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.