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Math Help - Proof by counter-example

  1. #1
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    Proof by counter-example

    How can i prove using a counter example that:

    The number n^3 − n^2 + n + 4 is divisible by 5, for all positive integers n is false ?
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  2. #2
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    Re: Proof by counter-example

    you could evaluate it for n=4....
    Thanks from topsquark and michele
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  3. #3
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    Re: Proof by counter-example

    It appears that numbers of the form n = 5i+4 and n = 5i+5, i = 0, 1, 2, \cdots don't work.
    Thanks from topsquark and michele
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    Re: Proof by counter-example

    Thanks guys but I never did it before so I would appreciate if I could have an example followed by the information of how it is done and what is happening during the process.
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  5. #5
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    Re: Proof by counter-example

    You were told exactly what to do! Set n= 4 in the given formula and do the calculation. What do you get?

    Do you know what 'counter example' means?
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  6. #6
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    Re: Proof by counter-example

    Ok what I have done is:

    if n = 5

    then

    5^3 - 5^2 + 5 + 4 = 109

    Thus 109 is not divisible by 5

    Is that the right way of present my answer or there is any specific notation to do this?
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  7. #7
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    Re: Proof by counter-example

    Quote Originally Posted by michele View Post
    Ok what I have done is:

    if n = 5

    then

    5^3 - 5^2 + 5 + 4 = 109

    Thus 109 is not divisible by 5

    Is that the right way of present my answer or there is any specific notation to do this?
    Your logic as written isn't quite right. Just remove the "thus" before "109".

    and then add

    "Thus $n^3-n^2+n+4$ is not divisible by $5$ for all $n$"
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  8. #8
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    Re: Proof by counter-example

    Quote Originally Posted by michele View Post
    Ok what I have done is:

    if n = 5

    then

    5^3 - 5^2 + 5 + 4 = 109

    Thus 109 is not divisible by 5
    As romsek said, "thus" is the wrong word. It implies that the fact that 109 is not divisible by 5 follows from what went before. In fact, "what went before" shows that 109 is of the from "n^3- n^2+ n+ 4" for n= 5. The fact that it is not divisible by 5 follows from the fact that 109= 5(21)+ 4.

    Is that the right way of present my answer or there is any specific notation to do this?
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