as stated in the title - Find the smallest integer k such that 600k is a cube number. Thanks ever so much!

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- May 14th 2009, 11:08 AMBabyMilo[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!

- May 14th 2009, 11:14 AMMoo
Hello,

The trick for this is to factor 600 :

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

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

So for $\displaystyle 2^3$, it's okay.

If 3 is in the prime decomposition of the cube number, then $\displaystyle 3^3$ has to. So keep a factor $\displaystyle 3^2$

Same reasoning for 5. Keep a factor 5.

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

This is a rather intuitive thing. I'm sorry if I'm not providing a formal proof :s - May 14th 2009, 11:16 AMIsomorphism
I think you mean positive integer k:

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

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 $\displaystyle k = 3^2 \times 5 = 45$

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

Can you see why this must be the smallest integer k? - May 14th 2009, 11:35 AMBabyMilo
Fully understood! Thanks both of you!