Note that $n \equiv S(n) \bmod 9$, so $S(n+2)-S(n) \equiv n+2-n\equiv2 \bmod 9$. So, since $S(n)=691 \equiv 7 \bmod 9 $, we have that $S(n+2) \equiv 9 \equiv 0\bmod 9$. Then, only one of the answer choices is congruent to $0 \bmod 9$, which is $(A)=9$. 
Every positive integer is congruent modulo $9$ to the sum of its decimal digits.  Now, let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507)=13$. For a particular positive integer $n, S(n)=691$. Which of the following could be the value of $S(n+2)$? (Adapted From 2017 AMC 12A Problems, Question \#18) 

- $$9$$ 
- $$17$$ 
- $$143$$ 
- $$116$$ 
- $$6$$ 