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Math Help - Find the smallest integer k such that 600k is a cube number.

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    [Solved] Find the smallest integer k such that 600k is a cube number.

    as stated in the title - Find the smallest integer k such that 600k is a cube number. Thanks ever so much!
    Last edited by BabyMilo; May 14th 2009 at 12:40 PM.
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  2. #2
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    Hello,
    Quote Originally Posted by BabyMilo View Post
    as stated in the title - Find the smallest integer k such that 600k is a cube number. Thanks ever so much!
    The trick for this is to factor 600 :

    600=4\times 150=8\times 3\times 5^2=2^3 \times 3\times 5^2

    But in a cube number, the prime decomposition is in the form p_1^{3\alpha_1}\times p_2^{3\alpha_2}\times\dots

    So for 2^3, it's okay.
    If 3 is in the prime decomposition of the cube number, then 3^3 has to. So keep a factor 3^2
    Same reasoning for 5. Keep a factor 5.

    k=3^2\times 5=45 is the smallest integer such that 600k is a cube number.


    This is a rather intuitive thing. I'm sorry if I'm not providing a formal proof :s
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    Quote Originally Posted by BabyMilo View Post
    as stated in the title - Find the smallest integer k such that 600k is a cube number. Thanks ever so much!
    I think you mean positive integer k:

    600k = 2^3 \times3 \times 5^2 \times k

    Now we see that 2 is already cubed. If we throw a couple of threes in k and one 5, we will get 3 and 5 cubed...

    So choose k = 3^2 \times 5 = 45

    600k = 2^3 \times3 \times 5^2 \times 3^2 \times 5 = 2^3 \times 3^3 \times 5^3 = (2 \times 3 \times 5)^3 = 30^3

    Can you see why this must be the smallest integer k?
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    Fully understood! Thanks both of you!
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