Three bells ring at an interval of 4, 7 and 12 seconds. They start to ring together at 7.00 a.m. When will they ring together again ?

Answer

To solve the problem of when the three bells will ring together again after starting at 7:00 a.m., we need to find the Least Common Multiple (LCM) of the intervals at which they ring: 4 seconds, 7 seconds, and 12 seconds.

### Step-by-Step Solution:

1. **Identify the intervals**: The three bells ring at intervals of 4 seconds, 7 seconds, and 12 seconds.

2. **Find the LCM**: We need to find the LCM of the numbers 4, 7, and 12. We can use the prime factorization method to do this.

- **Prime factorization**:
- 4 = 2 × 2 = \(2^2\)
- 7 = 7 = \(7^1\)
- 12 = 2 × 2 × 3 = \(2^2 × 3^1\)

3. **Determine the highest power of each prime factor**:
- For the prime number 2, the highest power is \(2^2\) (from 4 and 12).
- For the prime number 3, the highest power is \(3^1\) (from 12).
- For the prime number 7, the highest power is \(7^1\) (from 7).

4. **Calculate the LCM**: Multiply the highest powers of all prime factors together:
\[ LCM = 2^2 × 3^1 × 7^1 = 4 × 3 × 7 \]
- First, calculate \(4 × 3 = 12\).
- Then, calculate \(12 × 7 = 84\).

5. **Convert seconds to minutes and seconds**: The LCM is 84 seconds. To convert this into minutes and seconds:
- 84 seconds = 1 minute and 24 seconds.

6. **Determine the next ringing time**: Since the bells start ringing together at 7:00 a.m., we add 1 minute and 24 seconds to this time:
- 7:00 a.m. + 1 minute 24 seconds = 7:01:24 a.m.

### Final Answer:
The three bells will ring together again at **7:01:24 a.m.**

---