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Thread: Natural number

  1. #1
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    Natural number

    Find a where 2(1/a+1/(a+1)) is a natural number.
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  2. #2
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    Re: Natural number

    Quote Originally Posted by louis33 View Post
    Find a where 2(1/a+1/(a+1)) is a natural number.
    $a=1 \implies 2\left(\dfrac{1}{a} + \dfrac{1}{a+1} \right) = 3$
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    Re: Natural number

    Yeah, but I'd like to know how you got to that result in the first place. You can't just give me the answer
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    Re: Natural number

    I just tried $a=1$ and it worked ...
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    Re: Natural number

    m=2\left(\frac{1}{a}+\frac{1}{a+1}\right)

    if m is an integer then m a-4=\frac{-2}{a+1} must be an integer
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    Re: Natural number

    How did you get to that equation?
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  7. #7
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    Re: Natural number

    Quote Originally Posted by louis33 View Post
    Find a where 2(1/a+1/(a+1)) is a natural number.
    It should be obvious that double the sum of two small fractions will be less than 1. In fact

    $1 = 2 \left ( \dfrac{1}{4} + \dfrac{1}{4} \right ) \implies 1 > 2 \left ( \dfrac{1}{4 + k} + \dfrac{1}{5 + k} \right ) \text { if } k \ge 0.$

    So you have exactly 3 numbers to test, 1, 2, and 3. And 2 and 3 don't work.
    Last edited by JeffM; Nov 26th 2017 at 11:20 AM.
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