Emmy says to Max, "I ordered $36$ math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was $\$ \underline A~\underline B~\underline B.\underline B~\underline A$, where $A$ and $B$ are digits and $A \neq 0$." After a pause, Max says, "That was a good price." What is $A + B$?
$$\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$$
Difficulty: 27/100 - A relatively straightforward problem utilizing divisibility rules, but intimidating for its position.
Core Concepts: divisibility rules, modular arithmetic, system of equations
Challenges: Competitors may waste time bashing combinations if unaware of divisibility rules, or solving a system of modular congruences.
Techniques: Employing the divisibility rules for $9$ and $4$ allow you to reduce single-digit results for $A$ and $B$.
Error-prone Steps: Incorrectly applying divisibility rules.
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2-3 min
Beginner: ≤ 4 min