[Algebra]
Problem: Let $T=2018$, and append the digits of $T$ to $\underline{A} \underline{A} \underline{B}$ (for example, if $T=17$, then the result would be $\underline{1} \underline{\underline{A}} \underline{A} \underline{B}$ ). If the resulting number is divisible by 11 , compute the largest possible value of $A+B$.

Solution: Let $R$ be the remainder when $T$ is divided by 11 . Note that the alternating sum of the digits of the number must be divisible by 11 . This sum will be congruent $\bmod 11$ to $B-A+A-R=$ $B-R$, thus $B=R$. Because $A$ 's value is irrelevant, to maximize $A+B$, set $A=9$ to yield $A+B=9+R$. For $T=2018, R=5$, and the answer is $9+5=\mathbf{1 4}$.

Final answer: 14