2021 Fall AMC 12B Problem 3

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\qquad\textbf{(B)} \: 30\qquad\textbf{(C)} \: 60\qquad\textbf{(D)} \: 100\qquad\textbf{(E)} \: 120

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