Four bells begin to all after 3, 8, 20 and 24 sec respectively. If they strike together at 12 noon, they will strike together again at

Options

A. `1` pm
B. `2` pm
C. 12.30 pm
D. 2 min past 12 noon

Answer

To solve the problem of when the four bells will strike together again after initially striking at 12 noon, we need to find the least common multiple (LCM) of the times after which each bell strikes: 3 seconds, 8 seconds, 20 seconds, and 24 seconds.

### Step-by-Step Solution:

1. **Identify the times after which each bell strikes:**
- Bell 1: 3 seconds
- Bell 2: 8 seconds
- Bell 3: 20 seconds
- Bell 4: 24 seconds

2. **Find the LCM of the given times (3, 8, 20, 24):**
- Start by finding the prime factorization of each number:
- 3 = \(3^1\)
- 8 = \(2^3\)
- 20 = \(2^2 \times 5^1\)
- 24 = \(2^3 \times 3^1\)

3. **Determine the highest power of each prime factor:**
- For the prime number 2: the highest power is \(2^3\) (from 8 and 24).
- For the prime number 3: the highest power is \(3^1\) (from 3 and 24).
- For the prime number 5: the highest power is \(5^1\) (from 20).

4. **Calculate the LCM using the highest powers:**
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
- First, calculate \(8 \times 3 = 24\).
- Then, calculate \(24 \times 5 = 120\).

5. **Convert the LCM from seconds to minutes:**
- Since the LCM is 120 seconds, we convert it to minutes:
\[ 120 \text{ seconds} = \frac{120}{60} = 2 \text{ minutes} \]

6. **Determine the time they will strike together again:**
- If they strike together at 12 noon, they will strike together again after 2 minutes.
- Therefore, the next time they will strike together is at:
\[ 12:00 \text{ noon} + 2 \text{ minutes} = 12:02 \text{ PM} \]

### Final Answer:
The four bells will strike together again at **12:02 PM**.