Three hells chime at intervals of 48,60 and 90 min,respectively.If all the three bells chime together at 10:00 am, at what time will all the three chime again that day?

Options

A. `1:00 PM`
B. `2:00` PM
C. `8:00` PM
D. `10:00 `PM

Answer

To solve the problem of when the three bells will chime together again after 10:00 AM, we need to find the least common multiple (LCM) of the intervals at which the bells chime: 48 minutes, 60 minutes, and 90 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The intervals at which the bells chime are 48 minutes, 60 minutes, and 90 minutes.

2. **Prime Factorization**:
- For 48:
- \(48 = 2^4 \times 3^1\)
- For 60:
- \(60 = 2^2 \times 3^1 \times 5^1\)
- For 90:
- \(90 = 2^1 \times 3^2 \times 5^1\)

3. **Determine the LCM**:
- To find the LCM, take the highest power of each prime factor from the factorizations:
- For \(2\): highest power is \(2^4\) (from 48)
- For \(3\): highest power is \(3^2\) (from 90)
- For \(5\): highest power is \(5^1\) (from both 60 and 90)
- Therefore, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 5^1 \]

4. **Calculate the LCM**:
- Calculate \(2^4 = 16\)
- Calculate \(3^2 = 9\)
- Calculate \(5^1 = 5\)
- Now multiply these together:
\[ LCM = 16 \times 9 \times 5 \]
- First, calculate \(16 \times 9 = 144\)
- Then, calculate \(144 \times 5 = 720\)
- Thus, the LCM is \(720\) minutes.

5. **Convert LCM to Hours**:
- Since \(720\) minutes is equivalent to \(720 \div 60 = 12\) hours.

6. **Find the Next Chime Time**:
- If the bells chime together at 10:00 AM, adding \(12\) hours gives:
\[ 10:00 \text{ AM} + 12 \text{ hours} = 10:00 \text{ PM} \]

### Final Answer:
The three bells will chime together again at **10:00 PM**.