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Math Help - by direct method

  1. #1
    rcs
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    by direct method

    Prove: if x^2 is divisible by 3 then x is divisible by 3

    Thanks so much anybody who can help me on this.
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    MHF Contributor chisigma's Avatar
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    Re: by direct method

    Fundamental Theorem of Arithmetic -- from Wolfram MathWorld

    ... in normal practice x indicates a real variable, for integer numers usually letters like n, m, i, j or k are used...

    Kind regards

    \chi \sigma
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    Re: by direct method

    how will i use n m i k for the x^2 and x to be divisible by 3... shall i have to multiply each them by 3?
    thanks
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    MHF Contributor chisigma's Avatar
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    Re: by direct method

    What I say is that for an integer the letter 'n' is usually preferred to tghe letter 'x' and Your question should be written as: if 3| n^{2} then 3| n...

    Kind regards

    \chi \sigma
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    Re: by direct method

    If x^2 is divisible by 3, then its prime factorization contains at least one 3. Since 3 is a prime number, it follows that x must also have a factor of 3.

    (Since the question is about "divisibility", it is clear that x must be an integer so using "x" rather than "n" doesn't bother me.)
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    Re: by direct method

    this still makes me so hard... please can anyone do this proof for me?

    thanks.
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    Re: by direct method

    Quote Originally Posted by rcs View Post
    this still makes me so hard... please can anyone do this proof for me?

    thanks.
    Post #5 gives you the answer. What don't you understand?
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    Re: by direct method

    how i wish it could be shown how it is being proved..

    thanks
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    Grand Panjandrum
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    Re: by direct method

    Quote Originally Posted by rcs View Post
    how i wish it could be shown how it is being proved..

    thanks
    Consider the fundamental theorem of arithmetic (unique prime factorisation) applied to both x and x^2

    CB
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    Re: by direct method

    This is very sticklery, but I feel it's important to point out that this is a particular case of Euclid's lemma which is instrumental in proving the fundamental theorem of arithmetic.
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  11. #11
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    Re: by direct method

    You want to prove: If \displaystyle x^2 is divisible by 3 then \displaystyle x is divisible by 3.

    The contrapositive of this is: If \displaystyle x is not divisible by 3 then \displaystyle x^2 is not divisible by 3.

    To prove the contrapositive, note that any integer can be written as \displaystyle 3n, 3n+1 or \displaystyle 3n+2, where \displaystyle n is some other integer. Since we have said that \displaystyle x is not divisible by \displaystyle 3, then we can let \displaystyle x = 3n+1 or \displaystyle x = 3n+2.

    Case 1:

    \displaystyle \begin{align*} x &= 3n + 1 \\ x^2 &= (3n + 1)^2 \\ x^2 &= 9n^2 + 6n + 1 \\ x^2 &= 3(3n^2 + 2n) + 1 \end{align*}

    which is not divisible by 3.

    Case 2:

    \displaystyle \begin{align*} x &= 3n + 2 \\ x^2 &= (3n + 2)^2 \\ x^2 &= 9n^2 + 12n + 4 \\ x^2 &= 9n^2 + 12n + 3 + 1 \\ x^2 &= 3(3n^2 + 4n + 1) + 1 \end{align*}

    which is not divisible by 3.

    Since we have shown that if \displaystyle x is not divisible by 3, then \displaystyle x^2 is not divisible by 3, the contrapositive is also true.
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  12. #12
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    Re: by direct method

    There you are my Prove It! you are the best, you make me see what i wanted to see, thanks a lot, thanks for making me understand again.

    more power


    razel c. summer
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    Re: by direct method

    Quote Originally Posted by rcs View Post
    There you are my Prove It! you are the best, you make me see what i wanted to see, thanks a lot, thanks for making me understand again.

    more power


    razel c. summer
    You are more than welcome. Just one thing though, you should note that proving a statement by proving the contrapositive is not considered a direct proof, but rather an indirect proof. However, I think it's important to see how often it is easier to prove something indirectly than directly.
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  14. #14
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    Re: by direct method

    taken. thanks again
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