Four bells ring at intervals of 4, 6, 8 and 14 seconds. They start ringing simultaneously at 12.00 O' clock. At what time will they again ring simultaneously?

Options

A. 12 hrs. 2 min. 48 sec.
B. 12 hrs. 3 min.
C. 12 hrs. 3 min. 20 sec
D. 12 hrs. 3 min. 44 sec.

Answer

To solve the problem of when the four bells will ring simultaneously again after starting at 12:00, we need to find the Least Common Multiple (LCM) of the intervals at which they ring: 4 seconds, 6 seconds, 8 seconds, and 14 seconds.

### Step-by-Step Solution:

1. **List the intervals**: The bells ring at intervals of 4 seconds, 6 seconds, 8 seconds, and 14 seconds.

2. **Find the prime factorization of each interval**:
- 4 = 2²
- 6 = 2¹ × 3¹
- 8 = 2³
- 14 = 2¹ × 7¹

3. **Identify the highest power of each prime factor**:
- For the prime number 2: The highest power is 2³ (from 8).
- For the prime number 3: The highest power is 3¹ (from 6).
- For the prime number 7: The highest power is 7¹ (from 14).

4. **Calculate the LCM**:
- LCM = 2³ × 3¹ × 7¹
- LCM = 8 × 3 × 7
- LCM = 24 × 7
- LCM = 168 seconds

5. **Convert seconds into minutes**:
- 168 seconds = 2 minutes and 48 seconds.

6. **Determine the time when they will ring again**:
- The bells will ring together again 2 minutes and 48 seconds after 12:00.
- Therefore, they will ring together again at 12:02:48.

### Final Answer:
The bells will ring simultaneously again at **12:02:48**.