The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 a.m . Then at what time will they again change simultanously ?

Options

A. `8:7:22` hrs
B. `8:7:21` hrs
C. `8:7:12` hrs
D. `8:7:11` hrs

Explanation

To find out when the traffic lights will change simultaneously again after 8 a.m., we need to calculate the least common multiple (LCM) of the three time intervals: 48 seconds, 72 seconds, and 108 seconds.

### Step-by-Step Solution:

1. **Identify the time intervals**: The traffic lights change every 48 seconds, 72 seconds, and 108 seconds.

2. **Find the prime factorization of each interval**:
- 48 = 2^4 × 3^1
- 72 = 2^3 × 3^2
- 108 = 2^2 × 3^3

3. **Determine the LCM**:
- For LCM, we take the highest power of each prime factor from the factorizations:
- For 2: the highest power is 2^4 (from 48)
- For 3: the highest power is 3^3 (from 108)
- Therefore, LCM = 2^4 × 3^3

4. **Calculate the LCM**:
- 2^4 = 16
- 3^3 = 27
- Now, multiply these two results:
- LCM = 16 × 27 = 432 seconds

5. **Convert seconds to minutes**:
- To convert 432 seconds into minutes, divide by 60:
- 432 ÷ 60 = 7 minutes and 12 seconds

6. **Add the time to the initial time**:
- The initial time is 8:00 a.m.
- Adding 7 minutes and 12 seconds to 8:00 a.m. gives us:
- 8:00 a.m. + 7 minutes = 8:07 a.m.
- Adding 12 seconds results in 8:07:12 a.m.

### Final Answer:
The traffic lights will change simultaneously again at **8:07:12 a.m.**