methods.

    Factor: .x(x + 1) .= .0 .(mod 5)

    Then: .x .= .0 .(mod 5) . . x .= .5m, for some integer m.

    And: .x+1 .= .0 .(mod 5) . . x .= .5n - 1, for some integer n.

    One thing Soroban did not mention is that, the ring of integers multiplication modulo n>1 has no zero-divisors meaning that if, (where n is a prime or power of a prime).
    xy=0
    Then,
    x=0 or y=0
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  4. #4
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    Quote Originally Posted by MKLyon View Post
    (x^2)+x=0 in Z6.
    Same idea but more complex because the ring,
    Z6 has zero divisors.

    Thus,
    x(x+1)=0
    Thus,
    x=0, thus, x=0
    x+1=0, thus, x=5
    x=2 and x+1=3, thus, x=2
    x=3 and x+1=2 no solution.
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