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 find the least common multiple (LCM) of the time intervals of the three traffic lights: 24 seconds, 36 seconds, and 54 seconds.

### Step-by-Step Solution:

**Step 1: Find the LCM of 24, 36, and 54.**

To find the LCM, we can use the prime factorization method.

- **Prime factorization of 24:**
- 24 = 2 × 2 × 2 × 3 = \(2^3 \times 3^1\)

- **Prime factorization of 36:**
- 36 = 2 × 2 × 3 × 3 = \(2^2 \times 3^2\)

- **Prime factorization of 54:**
- 54 = 2 × 3 × 3 × 3 = \(2^1 \times 3^3\)

**Step 2: Determine the highest power of each prime factor.**

- For the prime number 2: The highest power is \(2^3\) (from 24).
- For the prime number 3: The highest power is \(3^3\) (from 54).

**Step 3: Calculate the LCM.**

Now, we multiply the highest powers of all prime factors together:
\[ \text{LCM} = 2^3 \times 3^3 = 8 \times 27 = 216 \]

So, the LCM of 24, 36, and 54 is 216 seconds.

**Step 4: Convert 216 seconds into minutes and seconds.**

To convert seconds into minutes:
- 216 seconds = 3 minutes and 36 seconds (since 216 ÷ 60 = 3 remainder 36).

**Step 5: Add the LCM to the initial time.**

The initial time is 10:15:00 AM. We need to add 3 minutes and 36 seconds to this time.

- Adding 3 minutes:
- 10:15:00 AM + 3 minutes = 10:18:00 AM.

- Adding 36 seconds:
- 10:18:00 AM + 36 seconds = 10:18:36 AM.

Thus, the traffic lights will change simultaneously again at **10:18:36 AM**.

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

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