In a shop. There are three clocks which chime at intervals of 15,20 and 30 minutes respectively. They all chimetogether at 10 a.m. At what time will they all chime together again ?

Answer

To solve the problem of when the three clocks will chime together again, we need to find the Least Common Multiple (LCM) of their chiming intervals: 15 minutes, 20 minutes, and 30 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The chiming intervals of the clocks are 15 minutes, 20 minutes, and 30 minutes.

2. **Find the prime factorization**:
- For 15:
- \(15 = 3 \times 5\)
- For 20:
- \(20 = 2^2 \times 5\)
- For 30:
- \(30 = 2 \times 3 \times 5\)

3. **List the highest powers of each prime factor**:
- The prime factors involved are 2, 3, and 5.
- The highest powers are:
- \(2^2\) (from 20)
- \(3^1\) (from 15 or 30)
- \(5^1\) (from any of the three)

4. **Calculate the LCM**:
- LCM = \(2^2 \times 3^1 \times 5^1\)
- LCM = \(4 \times 3 \times 5\)
- LCM = \(12 \times 5 = 60\)

5. **Interpret the result**:
- The LCM of 15, 20, and 30 is 60 minutes. This means that all three clocks will chime together again after 60 minutes.

6. **Determine the time**:
- They all chimed together at 10:00 AM.
- Adding 60 minutes (1 hour) to 10:00 AM gives us 11:00 AM.

### Final Answer:
The three clocks will chime together again at **11:00 AM**.