# Find the smallest integer k such that 600k is a cube number.

• May 14th 2009, 11:08 AM
BabyMilo
[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 AM
Moo
Hello,
Quote:

Originally Posted by BabyMilo
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
• May 14th 2009, 11:16 AM
Isomorphism
Quote:

Originally Posted by BabyMilo
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?
• May 14th 2009, 11:35 AM
BabyMilo
Fully understood! Thanks both of you!