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Difficulty: HardFactors, Multiples, and Primes

Three automated beacons are programmed to flash at specific intervals. Beacon A flashes every 2424 seconds, Beacon B flashes every 3636 seconds, and Beacon C flashes every 4545 seconds. If all three beacons flash simultaneously at exactly 8:00:00 PM, how many more times will they flash together within exactly one hour?

  1. A
    6
  2. 10Answer
  3. C
    330
  4. D
    1,200

Answer

10
To determine when multiple periodic events occur at the exact same time, we must find their Least Common Multiple (LCM). Using prime factorization, we find the LCM of 2424, 3636, and 4545 by taking the highest power of each distinct prime factor present: 23×32×5=3602^3 \times 3^2 \times 5 = 360 seconds. This means the beacons align and flash together once every 360360 seconds. Because the total timeframe is exactly one hour, we convert this duration to seconds (60 minutes×60 seconds/minute=3,600 seconds60 \text{ minutes} \times 60 \text{ seconds/minute} = 3,600 \text{ seconds}). Finally, dividing the total time by the simultaneous interval (3,600÷3603,600 \div 360) results in exactly 1010 additional synchronized flashes.

Step-by-Step Solution

1
Determine the prime factorization of each beacon's flashing interval.
24=23×324 = 2^3 \times 3, 36=22×3236 = 2^2 \times 3^2, and 45=32×545 = 3^2 \times 5
Finding the prime factors is necessary to calculate the Least Common Multiple (LCM), which dictates when the independent events align.
2
Calculate the Least Common Multiple (LCM) of the three intervals.
LCM=23×32×5=8×9×5=360\text{LCM} = 2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360 seconds
The LCM represents the exact time interval in seconds at which all three beacons will flash together again.
3
Convert the total given time window (one hour) into the same units (seconds).
1 hour=60 minutes×60 seconds/minute=3,600 seconds1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3,600 \text{ seconds}
Both the flashing interval and the total time frame must be in identical units before dividing.
4
Divide the total time window by the LCM interval.
3,600÷360=103,600 \div 360 = 10
This division yields the total number of times the beacons will synchronize during the designated one-hour period.

Key Concept

Applying the Least Common Multiple (LCM) and unit conversions to solve periodic real-world alignment problems.
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Topics

Factors, Multiples, and Primes

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