The Number Game


The product of three consecutive numbers when divided by each of them in turn, the sum of the three quotients will be 74.
What are the numbers?

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  • 2 Answer(s)

    4, 5 and 6.

    John123 Expert Answered on 26th July 2015.
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    Let the three consecutive numbers be x, x+1, and x+2.

    When we divide the product of these three numbers by the first number x, we get: (x)(x+1)(x+2)/x = (x+1)(x+2)
    When we divide the product by the second number x+1, we get: (x)(x+1)(x+2)/(x+1) = (x)(x+2)/(x+1)
    When we divide the product by the third number x+2, we get: (x)(x+1)(x+2)/(x+2) = (x)(x+1)
    According to the problem, the sum of these three quotients is 74, so we can write the equation:
    (x+1)(x+2) + (x)(x+2)/(x+1) + (x)(x+1) = 74
    Expanding the terms and simplifying, we get:
    3x^2 + 6x – 72 = 0

    Dividing by 3 and factoring, we get:  x^2 + 2x – 24 = 0
    (x+6)(x-4) = 0

    So x = -6 or x = 4. Since the numbers are consecutive, x must be positive, so x = 4.

    Therefore, the three consecutive numbers are 4, 5, and 6.

    Moshe Expert Answered on 2nd March 2023.
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