Answer:
7:48 p.m.
Step-by-step explanation:
You want to know when lights with periods 36 s, 64 s, and 110 s will change simultaneously again if they do so at 11 a.m..
LCM
The lights will change simultaneously after a time that is the least common multiple (LCM) of the periods of the lights.
The least common multiple is the product of the unique factors in each of the numbers:
36 = 2² · 3²
64 = 2⁶
110 = 2 · 5 · 11
The unique factors are 2⁶, 3², 5, and 11. Their product is 31680. The lights will change simultaneously again after 31680 seconds.
Time conversion
We can convert this to hours, minutes, and seconds by repeated division by 60:
31680 / 60 = 528 r 0
528 / 60 = 8 r 48
31680 seconds is 8 hours, 48 minutes, and no seconds.
Clock time
Adding that time to 11 a.m. gives a clock time of ...
11:00 +8:48 = 19:48 = 7:48 p.m.
The lights will change simultaneously again at 7:48 p.m..
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Additional comment
The lights will change simultaneously at 11 a.m. again after 11 days.
