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, …, (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?

2018 AMC 12B Problem 19 — Solution

2018 AMC 12B Problem 19

Below is the professionally curated solution for Problem 19 of the 2018 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12B solutions, or check the answer key.

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Concepts:divisibility greatest common divisor prime factorization

Difficulty rating: 2170

19.

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 $323?$

$324$

$330$

$340$

$361$

$646$

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.

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