2018 AMC 12B Problem 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,\dfrac{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?

\textbf{(A) } 324 \qquad \textbf{(B) } 330 \qquad \textbf{(C) } 340 \qquad \textbf{(D) } 361 \qquad \textbf{(E) } 646

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.