Does this simple proof-by-contradiction, also require contrapositive? Simple exercise 6.2 in Hammack's Book of Proof. "Use proof by contradiction to prove"
"Suppose $n$ is an integer. If $n^2$ is odd, then $n$ is odd"
So my approach was:
Suppose instead, IF $n^2$ is odd THEN $n$ is even
Alternatively, then you have the contrapositive, IF $n$ is not even ($n$ is odd), then $n^2$ is not odd ($n^2$ is even).
$n = 2k+1$  where $k$ is an integer.  (definition of odd)
$n^2 = (2k+1)^2$
$n^2 = 4k^2 + 4k + 1$
$n^2 = 2(2k^2 + 2k) + 1$
$n^2 = 2q + 1$ where $q = 2k^2 + 2k$
therefore $n^2$ is odd by definition of odd.
Therefore we have a contradiction. Contradictory contrapositive proposition said $n^2$ is not odd, but the derivation says $n^2$ is odd.  Therefore the contradictory contrapositive is false,  therefore the original proposition is true.
Not sure if this was the efficient/correct way to prove this using Proof-By-Contradiction.

Your proof looks correct to me, but I would like to share with you my strategy for proof by contradiction.
Consider the if/then statement $p\Rightarrow q$. In your case, $p$ represents "$n^{2}$ is odd" and $q$ represents "$n$ is odd". To achieve the proof by contradiction, we want to show that when $p$ is true, then it is impossible for $\neg q$, that is to say $n$ is even, to also be true. So we will assume that $p$ and $\neg q$ are both true (i.e., at the same time).
Applying this to your particular problem, we have $n^{2}$ is odd and $n$ is even. By definition, $n=2k$ for some integer $k$. Then $$n^{2}=(2k)^{2}=4k^{2}=2\cdot(2k^{2}).$$ Since the integers are multiplicatively closed, we have that $n^{2}$ is $2$ times an integer. Hence $n^{2}$ is even, which contradicts the assumption that $n^{2}$ is odd. 
This tells us two things:


*

*The logical statement "$n^{2}$ is odd and $n$ is even" is a false statement.

*The logical statement "$n$ is even implies $n^{2}$ is even" is a true statement. This is the contrapositive ($\neg q\Rightarrow\neg p$) of the original statement, which always has the same truth value as the original statement.

