(i) Let $d = \gcd(a, b)$. There exist two positive integers $x$ and $y$ which are relatively prime such that $a = dx$, $b = dy$. Using this fact in the initial equality we find $x + y + n = xy$, that is, $(x - 1)(y - 1) = n + 1$. Since $x$ and $y$ are relatively prime numbers, at least one of them is odd. Then $(x - 1)(y - 1)$ is even, so $n$ is odd.

(ii) As before, we are led to the equation $(x - 1)(y - 1) = 2008 = 2^3 \cdot 251$. Since $x$ and $y$ are relatively prime, the possibilities for $(x, y)$ are $(2, 2009)$, $(9, 252)$, $(252, 9)$, $(2009, 2)$. The complete set of solutions $(a, b)$ is
$$$$
\{(2d, 2009d), (9d, 252d), (252d, 9d), (2009d, 2d) : d \ge 1\}.