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

Question

Accepted Answer

To determine when the three clocks will chime together again, we need to find the least common multiple (LCM) of their chime intervals: 15 minutes, 20 minutes, and 30 minutes.

1. **Finding the LCM:**
   - Prime factorization of 15: $15 = 3 \times 5$
   - Prime factorization of 20: $20 = 2^2 \times 5$
   - Prime factorization of 30: $30 = 2 \times 3 \times 5$

The LCM is found by taking the highest power of each prime number that appears in the factorizations:
   - Highest power of 2: $2^2$
   - Highest power of 3: $3^1$
   - Highest power of 5: $5^1$

Therefore, the LCM is $2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60$.

2. **Explanation of the LCM:**
   - The LCM of 60 minutes means that all three clocks will chime together every 60 minutes.

3. **Calculating the next chime time:**
   - Since the clocks all chime together at 10 a.m., adding 60 minutes to this time gives us 11 a.m.

4. **Justification for Option 4:**
   - Option 4 states that the clocks will chime together again at 11 a.m., which is exactly 60 minutes after 10 a.m. This matches our calculation based on the LCM.

5. **Explanation of why other options are incorrect:**
   - **Option 1 (10:45 a.m.):** This is 45 minutes after 10 a.m., which is not a multiple of the LCM (60 minutes). Therefore, the clocks will not chime together at this time.
   - **Option 2 (10:50 a.m.):** This is 50 minutes after 10 a.m., which is also not a multiple of the LCM (60 minutes). Thus, the clocks will not chime together at this time.
   - **Option 3 (11:15 a.m.):** This is 75 minutes after 10 a.m., which again is not a multiple of the LCM (60 minutes). Hence, the clocks will not chime together at this time.

In conclusion, the correct answer is Option 4, 11 a.m., as it is the only time when all three clocks will chime together again after 10 a.m., based on the calculated LCM of their intervals.