Final answer:
The two lighthouses will flash together again at 10:03:20 pm, as the Least Common Multiple of their flashing intervals is 200 seconds.
Step-by-step explanation:
To find out when two lighthouses flash together again, we need to calculate the Least Common Multiple (LCM) of their flashing intervals. One lighthouse flashes every 40 seconds, and another flashes every 50 seconds. The LCM of 40 and 50 is the smallest number that both 40 and 50 can divide into without leaving a remainder. Calculating the LCM of 40 and 50, we find it to be 200 seconds. Therefore, since they flash together at 10 pm, they will flash together again 200 seconds later. The first number that is common to both lists is 200. Therefore, the lighthouses will flash together again after 200 seconds, or 3 minutes and 20 seconds.
200 seconds is equal to 3 minutes and 20 seconds, so adding this to 10 pm, we get that the lighthouses will flash together again at 10:03:20 pm.