Check My Proof: if $a$ is even then $9(a +5)$ is odd

Question

Just want to make sure I am doing this right.

If $a$ is an even integer then $9(a + 5)$ is odd

If $a$ is even, $a = 2k$, therefore, $9(a + 5) = 9(2k + 5)$ which is odd.

Answer 1

You need to justify the assertion that $9(2k + 5)$ is odd.

If $a$ is even, then $9(a + 5)$ is odd.

If $a$ is even, then there exists an integer $k$ such that $a = 2k$. Hence, \begin{align*} 9(a + 5) & = 9(2k + 5)\\ & = 18k + 45\\ & = 18k + 44 + 1\\ & = 2(9k + 22) + 1 \end{align*} Since the integers are closed under addition and multiplication, $9k + 22$ is an integer. Therefore, $9(a + 5)$ has the form $2m + 1$ for some integer $m$, so it is odd.