Three bells rings at the interval of 10, 15 and 20 min, respectively. If they all ring at 11 am at once, then What time they will ring together again?

Options

A. `1:00` pm
B. `1:20` pm
C. `12:00` noon
D. `1:30` pm

Answer

To solve the problem of when the three bells will ring together again after ringing at 11 am, we need to find the least common multiple (LCM) of their ringing intervals, which are 10 minutes, 15 minutes, and 20 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The intervals at which the bells ring are 10 minutes, 15 minutes, and 20 minutes.

2. **Find the LCM**:
- The first step in finding the LCM is to factor each number into its prime factors:
- 10 = 2 × 5
- 15 = 3 × 5
- 20 = 2² × 5
- Next, identify the highest power of each prime factor that appears in these factorizations:
- For the prime number 2, the highest power is 2² (from 20).
- For the prime number 3, the highest power is 3¹ (from 15).
- For the prime number 5, the highest power is 5¹ (appears in all).
- Now, multiply these highest powers together to find the LCM:
- LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5

3. **Calculate the LCM**:
- First, calculate 4 × 3 = 12
- Then, calculate 12 × 5 = 60
- Therefore, the LCM of 10, 15, and 20 is 60 minutes.

4. **Determine the next ringing time**:
- Since all three bells ring together at 11 am, we add the LCM (60 minutes) to this time.
- 11 am + 60 minutes = 12 noon.

5. **Conclusion**:
- The three bells will ring together again at 12 noon.

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