XY solution

here,

$xy+9(x+y)=2006$

$xy+9y=2006-9x$

$y(x+9)=2087-9(x+9)$

$y=\frac{2087-9(x+9)}{x+9}$

$y=\frac{2087}{x+9}-9$

Here $2087$ is a prime number.So it is only divisible by $1$ or $2087$

Since $y$ is integer. So $x+9$ is must be $\pm1$ or $\pm 2087$

$\therefore x=-8, -10, -2096$ or $2078$

Here we get $4$ solution for $x$ .

So, ans is $4$ .