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Math Help - Perfect square sum problem

  1. #1
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    Question Problem

    Extend the fraction 163/101 with a number such that the resulting fraction will have a numerator and a denominator whose sum are square numbers. Please provide the smallest of such number.

    I've had some problems with this task, hope that you can help me.
    Last edited by emilhp; December 8th 2011 at 11:03 AM.
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  2. #2
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    re: Perfect square sum problem

    Quote Originally Posted by emilhp View Post
    Extend the fraction 163/101 with a number such that the resulting fraction will have a numerator and a denominator whose sum are square numbers. Please provide the smallest of such number.
    let \frac{163}{101} = \frac{a}{c}

    \frac{a}{c} = \frac{ab}{cb}

    sum of the numerator and denominator ...

    ab+cb = b(a+c)

    for the sum b(a+c) to be a perfect square, 264b  = (2^3 \cdot 3 \cdot 11 \cdot b) would have to be a perfect square ...
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  3. #3
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    re: Perfect square sum problem

    Isn't extending a fraction just multiply it by a number? If that is the case, wouldn't the answer be

    (\frac{161}{101})(\frac{161}{101})=(\frac{161^2}{1  01^2}) or \frac{25921}{10201}
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  4. #4
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    Re: Perfect square sum problem

    Quote Originally Posted by skeeter View Post
    let \frac{163}{101} = \frac{a}{c}

    \frac{a}{c} = \frac{ab}{cb}

    sum of the numerator and denominator ...

    ab+cb = b(a+c)

    for the sum b(a+c) to be a perfect square, 264b  = (2^3 \cdot 3 \cdot 11 \cdot b) would have to be a perfect square ...



    How would you find b?
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  5. #5
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    Re: Perfect square sum problem

    Quote Originally Posted by emilhp View Post
    How would you find b?
    come on, I have to leave something for you to think about ...
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  6. #6
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    Re: Perfect square sum problem

    Quote Originally Posted by mdhafn View Post
    Isn't extending a fraction just multiply it by a number? If that is the case, wouldn't the answer be

    (\frac{161}{101})(\frac{161}{101})=(\frac{161^2}{1  01^2}) or \frac{25921}{10201}
    If the OP wanted to square the original fraction, this would be the case.
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