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Thread: Help me to solve it, please.

  1. #1
    uto
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    Help me to solve it, please.

    A unit fraction is a fraction of the form 1/n where n is a positive integer. Note that the unit unifraction 1/11 can be written as the sum of two unit fractions in the following three ways:
    1/11=1/12+1/132=1/22+1/22=1/132+1/12
    Are there any other ways of decomposing
    1/11 into the sum of two unit fractions? In how many ways can we write 1/60 as the sum of two unit fractions? More generally, in how many ways can the unit fraction 1/n be written as the sum of two unit fractions?
    In other words, how many ordered pairs (
    a, b) of positive integers a, b are there for which 1/n=1/a+1/b?
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  2. #2
    MHF Contributor
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    Re: Help me to solve it, please.

    For 1/60 = sum of 2 unit fractions, there are 23 ways:
    1: 1/61 + 1/3660
    2: 1/62 + 1/1860
    3: 1/63 + 1/1260
    ...
    21:1/108 + 1/135
    22: 1/110 + 1/132
    23: 1/120 + 1/120

    If you insist that switching 'em around is another set, then add 22, for 45.

    Nothing magic: used dumb code on a double looper....

    1/60 = 1/a + 1/b
    ab / (a + b) = 60
    Last edited by DenisB; Mar 10th 2018 at 09:50 AM.
    Thanks from uto
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  3. #3
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    Re: Help me to solve it, please.

    the number of solutions equals the number of positive divisors of $n^2$

    Proof: $n^2=(a-n)(b-n)$

    $n=60$, $60^2=2^4 3^2 5^2$ so $45$ solutions

    $n=11$ three solutions
    Thanks from topsquark and uto
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