Four bells ring at the intervals of 5, 6, 8 and 9 seconds. All the bells ring simultaneously 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 after starting together, we need to calculate the Least Common Multiple (LCM) of their ringing intervals: 5 seconds, 6 seconds, 8 seconds, and 9 seconds.

### Step-by-Step Solution:

1. **Identify the intervals**: The intervals at which the bells ring are 5 seconds, 6 seconds, 8 seconds, and 9 seconds.

2. **Find the prime factorization of each interval**:
- 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. **Multiply these highest powers together to find the LCM**:
\[ \text{LCM} = 2^3 × 3^2 × 5^1 \]
- Calculate \(2^3 = 8\)
- Calculate \(3^2 = 9\)
- Calculate \(5^1 = 5\)

5. **Now multiply these results**:
\[ 8 × 9 = 72 \]
\[ 72 × 5 = 360 \]

6. **Convert seconds to minutes**: Since the LCM is 360 seconds, we convert this to minutes:
\[ 360 \text{ seconds} ÷ 60 = 6 \text{ minutes} \]

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

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