In a fire range, 4 shooters are firing at their respective targets.The first, the second, the third and the fourth shooters hit the target once in every 5s,6s, 7s adn 8s respectively. If all of them hit their target at 9:00 am, when will they hit their target together again ?

Options

A. `9:04` am
B. `9:08` am
C. `9:14` am
D. None of these

Answer

To solve the problem of when all four shooters will hit their targets together again, we need to find the least common multiple (LCM) of the times at which each shooter hits their target. The shooters hit their targets every 5 seconds, 6 seconds, 7 seconds, and 8 seconds respectively.

### Step-by-Step Solution:

1. **Identify the Time Intervals**:
- Shooter 1: Hits every 5 seconds
- Shooter 2: Hits every 6 seconds
- Shooter 3: Hits every 7 seconds
- Shooter 4: Hits every 8 seconds

2. **Find the LCM of the Intervals**:
- We need to find the LCM of 5, 6, 7, and 8.
- **Prime Factorization**:
- 5 = \(5^1\)
- 6 = \(2^1 \times 3^1\)
- 7 = \(7^1\)
- 8 = \(2^3\)
- **Take the highest power of each prime**:
- From 5: \(5^1\)
- From 6: \(2^3\) and \(3^1\)
- From 7: \(7^1\)

3. **Calculate the LCM**:
- LCM = \(2^3 \times 3^1 \times 5^1 \times 7^1\)
- LCM = \(8 \times 3 \times 5 \times 7\)
- First, calculate \(8 \times 3 = 24\)
- Then, \(24 \times 5 = 120\)
- Finally, \(120 \times 7 = 840\)
- Therefore, the LCM is 840 seconds.

4. **Convert Seconds to Minutes**:
- To convert 840 seconds into minutes, divide by 60.
- \(840 \div 60 = 14\) minutes.

5. **Determine the Next Time They Hit Together**:
- They all hit their targets together at 9:00 AM.
- Adding 14 minutes to 9:00 AM gives us 9:14 AM.

### Final Answer:
The shooters will hit their targets together again at **9:14 AM**.

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