If it is known that $x^3$ is even, can we say that $x$ is even? It seems to be the case because an odd*odd*odd=odd (if we are dealing with natural numbers). But is there a proof?
It sounds like you've more or less given a proof without realizing it. Proof by contrapositive
On top of the mathematical proofs, you can try this yourself with very high numbers with a bit of code.
from __future__ import division for i in range(0,1000): if type((i*i*i)/2)==int: print(i) else: print("Done")
With any number you input, you can see it will never find any number that this statement does not hold true for.
http://labs.codecademy.com/ You can try the python code out here if you like.