The value of the two-digit number $\underline{a}~\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}~\underline{a}$ in base nine. What is $a+b?$
$$\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 14$$
Difficulty: 15/100 — Dependent solely on the knowledge of base numbers, trivial for those who know.
Core Concepts: base numbers, Diophantine equation
Challenges: If the contestant is unfamiliar with base numbers, it is impossible. Finding the smallest $a$ and $b$ once you reduce the equation.
Techniques: Set up an equation relating the two numbers.
Error-prone Steps: Considering the number $86$ while not realizing that $8$ is not a digit in base $7$.
Ideal Time:
Experienced: ≤ 20 sec
Intermediate: ≤ 30-60 sec
Beginner: ≤ 2 min