Answer
See below.
Work Step by Step
(a) To prove the statement by contraposition, we need to prove "for any integers $m,n$, if $m,n$ are not both even or both odd, then $m+n$ will not be even". To prove, let $m=2k$ and $n=2p+1$ (or vice versa), where $k$and $p$ are integers. We have $m+n=2(k+p)+1$ which is an odd number (not even). Thus we proved the statement by contraposition.
(b) To prove the statement by contradiction, suppose there exist integers $m,n$, such that $m+n$ is even while $m,n$ are not both even or both odd". Let $m=2k$ and $n=2p+1$ (or vice versa), where $k$ and $p$ are integers. We have $m+n=2(k+p)+1$ which is odd and contradicts to the condition that $m+n$ is even. Thus, we proved the statement by contradiction.
