The traffic lights at three different road crossings change after 24 seconds, 36 seconds and 54 seconds respectively. If they all change simultaneously at 10:15 :00 AM, then at what time will they again change simultaneously?

Options

A. `10:16:54 AM`
B. `10:18:36 AM`
C. `10:17:02 AM`
D. `10:22:12 AM`

Answer

To solve the problem of when the traffic lights will change simultaneously again after initially changing at 10:15:00 AM, we need to follow these steps:

### Step 1: Identify the time intervals for each traffic light
The traffic lights change at intervals of:
- Light 1: 24 seconds
- Light 2: 36 seconds
- Light 3: 54 seconds

### Step 2: Calculate the Least Common Multiple (LCM)
To find out when all three lights will change simultaneously again, we need to calculate the LCM of the three intervals (24, 36, and 54).

1. **Prime factorization**:
- 24 = 2^3 × 3^1
- 36 = 2^2 × 3^2
- 54 = 2^1 × 3^3

2. **Take the highest power of each prime**:
- For 2: max(3, 2, 1) = 3 → 2^3
- For 3: max(1, 2, 3) = 3 → 3^3

3. **Calculate the LCM**:
\[ LCM = 2^3 × 3^3 = 8 × 27 = 216 \text{ seconds} \]

### Step 3: Convert seconds into minutes and seconds
Next, we convert 216 seconds into minutes and seconds:
- 216 seconds = 3 minutes and 36 seconds (since 216 ÷ 60 = 3 minutes and 36 seconds).

### Step 4: Add the time interval to the initial time
Now we add the time interval to the initial time of 10:15:00 AM:
- 10:15:00 AM + 3 minutes 36 seconds = 10:18:36 AM.

### Final Answer
The traffic lights will change simultaneously again at **10:18:36 AM**.

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