First, we note that the values of $x$ and $y$ cannot be equal since they are integers and $x+y$ is odd. Next, we look at the case when $x>y$. We list the fifteen possible pairs of values for $x$ and $y$ and the corresponding values of $x y$. Therefore, the largest possible value for $x y$ is 240. Note that the largest value occurs when $x$ and $y$ are as close together as possible.