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Three bells toll every 6 minutes, 8 minu...

Three bells toll every 6 minutes, 8 minutes and 12 minutes respectively. If they all start tolling at 8.00 a.m. at what time will they toll together again?

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To solve the problem of when the three bells will toll together again after starting at 8:00 a.m., we need to find the least common multiple (LCM) of the intervals at which each bell tolls. The bells toll every 6 minutes, 8 minutes, and 12 minutes respectively. ### Step-by-Step Solution: 1. **Identify the Tolling Intervals**: - Bell 1 tolls every 6 minutes. - Bell 2 tolls every 8 minutes. - Bell 3 tolls every 12 minutes. 2. **Find the LCM of the Intervals**: - We will use the prime factorization method to find the LCM. - First, we factor each number: - 6 = 2 × 3 - 8 = 2^3 - 12 = 2^2 × 3 - Next, we take the highest power of each prime factor: - For 2, the highest power is 2^3 (from 8). - For 3, the highest power is 3^1 (from 6 and 12). - Now, we calculate the LCM: - LCM = 2^3 × 3^1 = 8 × 3 = 24 minutes. 3. **Calculate the Next Toll Time**: - The bells toll together at 8:00 a.m. and will toll together again after 24 minutes. - To find the time, we add 24 minutes to 8:00 a.m. - 8:00 a.m. + 24 minutes = 8:24 a.m. ### Final Answer: The three bells will toll together again at **8:24 a.m.**
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