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Math Help - if x and n are positive integers....

  1. #1
    Newbie everydayangel's Avatar
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    Question if x and n are positive integers....

    I'm stumped on this.. I don't remember this stuff and I need help. Not even sure it is in the right forum, but my Trig teacher gave us this on a worksheet so I tried my luck..
    (Moderator - looked for a mod or admin online but none on to ask where to post this - please move to correct forum for me, if not in correct one.. thanks)

    If x and n are positive integers with x > n and Xⁿ - Xⁿ⁻ - Xⁿ⁻ = 2009, find x + n.

    Choices:
    A - 10
    B - 11
    C - 12
    D - 13
    E - 14

    If anyone is able to answer this, can you explain how you got your answer so I can see what I should be doing here? I really appreciate it. Thanks..

    Ang
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  2. #2
    Super Member
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    Interesting Question. I have a solution that doesn't involve trig. There may be a way doing this using trig, but I'm not sure how.

    First off we have  x^n - x^{n-1} - x^{n-2} = 2009

    Factorising  x^{n-2} gives us

     x^{n-2} (x^2 - x - 1) = 2009

    Since x is a positive integer,  x^2 - x - 1 is also an integer. This means  x^{n-2} must also be an integer (but only if n > 1)

    Now we note that the prime factorisation of  2009 = 7^2 \cdot 41.

    Because  x^{n-2} is an integer only if n > 1, we can rule out the prime factor 41 as a possible value for x (it doesn't have a power > 1). This means x = 7. Logically,  x^2 - x - 1 is 41 (substitute 7 into this to prove it to yourself)

    Now divide both sides by 41 and substitute x = 7 to get  7^{n-2} = 49

    Clearly n = 4.

    So we now we have x = 7 and n = 4. The answer to the question is B - 11.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by everydayangel View Post
    I'm stumped on this.. I don't remember this stuff and I need help. Not even sure it is in the right forum, but my Trig teacher gave us this on a worksheet so I tried my luck..
    (Moderator - looked for a mod or admin online but none on to ask where to post this - please move to correct forum for me, if not in correct one.. thanks)

    If x and n are positive integers with x > n and Xⁿ - Xⁿ⁻ - Xⁿ⁻ = 2009, find x + n.

    Choices:
    A - 10
    B - 11
    C - 12
    D - 13
    E - 14

    If anyone is able to answer this, can you explain how you got your answer so I can see what I should be doing here? I really appreciate it. Thanks..

    Ang
    Try exhaustive search, start with x=14 ... 2, n=x-1 ... 1


    CB
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