The traffic light at three different road crossing change after every 48 sec., 72 sec. and 108 sec . Respectively . If that all change simultaneously at `8:20:00` hrs. then will again change simultaneously at-

Options

A. `8:27:12` hrs
B. `8:27:24` hrs
C. `8:27:36` hrs
D. `8:27:48` hrs

Answer

To solve the problem of finding when the traffic lights will change simultaneously again after starting together at 8:20:00, we need to find the least common multiple (LCM) of the time intervals of the traffic lights (48 seconds, 72 seconds, and 108 seconds).

### Step-by-Step Solution:

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

2. **Find the prime factorization**:
- For 48:
- \(48 = 2^4 \times 3^1\)
- For 72:
- \(72 = 2^3 \times 3^2\)
- For 108:
- \(108 = 2^2 \times 3^3\)

3. **Determine the LCM**:
- The LCM is found by taking 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, the LCM is:
\[ LCM = 2^4 \times 3^3 = 16 \times 27 = 432 \text{ seconds} \]

4. **Convert 432 seconds to minutes and seconds**:
- 432 seconds can be converted to minutes:
- \(432 \div 60 = 7\) minutes and \(12\) seconds (since \(432 - 420 = 12\)).

5. **Add the LCM to the initial time**:
- The initial time is 8:20:00.
- Adding 7 minutes and 12 seconds:
- \(8:20:00 + 00:07:12 = 8:27:12\).

6. **Final Answer**:
- The traffic lights will change simultaneously again at **8:27:12**.