Answer
See proof.
Work Step by Step
Suppose $m+n$ and $n+p$ are even integers where $m,n,p$ are integers. Since $m+n$ is even then $m$ and $n$ must either both be even or both be odd by exercises 1 and 2 of this section. Likewise $n$ and $p$ must both be even or both be odd.
Suppose $m$ and $n$ are odd, then $p$ must be odd since $n$ is odd. Then, $m+p$ is even since both $m$ and $p$ are odd (by exercise 1).
On the other hand, if $m$ and $n$ are even, then $p$ must be even since $n$ is even. Then, $m+p$ is even since both $m$ and $p$ are even (by exercise 2).
We used a direct proof here.
