Hello,

The trick for this is to factor 600 :

But in a cube number, the prime decomposition is in the form

So for , it's okay.

If 3 is in the prime decomposition of the cube number, then has to. So keep a factor

Same reasoning for 5. Keep a factor 5.

is the smallest integer such that is a cube number.

This is a rather intuitive thing. I'm sorry if I'm not providing a formal proof :s