The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds . If they start changing simultaneously at 8 a.m., after how much time will they change again simultaneously ?

Answer

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

### Step-by-Step Solution:

1. **Identify the time intervals**:
- The first traffic light changes every 48 seconds.
- The second traffic light changes every 72 seconds.
- The third traffic light changes every 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 the 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 together:
- LCM = 16 × 27 = 432 seconds.

5. **Convert seconds into minutes and seconds**:
- To convert 432 seconds into minutes, divide by 60:
- 432 ÷ 60 = 7 minutes with a remainder of 12 seconds.
- Therefore, 432 seconds = 7 minutes and 12 seconds.

6. **Determine the next simultaneous change time**:
- If the traffic lights start changing at 8 a.m., they will change again simultaneously after 7 minutes and 12 seconds.
- Adding this to 8:00 a.m. gives:
- 8:00 a.m. + 7 minutes and 12 seconds = 8:07:12 a.m.

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