Three bells toll at intervals of 12 minutes ,15 minutes and 18 minutes respectively, if they start tolling together, after what time will they next toll together

Answer

To find out when the three bells will next toll together, we need to calculate the Least Common Multiple (LCM) of their tolling intervals: 12 minutes, 15 minutes, and 18 minutes.

### Step-by-Step Solution:

1. **Identify the intervals**: The intervals at which the bells toll are 12, 15, and 18 minutes.

2. **Prime Factorization**:
- For 12:
- \(12 = 2^2 \times 3^1\)
- For 15:
- \(15 = 3^1 \times 5^1\)
- For 18:
- \(18 = 2^1 \times 3^2\)

3. **Determine the LCM**:
- To find the LCM, we take the highest power of each prime number that appears in the factorizations:
- For the prime number 2: The highest power is \(2^2\) (from 12).
- For the prime number 3: The highest power is \(3^2\) (from 18).
- For the prime number 5: The highest power is \(5^1\) (from 15).
- Therefore, the LCM is calculated as follows:
\[ LCM = 2^2 \times 3^2 \times 5^1 \]

4. **Calculate the LCM**:
- First, calculate \(2^2 = 4\).
- Next, calculate \(3^2 = 9\).
- Now, multiply these results with 5:
\[ LCM = 4 \times 9 \times 5 \]
- Calculate \(4 \times 9 = 36\).
- Finally, calculate \(36 \times 5 = 180\).

5. **Conclusion**:
- The LCM of 12, 15, and 18 is 180 minutes.
- Therefore, the three bells will next toll together after **180 minutes**.