Seven bells ring at intervals of 2,3,4,6,8,9 and 12 min,respectively.They started ringing simultaneously at 5:00 in the morning. What will be the next time when they all ring simultaneously?

Answer

To find the next time when all seven bells ring simultaneously after starting at 5:00 AM, we need to calculate the Least Common Multiple (LCM) of their ringing intervals: 2, 3, 4, 6, 8, 9, and 12 minutes.

### Step-by-Step Solution:

1. **List the Intervals**:
The intervals at which the bells ring are: 2, 3, 4, 6, 8, 9, and 12 minutes.

2. **Find the Prime Factorization**:
- 2 = 2
- 3 = 3
- 4 = 2²
- 6 = 2 × 3
- 8 = 2³
- 9 = 3²
- 12 = 2² × 3

3. **Identify the Highest Powers of Each Prime**:
- For prime number 2: The highest power is 2³ (from 8).
- For prime number 3: The highest power is 3² (from 9).

4. **Calculate the LCM**:
The LCM is found by multiplying the highest powers of all prime factors:
\[ LCM = 2³ × 3² = 8 × 9 = 72 \text{ minutes} \]

5. **Add the LCM to the Start Time**:
The bells started ringing at 5:00 AM. We need to add 72 minutes to this time:
- 72 minutes = 1 hour and 12 minutes.
- Therefore, 5:00 AM + 1 hour 12 minutes = 6:12 AM.

### Conclusion:
The next time when all the bells will ring simultaneously is **6:12 AM**.

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