[Math] prove if 7n+4 is even then n is even

Question

I'm new to proofs and I wanted to verify that this proof is sound:

If $7n+4$ is even then $n$ is even

Since we know $7n+4$ is even then $7n+4 = 2k$ for some integer $k$

$$\begin{align}7n + 4 &= 2k
\\7n &= 2k-4
\\ n &= (2k-4)/7
\\ n &= 2((k-2)/7)\end{align}$$

Therefore we have shown that $n$ is even if $7n+4$ is even

Is it sound to come to that conclusion using the antecedant?

Answer 1

Dividing by the $7$ is not OK when working in the integers.

Hint. Your second line is good. What could you conclude if $n$ were odd?