Wednesday, October 7, 2009

1/x + 1/y = 1/210

Find the number of integer solutions to 1/x + 1/y = 1/210.
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  1. DavidOctober 8, 2009 at 4:51 PM

    The answer is 161.

    1/x + 1/y = 1/210
    (x+y)/(xy) = 1/210
    xy = 210(x+y)
    xy - 210x - 210y = 0
    (x - 210)(y - 210) = 210^2

    Now, 210^2 has 81 positive divisors and 81 negative divisors. So there are 162 possible values for x - 210, and y - 210 is determined by x - 210. However, we must exclude the case where x - 210 = y - 210 = 210, because this leads to division by zero.

    Therefore the number of integer solutions is 2*81-1 = 161.

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