Answer :
Final Answer:
The three lights flash together every 2 minutes (120 seconds), starting at 10:32 p.m. To find when they flash together after midnight, we add 2 hours, accounting for the 30 minutes that have already passed, resulting in 1:10 a.m. as the next time they flash together.
Explanation:
To find out when the three lights will flash together again, we need to determine the least common multiple (LCM) of their respective flashing intervals: 6 seconds, 8 seconds, and 10 seconds.
First, we find the LCM of 6, 8, and 10. To do this, we break down each number into its prime factors:
- 6 = 2 x 3
- 8 = 2 x 2 x 2
- 10 = 2 x 5
Next, we identify the highest power of each prime factor that appears in these numbers:
- 2^3 (from 8)
- 3^1 (from 6)
- 5^1 (from 10)
Now, we calculate the LCM by multiplying these highest powers of prime factors:
LCM = 2^3 x 3^1 x 5^1 = 8 x 3 x 5 = 120 seconds.
So, the lights will flash together again every 120 seconds, which is equivalent to 2 minutes. To find out what time they will flash together after 10:30 p.m., we add 2 minutes to 10:30 p.m., resulting in 10:32 p.m. as the first time the lights will flash together. However, we need to find when they will flash together again after midnight.
At midnight (12:00 a.m.), we add another 2 minutes, bringing us to 12:02 a.m. The lights will continue to flash together every 2 minutes, so we add 2 hours to this time to find the final answer:
12:02 a.m. + 2 hours = 2:02 a.m.
However, since we started at 10:30 p.m., we must account for the 30 minutes that have already passed, making the complete time:
2:02 a.m. + 30 minutes = 2 hours and 32 minutes after 10:30 p.m.
So, the lights will flash together again at 2 hours and 32 minutes after 10:30 p.m., which is 1:10 a.m.
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