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Math Help - No. of solutions..

  1. #1
    Member great_math's Avatar
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    No. of solutions..

    how many solutions are there in N*N to the equation 1/x + 1/y = 1/1995 ?
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  2. #2
    MHF Contributor
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    I am afraid that it is not correct

    (x-1995)(y-1995) = -1995(x+y) + xy + 1995^2

    and 1995^2 = 3^2 \cdot 5^2 \cdot 7^2 \cdot 19^2
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  3. #3
    Lord of certain Rings
    Isomorphism's Avatar
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    Well, let me try again:

    Attempt #2:

    \frac1{x} + \frac1{y} = \frac1{1995} \Leftrightarrow (x - 1995)(y - 1995) = 1995^2 = 3^2 5^2 7^2 19^2

    We observe that if x - 1995 is negative then y - 1995 will also be. But at least one of the factors has to less than -1995, giving us negative solutions, which we do not want.

    Thus both x - 1995 and y - 1995 are positive.

    Since we have 81 factors for 1995^2 ,we have 81 solutions for x,y.
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