Proof by contradiction that if n is a positive integer, then n is odd if and only if 5n + 6 is odd

Let us prove by contradiction that if $n$ is a positive integer, then $n$ is odd if and only if $5n + 6$ is odd.

Let $n$ is not odd. Then $n$ is even, and hence $n=2k,\ k\in\mathbb N.$ It follows that $5n+6=5(2k)+6=2(5k+3)$, and hence $5n+6$ is even, that is $5n + 6$ is not odd.

On the other hand, let $5n+6$ is not odd. Then $5n+6$ is even, and hence $5n+6=2t,\ t\in\mathbb N.$

It follows that $5n=2t-6=2(t-3)$, and hence $5n$ is even. Therefore, $n$ is also even, that is $n$ is not odd.