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Difficulty: MediumGreatest Common Divisor (GCD) and Least Common Multiple (LCM)

Three automated security beacons flash at regular intervals of 1818 seconds, 2424 seconds, and 4242 seconds, respectively. If all three beacons flash simultaneously at 8:00:00 AM, how many times will all three beacons flash simultaneously between 8:01:00 AM and 9:00:00 AM, inclusive?

Answer: 7

Answer

7
The least common multiple of 18, 24, and 42 is 504 seconds. The timeframe between 8:01:00 AM and 9:00:00 AM inclusive represents elapsed times from 60 seconds to 3600 seconds after 8:00:00 AM. Dividing 3600 by 504 gives 7 full periods (3528 seconds), all of which are greater than 60 seconds. Therefore, the beacons flash simultaneously 7 times within the specified period.

Step-by-Step Solution

1
Find the Least Common Multiple (LCM) of the three flash intervals.
LCM(18, 24, 42) = 504 seconds
The beacons flash simultaneously at time intervals that are integer multiples of the LCM of their individual cycle times.
2
Convert the target time interval (8:01:00 AM to 9:00:00 AM) into seconds after the initial flash at 8:00:00 AM (t = 0).
Range is from t = 60 seconds to t = 3600 seconds
8:01:00 AM is 60 seconds after 8:00:00 AM, and 1 hour (9:00:00 AM) equals 3600 seconds after 8:00:00 AM.
3
Determine the number of positive multiples of 504 within the interval [60, 3600].
7 multiples (504, 1008, 1512, 2016, 2520, 3024, 3528)
Dividing 3600 by 504 gives 7 with a remainder of 72. Since the first positive multiple (504) is greater than 60, all 7 multiples fall strictly within the specified window.

Key Concept

Application of Least Common Multiple (LCM) to periodic events
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Topics

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

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