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Math Help - Chinese Remainder Theorem Problem

  1. #1
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    Chinese Remainder Theorem Problem

    I'm not quite sure how to start on this problem.

    Let n_1,n_2,n_3 be mutually prime where n=n_1n_2n_3.

    Let b_1,b_2,b_3 be any integers,then there exists x such that x\equiv b_i\mod  n_i.

    Suppose f(x)=7x^3-x^2+2x-11.

    Suppose f(x)\equiv 0 has a solution mod n_i for i=1,2,3.

    Prove that f(x)\equiv 0 has a solution mod n.

    I don't need to find the solution, just show that one exists.


    Do I use the actual solution x_i of f(x)\equiv 0\mod n_i, or just the fact that it has a solution. Sorry, I'm just lost on this problem.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    For i=1,2,3 let f(x_i) \equiv 0 \mod n_i. Apply the CRT to find an integer a \equiv x_i \mod n_i for i=1,2,3. Then f(a) \equiv f(x_i) \equiv 0 \mod n_i for i=1,2,3 so f(a) \equiv 0 \mod n.
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