The traffic light at three different signal points change after every 15 seconds, 30 seconds and 45 seconds respectively. If all change simultaneously at 8: 15 :20 hours, then when will they again change simultaneously?

Options

A. 8:16:20 hours
B. 8:18:35 hours
C. 8:18:45 hours
D. 8:16:50 hours

Answer

To find out when the traffic lights will change simultaneously again, we need to determine the least common multiple (LCM) of the time intervals at which the lights change.

**Step 1: Identify the time intervals.**
The traffic lights change after every:
- Light 1: 15 seconds
- Light 2: 30 seconds
- Light 3: 45 seconds

**Step 2: Find the LCM of the time intervals.**
To find the LCM, we can use the prime factorization method.

- **15 seconds:**
- Prime factorization: \( 15 = 3 \times 5 \)

- **30 seconds:**
- Prime factorization: \( 30 = 2 \times 3 \times 5 \)

- **45 seconds:**
- Prime factorization: \( 45 = 3^2 \times 5 \)

**Step 3: Determine the highest power of each prime factor.**
- For \( 2 \): highest power is \( 2^1 \) (from 30)
- For \( 3 \): highest power is \( 3^2 \) (from 45)
- For \( 5 \): highest power is \( 5^1 \) (from both 15 and 30)

**Step 4: Calculate the LCM.**
Now, we multiply the highest powers of all prime factors:
\[ LCM = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 \]
Calculating this step-by-step:
- \( 2 \times 9 = 18 \)
- \( 18 \times 5 = 90 \)

So, the LCM of 15, 30, and 45 seconds is 90 seconds.

**Step 5: Calculate the next simultaneous change time.**
The lights change simultaneously at 8:15:20 hours. We need to add 90 seconds to this time.

- 90 seconds = 1 minute and 30 seconds.
- Adding this to 8:15:20:
- Add 1 minute: 8:16:20
- Add 30 seconds: 8:16:50

**Final Answer:**
The traffic lights will change simultaneously again at 8:16:50 hours.

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