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Thread: Solutions to congruence using Chinese remainder theorem

  1. #1
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    Solutions to congruence using Chinese remainder theorem

    f(x) = x^5 + 3x^2 + 4
    Find all solutions to the congruence f(x) congruent to 0 mod 12 by using the Chinese remainder theorem.

    I'm not sure exactly how to do this properly.

    I started by creating two congruences since 12 = 4 x 3.
    I manually tried entries 0 to 3 for mod 4 and entries 0 to 2 for mod 3.
    The only common solution is x=2, so I tried x = 2 as well as x= 5, 8, 11 (because of mod 3) and x= 6, 10 (because of mod 4) in the original polynomial.
    My final answer is x = 2, 5, or 8.
    1) Is this right?
    2) Whether or not it's right, how do I work this problem and present the solution properly?
    Thanks in advance.
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  2. #2
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    Re: Solutions to congruence using Chinese remainder theorem

    $\displaystyle x \equiv 2 \pmod 3$

    and

    $\displaystyle x \equiv 0,1,2 \pmod 4$

    Combining these we get 3 systems each of which can be solved using CRT

    your answer is correct
    Thanks from patrickmanning
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  3. #3
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    Re: Solutions to congruence using Chinese remainder theorem

    Thank you for the reply.

    "Combining these we get 3 systems each of which can be solved using CRT"

    So, the three systems are:
    1) x= 2mod3 and x= 2mod4 which gives the answer x = 2
    2) x= 2mod3 and x= 1mod4 which gives the answer x = 5
    3) x= 2mod3 and x= 0mod4 which gives the answer x = 8

    Right?
    Last edited by patrickmanning; Oct 6th 2018 at 10:20 AM.
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  4. #4
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    Re: Solutions to congruence using Chinese remainder theorem

    Correct

    $\displaystyle x \equiv 2,5,8 \pmod {12}$
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