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Math Help - Divisibility Proof

  1. #1
    AAM
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    Divisibility Proof

    Hi guys,

    I need to show that for x>=1 x^3-x^5 is divisible by 12.

    Not really sure how to start! :-s

    Many thanks in advance. x
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  2. #2
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    The trick you often use for these type of questions is proving that (x+1) and (x) give the same remainder.

    Easy example: 4^x is always divisible by 4. You take (x+1) which is: 4^(x+1) = 4*4^x. When dividing with 4 you can throw out the 4 and you're left with 4^x. Now if you prove that x=1 is divisible by four, you've also proven it for x+1 and x+2 etc.

    Your problem is a little harder ofcourse, and sometimes it's easier to use (x+2) or (x+3) instead of (x+1), but you'll just have to try a little. Hope this helped, and if you still can't solve it I'll try to give you a start.
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  3. #3
    AAM
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    Thanks josh_amsterdam :-)

    .... but I still don't know where to begin! :-s lol!
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  4. #4
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    Hello, AAM!

    That's a silly way to write the problem.
    Are they deliberately being annoying?


    Show that x^3-x^5 is divisible by 12 for x \geq 1
    Let N \:=\:x^3-x^5

    Then: . N \;=\;-x^3(x^2-1)\;=\;-x^3(x-1)(x+1)


    Let's ignore the leading minus-sign.

    \text{We have: }\:N \;=\;\underbrace{(x-1)\cdot x\cdot(x+1)}_{\text{3 consecutive integers}}\cdot x^2


    With 3 consecutive integers, one of them is a multiple of 3.
    . . Hence, N is divisible by 3.


    There are two cases: . \begin{array}{c}\text{(1) }x\text{ is even.} \\ \text{(2) }x\text{ is odd.}\; \end{array}


    \text{(1) If }x\text{ is even: }\;N \;=\;\underbrace{(x-1)}_{\text{odd}}\cdot\underbrace{x}_{\text{even}}\  cdot\underbrace{(x+1)}_{\text{odd}}\cdot\underbrac  e{x^2}_{\text{even}} \quad\hdots\quad \text{Hence, }N\,\text{is divisible by {\color{red}4}.}


    \text{(2) If }x\text{ is odd: }\:N \;=\;\underbrace{(x-1)}_{\text{even}} \cdot \underbrace{x}_{\text{odd}} \cdot\underbrace{(x+1)}_{\text{even}}\cdot\underbr  ace{x^2}_{\text{odd}} \quad\hdots\quad \text{Hence, }N\text{ is divisible by {\color{red}4}.}


    . . Therefore, N is divisible by 12.

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  5. #5
    AAM
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    Thank you Soroban! :-D
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