I am still new to proofs and just starting to get some practice in. Here is what the homework problem asks.
Prove that if $n$ is an integer and $3n+2$ is even, then $n$ is even.
Proof attempt by contradiction: Suppose $3n+2$ is even and $n$ is odd. If $n$ is odd, then $n=2k+1$ for some $k \in \mathbb{Z}$. Then, $3n+2=3(2k+1)+2=2(3k)+5$. Since $k \in \mathbb{Z}$, then $3k \in \mathbb{Z}$. Let $3k=m \Rightarrow 3n+2=2m+5$, which is odd. Thus, contradicting our assumption that $3n+2$ is even. $\blacksquare$
I feel as if I am missing some steps and/or grazing over something important. Any feedback, tips/suggestions, or words of wisdom would be greatly appreciated. Thank you in advance.
Best Answer
Yes, your proof is OK.
Say $n$ is odd, so $3n$ is odd, so $3n+2$ is odd, a contradiction.
Or you can do like this: since $2\mid 3n+2$ we have $2\mid (3n+2)-2 = 3n$. But $\gcd(3,2)=1$ so $2\mid n$.