Four different bells ring at intervals of 5, 6, 8 and 10 minutes respectively. If they ring together at 4 p.m. they will ring together again at-

Options

A. `5cdot30` p.m.
B. `6cdot00` p.m.
C. `7cdot00` p.m.
D. `8cdot10` p.m.

Answer

To solve the problem of when the four bells will ring together again after ringing at 4 p.m., we need to find the least common multiple (LCM) of the intervals at which they ring: 5 minutes, 6 minutes, 8 minutes, and 10 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The bells ring at intervals of 5, 6, 8, and 10 minutes.

2. **Find the LCM**:
- To find the LCM, we can use the prime factorization method.
- Prime factorization of each interval:
- 5 = 5^1
- 6 = 2^1 × 3^1
- 8 = 2^3
- 10 = 2^1 × 5^1
- Now, take the highest power of each prime number:
- For 2: highest power is 2^3 (from 8)
- For 3: highest power is 3^1 (from 6)
- For 5: highest power is 5^1 (from 5 and 10)
- Therefore, LCM = 2^3 × 3^1 × 5^1

3. **Calculate the LCM**:
- LCM = 8 × 3 × 5
- First, calculate 8 × 3 = 24
- Then, calculate 24 × 5 = 120
- So, the LCM of 5, 6, 8, and 10 is 120 minutes.

4. **Convert minutes to hours**:
- 120 minutes = 120 / 60 = 2 hours.

5. **Determine the next ringing time**:
- Since the bells ring together at 4 p.m., we add 2 hours to this time.
- 4 p.m. + 2 hours = 6 p.m.

### Final Answer:
The bells will ring together again at **6 p.m.**.