Three bells ring at an interval of 15, 20 & 30 minutes respectively. If all three bells ring together at 11 am, They when will they ring together again ?

Options

A. `11:30` AM
B. `12:00` PM
C. `12:30` PM
D. `1:00` PM

Answer

To find out when the three bells will ring together again after 11 am, we need to determine the least common multiple (LCM) of their ringing intervals: 15 minutes, 20 minutes, and 30 minutes.

### Step-by-Step Solution:

1. **Identify the ringing intervals**:
- Bell 1: 15 minutes
- Bell 2: 20 minutes
- Bell 3: 30 minutes

2. **Find the prime factorization of each interval**:
- 15 = 3 × 5
- 20 = 2² × 5
- 30 = 2 × 3 × 5

3. **Determine the highest power of each prime factor**:
- For the prime number 2: The highest power is 2² (from 20).
- For the prime number 3: The highest power is 3¹ (from 15 and 30).
- For the prime number 5: The highest power is 5¹ (from all three).

4. **Calculate the LCM**:
- LCM = 2² × 3¹ × 5¹
- LCM = 4 × 3 × 5
- LCM = 12 × 5
- LCM = 60 minutes

5. **Determine the next time they will ring together**:
- Since they all ring together at 11 am, we add the LCM (60 minutes) to this time.
- 11 am + 60 minutes = 12 pm (noon).

### Final Answer:
The three bells will ring together again at **12 pm**.