smallest of 3 integers

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The sum of the reciprocals of three consecutive positive integers is equal to 47 divided by the product of the integers. What is the smallest of the three integers?

  • smallest of 3 integers -

    1/(x-1) + 1/x + 1/(x+1) = 47/[(x-1)x(x+1)]
    put all over a common denominator of (x-1)x(x+1). The numerators then satisfy

    x(x+1) + (x-1)(x+1) + (x-1)x = 47
    3x^2 - 1 = 47
    3x^2 = 48
    x = 4

    check:
    1/3 + 1/4 + 1/5 = 47/60

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