There are three street lights which get on for one second after 30 S, 40 s and 50 s, respectively. If last time they were on simultaneously at 4:00 pm, At what time after 4: 00 pm all of them get on again simultaneously?

Options

A. `4:10` pm
B. `4:11` pm
C. `4:12` pm
D. `4:14` pm

Answer

To solve the problem, we need to find the least common multiple (LCM) of the time intervals at which the street lights turn on: 30 seconds, 40 seconds, and 50 seconds. Let's go through the steps to find the LCM and determine the time when all street lights will be on simultaneously again after 4:00 PM.

### Step-by-Step Solution:

1. **Identify the time intervals**: The street lights turn on at intervals of 30 seconds, 40 seconds, and 50 seconds.

2. **Find the LCM of 30, 40, and 50**:
- **Prime Factorization**:
- 30 = 2 × 3 × 5
- 40 = 2^3 × 5
- 50 = 2 × 5^2
- **Take the highest power of each prime factor**:
- For 2: highest power is 2^3 (from 40)
- For 3: highest power is 3^1 (from 30)
- For 5: highest power is 5^2 (from 50)
- **Calculate the LCM**:
\[ \text{LCM} = 2^3 × 3^1 × 5^2 = 8 × 3 × 25 \]
\[ = 24 × 25 = 600 \]

3. **Convert LCM from seconds to minutes**:
- Since 600 seconds is the time until they all turn on together again:
\[ \text{Minutes} = \frac{600 \text{ seconds}}{60 \text{ seconds/minute}} = 10 \text{ minutes} \]

4. **Add the time to the last simultaneous time**:
- The last time they were on together was at 4:00 PM. Adding 10 minutes:
\[ 4:00 \text{ PM} + 10 \text{ minutes} = 4:10 \text{ PM} \]

### Final Answer:
All three street lights will turn on simultaneously again at **4:10 PM**.

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