Prove that a number is even, given the cube is even 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.