In a marriage party, three different lights shine after every 10 seconds, 15 seconds and 25 seconds respectively. If all these lights change simultaneously at 6 : 25 : 00 hours, then the time when they will again shine simultaneously is

Options

A. `6:26:30` hours
B. `6:27:30` hours
C. `6:28:30` hours
D. None of these

Answer

To solve the problem of when the three lights will shine simultaneously again after initially shining together at 6:25:00, we need to find the least common multiple (LCM) of the intervals at which the lights shine.

### Step-by-Step Solution:

1. **Identify the shining intervals**:
- The first light shines every 10 seconds.
- The second light shines every 15 seconds.
- The third light shines every 25 seconds.

2. **Find the LCM of the intervals**:
- To find the LCM of 10, 15, and 25, we can start by finding the prime factorization of each number:
- 10 = 2 × 5
- 15 = 3 × 5
- 25 = 5 × 5
- Next, take the highest power of each prime number:
- The highest power of 2 is \(2^1\) (from 10).
- The highest power of 3 is \(3^1\) (from 15).
- The highest power of 5 is \(5^2\) (from 25).
- Now, multiply these together to find the LCM:
\[ LCM = 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \text{ seconds} \]

3. **Convert LCM to minutes and seconds**:
- 150 seconds can be converted to minutes:
\[ 150 \text{ seconds} = 2 \text{ minutes and } 30 \text{ seconds} \]

4. **Add the LCM to the initial time**:
- The initial time when the lights shine together is 6:25:00.
- Adding 2 minutes and 30 seconds to this time:
- 6:25:00 + 0:02:30 = 6:27:30

5. **Final Answer**:
- The lights will shine simultaneously again at **6:27:30**.