Four bells ring at the intervals of 5, 6,8 and 9 seconds . All the bells ring simulataneously at same time. They will again ring simultaneously after

Options

A. 6 minutes
B. 12 minutes
C. 18 minutes
D. 24 minutes

Answer

To find out when the four bells will ring simultaneously again, we need to calculate the Least Common Multiple (LCM) of the intervals at which they ring: 5 seconds, 6 seconds, 8 seconds, and 9 seconds.

### Step-by-Step Solution:

1. **Identify the numbers**: The intervals of the bells are 5, 6, 8, and 9 seconds.

2. **Find the prime factorization of each number**:
- 5 = 5^1
- 6 = 2^1 × 3^1
- 8 = 2^3
- 9 = 3^2

3. **Determine the highest power of each prime factor**:
- 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 5: The highest power is 5^1 (from 5).

4. **Calculate the LCM using the highest powers**:
\[ \text{LCM} = 2^3 × 3^2 × 5^1 \]

5. **Calculate each part**:
- \(2^3 = 8\)
- \(3^2 = 9\)
- \(5^1 = 5\)

6. **Multiply these values together**:
\[ \text{LCM} = 8 × 9 × 5 \]

7. **First, multiply 8 and 9**:
\[ 8 × 9 = 72 \]

8. **Now multiply the result by 5**:
\[ 72 × 5 = 360 \]

9. **Convert seconds to minutes**: Since the answer is in seconds, we convert it to minutes:
\[ 360 \text{ seconds} = \frac{360}{60} = 6 \text{ minutes} \]

### Final Answer:
The four bells will ring simultaneously again after **6 minutes**.