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Thread: Proving statements

  1. #1
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    Proving statements

    Hi guys, need help with two probs below


    1. Prove either true or false:

    3|(a^2+b^2) => 3|a and 3|b



    2. Prove that prime number 7t+1 can also be written as 14s+1


    Much appreciated!
    Last edited by lolcat; Oct 2nd 2011 at 02:57 PM.
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  2. #2
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    Re: Proving statements

    Hello, lolcat!

    1. Prove either true or false:

    . . $\displaystyle 3|(a^2+b^2) \;\Rightarrow\;3|a\,\text{ and }\,3|b$

    This is not a rigorous proof . . .


    Note the squares:

    .$\displaystyle \begin{array}{cccc} 0^2 &\equiv& 0 & \text{(mod 3)} \\1^2 &\equiv& 1 & \text{(mod 3)} \\ 2^2 &\equiv& 1 & \text{(mod 3)} \\ 3^2 &\equiv& 0 & \text{(mod 3)} \\ 4^2 &\equiv&1 & \text{(mod 3)} \\ 5^2 &\equiv& 1 & \text{(mod 3)} \\ 6^2 &\equiv&0 & \text{(mod 3)} \\ 7^2 &\equiv& 1 & \text{(mod 3)} \\ 8^2 &\equiv&1 & \text{(mod 3)} \\ 9^2 &\equiv& 0 & \text{(mod 3)} \end{array}$


    If $\displaystyle a^2+b^2$ is divisible by 3, then: .$\displaystyle a^2+b^2 \:\equiv\:0\text{ (mod 3)}$

    We see that the only way that $\displaystyle a^2+b^2 \:\equiv\:0\text{ (mod 3)}$
    . . is when both $\displaystyle a$ and $\displaystyle b$ are divisible by 3.




    2. Prove that prime number $\displaystyle P \,=\,7t+1$ can also be written as $\displaystyle 14s+1.$

    If $\displaystyle t$ is odd, $\displaystyle t \,=\,2m-1,$
    . . then: .$\displaystyle P \:=\:7(2m-1) + 1 \:=\:14m - 7 + 1 \:=\:14m - 6 \:=\:2(7m -3)$
    $\displaystyle P$ is not a prime.

    Hence, $\displaystyle t$ must be even, $\displaystyle t = 2s$

    Then: .$\displaystyle P \:=\:7(2s) + 1 \:=\:14s + 1$

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