AMC 12 2018b 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 $323$?

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