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Math Help - How do I solve this congruence?

  1. #1
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    How do I solve this congruence?

    x^3-9x^2+23x-15=0(mod 143)
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  2. #2
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    Quote Originally Posted by Willie_Trombone View Post
    x^3-9x^2+23x-15=0(mod 143)
    You need to solve x^3 - 9x^2 + 23x - 15\equiv 0(\bmod 11) and x^3 - 9x^2 + 23x - 15\equiv 0 (\bmod 13) simultaneously.

    In the first congruence we can factor, (x+6)(x+8)(x+10)\equiv 0(\bmod 11) this gives the solutions, x\equiv 1,3,5(\bmod 11). In the second congruence we can factor, (x+8)(x+10)(x+12)\equiv 0(\bmod 13) this gives the solutions, x\equiv 1,3,5(\bmod 13). Now use the Chinese remainder theorem to get possibly 9 solutions up to congruence modulo 11\cdot 13 = 143.
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