Congruence Classes modulo $13$

In modulo $13$ , $3^1 \equiv 3$ , $3^2 \equiv 9$ and $3^3 \equiv 27 \equiv 1$ , so for each number $n$ : $3^n \equiv 3^{3q} \times 3^r \equiv (3^3)^q \times 3^r \equiv 1^q \times 3^r \equiv 3^r$ , where $q$ and $r$ are respectively the quotient and the remainder of the Euclidean division of $n$ by $3$ , $0 \le r \le 2$ ; this means that $3^n \equiv 1, 3, 9$ , $N=3$ , $S=13$ and $N+S=\boxed{16}$