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Math Help - Alternative methods

  1. #1
    Senior Member I-Think's Avatar
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    Alternative methods

    I'm looking for alternative methods a question. My answer relies on a bit of guesswork and I don't think it's elegant, so I'm seeking other answers.
    I'll post my solution in another post

    Question
    Let x and y be integers. Find the value of 3x^2y^2 if
    y^2+3x^2y^2=30x^2+517
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  2. #2
    Senior Member I-Think's Avatar
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    My solution

    y^2+3x^2y^2=30x^2+517

    y^2-517=3x^2(10-y^2)

    Fact 1) The L.H.S. must be a multiple of 3
    Fact 2) If 0<y\leq{3}, then the R.H.S. is positive, but the L.H.S. is -ive, hence y>3
    Fact 3) If y>22, a similar situation occurs, so 3<y<22
    Fact 4) y cannot be a multiple of 3, as the L.H.S. would not be multiple of 3
    Fact 5) I have stopped determining constraints and begun to guess the answer

    Upon guessing, y=7 and x=2
    So the value of 3x^2y^2 follows
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  3. #3
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    I think,

    if you had 30x^2 + 517 could you not take x^2 to the other side?


    and if you did and then divided (y^2) + (3x^2)(y^2)/(x^2)

    Would that not cancel out the x2?

    Giving you y^2 + (3)(y^2) = 30 + 517
    ??

    I am probably breaking all kinds of rules but it seems that you loose x in that case and then you would have

    4y^2 = 537

    y^2 = 537/4

    y = square root of (537/4)


    I know I am all kinds of wrong I was just curious... Its ok to make a bit of fun of me I dont mind :P

    Brian
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  4. #4
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    Hello, I-Think!

    Let x and y be integers.
    Find the value of 3x^2y^2 if: . y^2+3x^2y^2\:=\:30x^2+517

    We have: . y^2(1+3x^2) \:=\:30x^2+517 \quad\Rightarrow\quad y^2 \:=\:\frac{30x^2+517}{3x^2+1} \quad\Rightarrow\quad y^2\;=\;10 + \frac{507}{3x^2+1} .[1]

    Since y^2 is an integer, 3x^2+1 must be a factor of 507.

    The factors of 507 are: . 1,\:3,\:13,\:39,\:169,\:507

    The only case in which x^2 is an integer is: . 3x^2+1 \:=\:13 \quad\Rightarrow\quad x^2 \,=\,4

    Sustitute into [1]: . y^2 \:=\:10 + \frac{507}{3(4)+1} \quad\Rightarrow\quad y^2 \:=\:49


    Therefore: . 3x^2y^2 \:=\:3(4)(49) \:=\:588

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