So, can I just put x times x is not a multiple of 2? Does that solve the problem?
you assumed x was odd in your proof, which is what you needed to prove. you were begging the question. in any case, the direct proof is the better and much easier way to do this, but since you need an indirect way, here is my suggestion. use lemmas.
LEMMA: (a) an odd integer times an odd integer is odd
............(b) an odd integer times an even integer is even
............(c) an even integer times an even integer is even
i leave the proof of the lemma to you (or just part (a) of the lemma, since that is what you are dealing with here )
now, we can make our indirect proof run smoothly.
Claim: if $\displaystyle x$ is even, then $\displaystyle x^2$ is even
Proof:
By the contrapositive, we need to show that if $\displaystyle x^2$ is odd, then $\displaystyle x$ is odd. This is a direct consequence of Lemma (a)
QED
any questions?
EDIT: Ah, i just saw Reckoner's nice solution. pick your favorite.