Five bells start ringing in the interval of 6, 7, 8, 9, 12 seconds respectively. after how many seconds they will ring at the same time ?

Options

A. 72
B. 612
C. 504
D. 318

Answer

To find out after how many seconds the five bells will ring at the same time, we need to calculate the Least Common Multiple (LCM) of the intervals at which they ring: 6, 7, 8, 9, and 12 seconds.

### Step-by-Step Solution:

1. **List the intervals**: The intervals at which the bells ring are 6, 7, 8, 9, and 12 seconds.

2. **Find the prime factorization of each number**:
- 6 = 2 × 3
- 7 = 7 (it is a prime number)
- 8 = 2^3
- 9 = 3^2
- 12 = 2^2 × 3

3. **Identify the highest powers of all prime factors**:
- For the prime number 2: The highest power is 2^3 (from 8).
- For the prime number 3: The highest power is 3^2 (from 9).
- For the prime number 7: The highest power is 7^1 (from 7).

4. **Multiply these highest powers together to find the LCM**:
\[ \text{LCM} = 2^3 × 3^2 × 7^1 \]

5. **Calculate each part**:
- 2^3 = 8
- 3^2 = 9
- 7^1 = 7

6. **Now multiply these results**:
\[ \text{LCM} = 8 × 9 × 7 \]

7. **Calculate step by step**:
- First, calculate 8 × 9 = 72
- Then, calculate 72 × 7 = 504

8. **Conclusion**: The least common multiple of 6, 7, 8, 9, and 12 is 504 seconds. Therefore, the bells will ring together after 504 seconds.

### Final Answer:
The bells will ring at the same time after **504 seconds**.