[Number Theory]
Problem: Suppose that neither of the three-digit numbers $M=\underline{4} \underline{A} \underline{6}$ and $N=\underline{1} \underline{B} \underline{7}$ is divisible by 9 , but the product $M \cdot N$ is divisible by 9 . Compute the largest possible value of $A+B$.

Solution: In order for the conditions of the problem to be satisfied, $M$ and $N$ must both be divisible by 3 , but not by 9 . Thus the largest possible value of $A$ is 5 , and the largest possible value of $B$ is 7 , so $A+B=\mathbf{1 2}$.

Final answer: 12