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Math Help - Smallest Integer

  1. #1
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    Smallest Integer

    What is the smallest integer greater than 1 that is both the square of an integer and the cube of an integer?
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  2. #2
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    Entered By Symmetry

    For the record, I deleted this message. How it got to you so quickly I don't know since I deleted it almost as soon as I posted it. I realized that I had read the question wrong: there is nothing in the problem that says the number has to be the square and cube of the SAME integer as I supposed when I did the above analysis.

    I believe the answer is 64 = 8^2 = 4^3, but I can only find that by trial and error. I presume someone will come along and show you how to do this.
    Last edited by symmetry; January 18th 2007 at 06:01 PM. Reason: Update
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  3. #3
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    Quote Originally Posted by symmetry View Post
    What is the smallest integer greater than 1 that is both the square of
    an integer and the cube of an integer?
    ---End Quote---
    You have x^3 = x^2.

    So
    x^3 - x^2 = 0

    x^2(x - 1) = 0

    So either x^2 = 0, which implies x = 0 or x - 1 = 0 which implies x =
    1.

    Since x = 0 and x = 1 are the only solutions there is no number (much
    less an integer!) greater than 1 with this property.

    -Dan
    For the record, I deleted this message. How it got to you so quickly I don't know since I deleted it almost as soon as I posted it. I realized that I had read the question wrong: there is nothing in the problem that says the number has to be the square and cube of the SAME integer as I supposed when I did the above analysis.

    I believe the answer is 64 = 8^2 = 4^3, but I can only find that by trial and error. I presume someone will come along and show you how to do this.

    -Dan
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