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Difficulty: MediumPrime Factorization, GCD, and LCM

Three decorative light signals flash at regular intervals of 1515 seconds, 2020 seconds, and 3636 seconds, respectively. If all three signals flash simultaneously at 12:00 PM, how many times will all three signals flash simultaneously between 12:01 PM and 1:00 PM, inclusive?

Answer: 20

Answer

20
The three signals flash together every 180 seconds (3 minutes), which is the least common multiple of 15, 20, and 36. Within the 60-minute interval from 12:01 PM to 1:00 PM inclusive, simultaneous flashes occur at minute 3, 6, 9, ..., up to minute 60, resulting in exactly 20 simultaneous flashes.

Step-by-Step Solution

1
Find the prime factorization of each interval in seconds
15=3⋅515 = 3 \cdot 5, 20=22⋅520 = 2^2 \cdot 5, 36=22⋅3236 = 2^2 \cdot 3^2
Prime factorization allows systematic calculation of the least common multiple.
2
Calculate the LCM of 15, 20, and 36
LCM=22⋅32⋅5=180\text{LCM} = 2^2 \cdot 3^2 \cdot 5 = 180 seconds
The LCM gives the minimum period of time after which all three signals flash together.
3
Convert the period into minutes and find the frequency in 60 minutes
180 seconds=3 minutes180\text{ seconds} = 3\text{ minutes}; 60÷3=2060 \div 3 = 20 flashes
The timeframe between 12:01 PM and 1:00 PM inclusive contains 60 minutes, yielding 20 multiples of 3 minutes.

Key Concept

Least Common Multiple (LCM) for periodic events
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Topics

Prime Factorization, GCD, and LCM

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