Teachers and students are playing a game together. Students clap after every 3 seconds, while the teacher claps after every 6 seconds. If both of them start the game together, when will both the teacher and students clap together the next time?
Two traffic lights at an intersection change at different intervals. The first light changes every 45 seconds, and the second light changes every 60 seconds. If both lights just changed together, in how many seconds will they change together again?
Two cleaning robots are set to clean at different intervals. Robot A cleans every 18 hours, and Robot B cleans every 24 hours. If both robots start cleaning at midnight, when will they clean together again?
Answer (2)
The teacher and students will clap together again in 6 seconds. The traffic lights will change together after 180 seconds. The cleaning robots will clean together again in 72 hours, or 3 days, after midnight.
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To solve these problems, we need to find the least common multiple (LCM) of the given time intervals.
Students and Teachers Clapping:
Students clap every 3 seconds.
The teacher claps every 6 seconds.
We find the LCM of 3 and 6.
The factors of 3 are: 1, 3. The factors of 6 are: 1, 2, 3, 6.
The LCM of 3 and 6 is 6.
Therefore, the teacher and students will clap together every 6 seconds.
Traffic Lights Changing:
The first light changes every 45 seconds.
The second light changes every 60 seconds.
We need to find the LCM of 45 and 60.
The prime factors of 45 are: 3 2 × 5 . The prime factors of 60 are: 2 2 × 3 × 5 .
To find the LCM, take the highest power of each prime factor: 2 2 , 3 2 , 5 .
L CM = 2 2 × 3 2 × 5 = 180 .
Therefore, the lights will change together every 180 seconds.
Cleaning Robots:
Robot A cleans every 18 hours.
Robot B cleans every 24 hours.
We find the LCM of 18 and 24.
The prime factors of 18 are: 2 × 3 2 . The prime factors of 24 are: 2 3 × 3 .
To find the LCM, take the highest power of each prime factor: 2 3 , 3 2 .
L CM = 2 3 × 3 2 = 72 .
Therefore, both robots will clean together every 72 hours, or 3 days, if they start cleaning together at midnight.
These steps give us the time when both events will synchronize based on their interval timings.
