This is a question from my textbook, and I'm pretty sure I have the answer, but I am having trouble writing out what I am thinking.
Prove: If $n$ is an integer and $3n+2$ is odd, then $n$ is odd.
Contrapositive: If $n$ is even, then $3n+2$ is even.
$n$ is even, by definition $n = 2k$, where $k$ is an integer.
Plugging in $2k$ into $3n+2$, we have $3(2k)+2$.
$3(2k)+2 = 6k+2 = 2(3k+1)$, which means $3n+2$ is even.
Since if $n$ is even, then $3n+2$ is even. Negating the conclusion of the original conditional statement implies that it's hypothesis "$3n+2$ is odd" is false, therefor if $3n+2$ is odd, $n$ is odd.
What am I missing, or what can I do to make this less awkward? Or am I wrong about the proof on the most basic levels?