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Thread: When n Is A Positive Integer

  1. #1
    Super Member harpazo's Avatar
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    When n Is A Positive Integer

    If n is a positive integer, then n(n + 1)(n + 2) is

    A. Even only when n is even
    B. Even only when n is odd
    C. Odd whenever n is odd
    D. Divisible by 3 only when n is odd
    E. Divisible by 4 whenever n is even

    I decided to let n = 2. I got 24, which is even.
    I then let n = 3. This produced 60, which is also even.

    The answer is E.

    When n is 3, 60 is the answer, which is divisible by 4.
    However, when n is 2, the answer is 24, which is also divisible by 4.

    Question:

    Why is the answer E and not B?
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    Re: When n Is A Positive Integer

    Quote Originally Posted by harpazo View Post
    If n is a positive integer, then n(n + 1)(n + 2) is
    A. Even only when n is even
    B. Even only when n is odd
    C. Odd whenever n is odd
    D. Divisible by 3 only when n is odd
    E. Divisible by 4 whenever n is even
    The answer is E.
    Question: Why is the answer E and not B?
    $(\forall n\in \mathbb{Z}^+)~[n(n+1)(n+2)\text{ is an even integer.}]$ That makes B. FALSE.
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    Super Member harpazo's Avatar
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    Re: When n Is A Positive Integer

    Quote Originally Posted by Plato View Post
    $(\forall n\in \mathbb{Z}^+)~[n(n+1)(n+2)\text{ is an even integer.}]$ That makes B. FALSE.
    Sorry but I am not familiar with your math symbols you posted in your reply.
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    Re: When n Is A Positive Integer

    Quote Originally Posted by harpazo View Post
    Sorry but I am not familiar with your math symbols you posted in your reply.
    $(\forall n\in \mathbb{Z}^+)~[n(n+1)(n+2)\text{ is an even integer.}]$ the upside down A is read "for all" n that is a positive integer ($n\in\mathbb{Z}^+$) the product $[n(n+1)(n+2)\text{ is an even integer.}]$
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    Re: When n Is A Positive Integer

    Quote Originally Posted by Plato View Post
    ...the upside down A is read "for all"
    So if there is no charge for an upside down A, then it's a "free for all"
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    Re: When n Is A Positive Integer

    Of three consecutive integers, there must be at least one even integer: "odd, even, odd" or "even, odd, even". And if there is at least one even integer in a product then there is a factor of 2 so the product is even.
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    Super Member harpazo's Avatar
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    Re: When n Is A Positive Integer

    Quote Originally Posted by HallsofIvy View Post
    Of three consecutive integers, there must be at least one even integer: "odd, even, odd" or "even, odd, even". And if there is at least one even integer in a product then there is a factor of 2 so the product is even.
    Sorry but I don't get it.
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    Super Member harpazo's Avatar
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    Re: When n Is A Positive Integer

    Quote Originally Posted by Plato View Post
    $(\forall n\in \mathbb{Z}^+)~[n(n+1)(n+2)\text{ is an even integer.}]$ the upside down A is read "for all" n that is a positive integer ($n\in\mathbb{Z}^+$) the product $[n(n+1)(n+2)\text{ is an even integer.}]$
    Sorry but I'm ready to read and/or apply advanced math symbols in my questions.
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