Proof by contrapositive:
[A] All integers are odd or even
[B] if at least one of x,y are even then xy must be even (you should prove this, but it is trivial).
Since xy is odd then the conditions in [B] have not occured, ie neither X nor Y is even
so, by [A], both X and Y are odd.
Proof by exhaustion:
there are only 4 possible cases for (x,y): (odd,odd),(even,odd),(odd,even),(even,even).
its easy to determine the evenness of xy in each case, and you can prove/disprove any statement you want from the results.