Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16$ and dividing $272760$ by $N$ leaves a remainder of $15$. What is the tens digit of $N$?
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$
Difficulty: 36/100 — Although admittedly a little guessable, the actual idea is a little tricky to find and the computation is slightly vexing.
Core Concepts: modular arithmetic, greatest common divisor, Euclidean algorithm
Challenges: Reinterpreting the condition to find a set of values for $N$ in each case.
Techniques: When the problem reduces to a greatest common divisor of big numbers, apply the Euclidean algorithm. Identifying the remainder to divisibility conversion.
Error-prone Steps: Simplifying the greatest common divisor incorrectly.
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2-3 min
Beginner: ≤ 5-6 min