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Difficulty: EasyFactors, Multiples, and Prime Factorization

Three church bells toll together at intervals of 99 minutes, 1212 minutes, and 1515 minutes respectively. If they all toll together at 8:00 AM8:00\text{ AM}, at what time will they next toll together?

  1. A
    8:03 AM
  2. B
    8:36 AM
  3. C
    10:00 AM
  4. 11:00 AMAnswer

Answer

11:00 AM
The bells will toll together after a duration equal to the Least Common Multiple (LCM) of their individual tolling intervals. The prime factorizations are 9=329 = 3^2, 12=22×312 = 2^2 \times 3, and 15=3×515 = 3 \times 5. Taking the highest power of each prime factor yields LCM(9,12,15)=22×32×5=180\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 180 minutes, which is equal to 33 hours. Adding 33 hours to 8:00 AM8:00\text{ AM} gives 11:00 AM11:00\text{ AM}.

Step-by-Step Solution

1
Find prime factorizations of each interval
9=329 = 3^2, 12=22×312 = 2^2 \times 3, and 15=3×515 = 3 \times 5
To compute the Least Common Multiple (LCM), express each number in prime factor form.
2
Calculate the LCM
LCM(9,12,15)=22×32×5=4×9×5=180 minutes\text{LCM}(9, 12, 15) = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180\text{ minutes}
Take the highest power of each prime factor involved.
3
Convert minutes to hours and add to the initial time
180 minutes=3 hours180\text{ minutes} = 3\text{ hours}; 8:00 AM+3 hours=11:00 AM8:00\text{ AM} + 3\text{ hours} = 11:00\text{ AM}
Determine the exact time the bells will next chime together.

Key Concept

Application of Least Common Multiple (LCM) in Simultaneous Periodic Events
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Topics

Factors, Multiples, and Prime Factorization

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