If $n$ is odd, then $n^2$ is odd
Claim:
If $n$ is odd, then $n^2$ is odd.
Proof:
Assume $n$ is odd. Then, by definition, there exists an integer $k$ such that:
$$ n=2k+1 $$Then:
$$ n^2=(2k+1)^2 =(2k+1)(2k+1) =4k^2+4k+1 =2(2k^2+2k)+1 $$Let $m=2k^2+2k$, which is an integer. Then:
$$ n^2=2m+1 $$Therefore, by direct example, $n^2$ is odd.
Claim:
If $n^2$ is odd, then $n$ is odd.
Proof:
Consider the contrapositive; if $n$ is odd, then $n^2$ is odd. We have already shown this to be true. Therefore, by contrapositive, $n$ is odd.