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Math Help - Basic proof

  1. #1
    Senior Member DivideBy0's Avatar
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    Basic proof

    Show that n(n^2 - 1) is divisible by 24 when n is an odd number.

    What I've got so far is a lemma that the square of any odd number is of the form 8q + 1.

    So I kinda have... n(8q + 1 - 1)=8nq

    This is where I got stuck... please help lol

    (good to see LaTeX up )
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by DivideBy0 View Post
    Show that n(n^2 - 1) is divisible by 24 when n is an odd number.

    What I've got so far is a lemma that the square of any odd number is of the form 8q + 1.

    So I kinda have... n(8q + 1 - 1)=8nq

    This is where I got stuck... please help lol

    (good to see LaTeX up )
    One way...

    n(n^2 - 1) = n(n + 1)(n - 1) = (n - 1)n(n + 1)

    So we have the product of three consectutive numbers, starting with an even number. So one of the three numbers will be divisible by 3, one of the two even numbers will be divisble by 4, and the remaining even number will be divisible by 2. Thus the product is divisible by 2 * 3 * 4 = 24.

    -Dan
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