The bells of three Churchs ring on the interval of 10, 12 and 15 minute respectively. If those ring at 8:30 A.M. together, then when will those ring next time together?

Options

A. 9:00 A.M.
B. 9:30 A.M.
C. 10:00 A.M.
D. 12:00 A.M.

Answer

To find out when the bells of the three churches will ring together again after 8:30 A.M., we need to calculate the least common multiple (LCM) of the intervals at which they ring: 10 minutes, 12 minutes, and 15 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The bells ring at intervals of 10 minutes, 12 minutes, and 15 minutes.

2. **Find the LCM**:
- **Prime factorization**:
- 10 = 2 × 5
- 12 = 2² × 3
- 15 = 3 × 5
- **Take the highest power of each prime**:
- For 2: highest power is 2² (from 12)
- For 3: highest power is 3¹ (from 12 and 15)
- For 5: highest power is 5¹ (from 10 and 15)
- **Calculate the LCM**:
- LCM = 2² × 3¹ × 5¹
- LCM = 4 × 3 × 5
- LCM = 12 × 5
- LCM = 60 minutes

3. **Determine the next time they will ring together**:
- They ring together at 8:30 A.M. and will ring together again after 60 minutes.
- 60 minutes = 1 hour.
- Therefore, 8:30 A.M. + 1 hour = 9:30 A.M.

### Final Answer:
The bells will ring together again at **9:30 A.M.**.

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