Question

Mary chose an even $4$ -digit number $n.$ She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \tfrac{n}{2}, n.$ At some moment Mary wrote $323$ as a divisor of $n.$ What is the smallest possible value of the next divisor written to the right of $323?$

$324$

$330$

$340$

$361$

$646$

Answer: C

Concepts:divisibility greatest common divisor prime factorization

Difficulty rating: 2170

Solution:

Let $d$ be the next divisor after $323.$ If $\gcd(d,323)=1,$ then $323d$ divides $n,$ forcing $n\ge323d\gt323^2\gt9999,$ impossible for a $4$ -digit number. So $d$ shares a prime factor with $323=17\cdot19.$

Then $d-323\ge\gcd(d,323)\ge17,$ so $d\ge340.$ Indeed $d=340=17\cdot20$ occurs for $n=17\cdot19\cdot20=6460,$ which is even and $4$ -digit.

Thus, the correct answer is C.

2018 AMC 12B Problem 19

Answer: C

Difficulty rating: 2170

Solution:

Problem 19 in Other Years