AMC 10 2021-fall-b Problem 4

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$ $\textbf{(A)}\: 10 \textbf{(B)} \: 30 \textbf{(C)} \: 60 \textbf{(D)} \: 100 \textbf{(E)} \: 120$