Light [tex] m [/tex] flashes every 4 minutes. Light [tex] n [/tex] flashes every 10 minutes. If the lights flashed together at 6:00 AM, at what time would they next flash together?

To solve this problem, we need to determine when both lights will flash together again, given that Light M flashes every 4 minutes and Light N flashes every 10 minutes. Since they both flashed together at an initial time, we are looking for the next time this happens.

Here's a step-by-step explanation:

1. Identify the Flash Intervals:
- Light M flashes every 4 minutes.
- Light N flashes every 10 minutes.

2. Determine the Least Common Multiple (LCM):
- To find out when both lights will flash together again, we need to calculate the Least Common Multiple (LCM) of the two intervals, 4 and 10 minutes. The LCM is the smallest positive number that is a multiple of both intervals.

3. Calculate the LCM of 4 and 10:
- The factors of 4 are 2 and 2.
- The factors of 10 are 2 and 5.
- To find the LCM, take the highest power of each prime factor present in 4 and 10:
- The highest power of 2 is 2² (from 4).
- The highest power of 5 is 5¹ (from 10).
- Multiply these together: [tex]\(2^2 \times 5^1 = 4 \times 5 = 20\)[/tex].

4. Result:
- The LCM of 4 and 10 is 20. Therefore, both lights will flash together every 20 minutes.

5. Calculate the Next Time They Flash Together:
- Since they initially flash together at G:00 am, they will next flash together 20 minutes later.

Conclusion:
The next time Light M and Light N will flash together after G:00 am is at G:20 am.