Question

Mary chose an even $4$ -digit number $n.$ She wrote down all the divisors of $n$ in increasing order from left to right: $1, 2, \ldots, \frac{n}{2}, n.$ 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 $323?$

$324$

$330$

$340$

$361$

$646$

Solution:

Factor $323 = 17 \cdot 19.$ Since it divides the even number $n,$ $n$ is a multiple of $2 \cdot 17 \cdot 19 = 646.$ For the next divisor $d,$ $n$ has to be a multiple of $\operatorname{lcm}(323, d).$ Something like $324$ or $330 = 2 \cdot 3 \cdot 5 \cdot 11$ shares no factor with $323,$ which pushes $n \ge 323 \cdot 324 > 9999,$ too big. But $d = 340 = 2^2 \cdot 5 \cdot 17$ gives $\operatorname{lcm}(323, 340) = 2^2 \cdot 5 \cdot 17 \cdot 19 = 6460,$ an even $4$ -digit number whose divisor list jumps straight from $323$ to $340.$ So the smallest possible next divisor is $340.$ Thus, C is the correct answer.

2018 AMC 10B Problem 21

Solution:

Problem 21 in Other Years