AMC 12 2019b Problem 14

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$

$\textbf{(A) }98 \textbf{(B) }100 \textbf{(C) }117 \textbf{(D) }119 \textbf{(E) }121$