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Thread: polynomial

  1. #1
    Junior Member ihmth's Avatar
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    polynomial

    Here's a problem that i can't answer, pls help:
    Find the remainder when 23^125 - 4*23^82 + 5*23^28 + 23^18 is divided by 23^2 + 23 +1.
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  2. #2
    Senior Member topher0805's Avatar
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    You can use long division, although it is a painful process. I would show you but I have to learn the LaTex involved in long division.

    Also, this is not a polynomial. There are no variables, so it is much easier than you think. Simply find the values of the two expressions, then use long division.
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  3. #3
    Junior Member ihmth's Avatar
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    Is this correct? btw the answer is 0

    23 / (23^2 + 24) = remainder: 23
    23^2 / (23^2 + 24) = remainder: 529
    23^3 / (23^2 + 24) = remainder: 1
    23^4 / (23^2 + 24) = remainder: 23
    ...
    ...
    ...
    so 23^{125} / (23^2 + 24) = remainder: 529
    ----125/3 = 41 r:2

    4*23^{82} / (23^2 + 24) = remainder: 23
    ----82/3 = 27 r:1

    5*23^{28} / (23^2 + 24) = Remainder: 23
    ----28/3 = 9 r:1

    23^{18} / (23^2 + 24) = Remainder: 1
    ----18/3 = 6 r:0

    then 529 - 4*23 + 5*23 + 1 = 553...which is divisible by
    23^2 + 23 + 1 = 553

    therefore the remainder is 0 / no remainder
    Last edited by ihmth; Feb 18th 2008 at 04:25 AM.
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