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Math Help - Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

  1. #1
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    Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.
    I wonder if there are other ways that can prove this.

    My way of proving this:
    Let P(n) be "5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33." for all positive integers n.
    When n = 1, 5^3+11^3+17^3 = 33*193
    Thus P(1) is true.
    Assume P(k) is true for some positive integers k.
    i.e. 5^(2k+1)+11^(2k+1)+17^(2k+1) = 33N, where N is an integer.
    When n = k+1,
    5^[2(k+1)+1] + 11^[2(k+1)+1] + 17^[2(k+1)+1]
    =5^(2k+3) + 11^(2k+3) + 17^(2k+3)
    =25[5^(2k+1)] + 121[11^(2k+1)] + 289[17^(2k+1)]
    =25[5^(2k+1)+11^(2k+1)+17^(2k+1)] + 96[11^(2k+1)] + 264[17^(2k+1)]
    =25(33N) + 99[11^(2k+1)] - 3[11^(2k+1)] + 264*17^(2k+1)
    =33[25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1)]
    Since N and k are integers,
    25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1) is an integer.
    Therefore, 25N + 3*11^(2k+1) - 11^(2k) + 8*17^(2k+1) is divisible by 33.
    Thus P(k+1) is true.
    By M.I., P(n) is true for all positive integers n.
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  2. #2
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    Re: Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    I think such type of questions are best dealt with by Mathematical Induction.
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  3. #3
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    Re: Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    Another way
    The given expression is divisible by 11 using the fact that 5 = -6 (mod 11) and 17 = 6 (mod 11)
    Similarly, we can use congruence mod 3 to show that the expression is divisible by 3
    Therefore it is divisible by 33
    Thanks from Shakarri
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  4. #4
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    Re: Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    Thanks, Idea!

    Are there no other ways of proving this?
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  5. #5
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    Re: Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    How many do you need? Proof by induction, which is what you did, is what I would have used but I agree that a proof by congruence is simpler.
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  6. #6
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    Re: Prove 5^(2n+1)+11^(2n+1)+17^(2n+1) is divisible by 33.

    I just want to know if there are other ways to prove it. Do you prove it by induction in the same way as I did? If not, I would like to know how you do it. Thanks!
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