2025 AMC 10A Problems/Problem 17

(Problem goes here)

Problem

Let be the unique positive integer such that dividing by leaves a remainder of and dividing by leaves a remainder of . What is the tens digit of ?

Solution 1

The problem statement implies and We want to find that satisfies both of these conditions. Hence, we can just find the greatest common divisor of the two numbers. so the answer is

~Tacos_are_yummy_1

Alternate Solution

We are given \(273436 \equiv 16 \pmod{N}\) and \(272760 \equiv 15 \pmod{N}.\) Subtracting the second equation from the first yields \(273436 - 272760 \equiv 16 - 15 \pmod{N}.\) so now, \(676 \equiv 1 \pmod{N}.\) Hence, \(N \mid (676 - 1) = 675.\) The prime factorization of 675 is \(675 = 5^2 \times 3^3.\) Therefore, \(N\) must be a divisor of 675. Since the remainders are less than \(N,\) we have \(N > 16.\) The largest divisor of 675 greater than 16 is \(45.\) The tens digit of 45 is 4. Thus, the answer is \(\boxed{\text{(E) }4}.\)

~WildSealVM/Vincent M.