2021 Fall AMC 10B Problem 22

For each integer n\geq 2, let S_n be the sum of all products jk, where j and k are integers and 1\leq j<k\leq n. What is the sum of the 10 least values of n such that S_n is divisible by 3?

\textbf{(A)}\ 196 \qquad\textbf{(B)}\ 197 \qquad\textbf{(C)}\ 198 \qquad\textbf{(D)}\ 199 \qquad\textbf{(E)}\ 200

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