Irska

(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\}.