All integers can be represented in three ways:
$$ \begin{align} n &= 2k+1 \\ n &= 4k \\ n &= 4k+2 \end{align} $$
If \(n\) is odd (\(2k+1\)), then \(n=(k)+(k+1)\), and both \(k\) and \(k+1\) are relatively prime.
If \(n=4k\), then \(n=(2k+1) + (2k-1)\), and both \(2k+1\) and \(2k-1\) are relatively prime, see this article for more detail.
If \(n=4k+2\), then \(n=(2k-1) + (2k+3)\), and both \(2k-1\) and \(2k+3\) are relatively prime.