Find the remainder.

We can quickly deduce that $13^{2009} \equiv (-5)^{2009} \mod{18}$ and $(-5)^{2009} = -(5^{2009})$

Substituting, we find that $5^{2009} + 13^{2009} \equiv 5^{2009} - 5^{2009} = \boxed{0} \mod{18}$

We know

For any positive integer $a,b,n,e$

$a^n+b^n \mod e=(a+c)^9 \mod e$

$\therefore 5^{2009}+13^{2009} \mod 18=(5+13)^{2009} \mod e$

and we know if $a=b\mod e$ then $a^n=b^n\mod e$

$\therefore (5+13)^{2009} \mod 18=0^{2009} \mod 18$

hence our answer is $\boxed{0}$