The question asks which one is impossible, all we need to do is find one possible way that the others are possible. After trying, when $n$ and $m$ are both even or odd, the calculation works, so $D$ is not correct. 
If $n$ and $m$ are integers and $n^{2}+m^{2}$ is even, which of the following is impossible? (2014 AMC 8 Problem, Question \#13) 
(1) $n$ and $m$ are even 
(2) $n$ and $m$ are odd 
(3) $n+m$ is even 
(4) $n+m$ is odd 
(5) none of these are impossible 