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Help 4 consecutive numbers divisible by 4

  1. Feb 25, 2010 #1
    help!! 4 consecutive numbers divisible by 4

    1. The problem statement, all variables and given/known data


    prove that for any integer n(n^2-1)(n+2) is divisible by 4??



    2. Relevant equations



    3. The attempt at a solution
    i know two of them are even, but how do i actually prove this??

    thanks
     
  2. jcsd
  3. Feb 25, 2010 #2
    Re: help!! 4 consecutive numbers divisible by 4

    If you know that two of them are even, then each is of the form 2a and 2b for some integers a and b. Then multiplying them together gives a multiple of 4.

    You can start this by factoring the formula given. Expand it completely and then factor out n. Then factor the resulting third degree polynomial. One of the factors is (x+1). Your first factor was n, the second is n+1, so I think you see where this is going. Then after factoring completely you can find the solution.
     
  4. Feb 26, 2010 #3
    Re: help!! 4 consecutive numbers divisible by 4

    If n is even, then so is (n+2), and the product is divisible by 4.

    If n is odd, then (n^2-1)=(n-1)(n+1) is a product of even numbers, so again the whole thing is divisible by 4.
     
  5. Feb 26, 2010 #4
    Re: help!! 4 consecutive numbers divisible by 4

    Try factoring n(n2-1)(n+2) and see if you notice something.
     
  6. Feb 27, 2010 #5

    HallsofIvy

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    Re: help!! 4 consecutive numbers divisible by 4

    If you mean "notice that this is (n-1)(n)(n+1)(n+2), the product of 4 consecutive integers, I suspect, from the title of this thread, that he already knew that!
     
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