Three men A, B and C in same direction from one point around a field start to run. A, B and C complete a cricle in 7 minutes, 8 minutes and 10 minutes respectively. When will they meet at the first time ?

Options

A. 4 hour
B. 5 hour
C. 3 hour
D. None of these

Answer

To find out when A, B, and C will meet for the first time while running around a field, we need to determine the least common multiple (LCM) of the times it takes each of them to complete a circle.

**Step-by-step Solution:**

1. **Identify the times taken by A, B, and C:**
- A completes a circle in 7 minutes.
- B completes a circle in 8 minutes.
- C completes a circle in 10 minutes.

2. **List the times:**
- The times are: 7 minutes, 8 minutes, and 10 minutes.

3. **Find the LCM of the three times:**
- To find the LCM, we can use the prime factorization method.
- Prime factorization:
- 7 is a prime number: \(7^1\)
- 8 can be factored as \(2^3\)
- 10 can be factored as \(2^1 \times 5^1\)

4. **Take the highest power of each prime:**
- For 2: The highest power is \(2^3\) (from 8).
- For 5: The highest power is \(5^1\) (from 10).
- For 7: The highest power is \(7^1\) (from 7).

5. **Calculate the LCM:**
- LCM = \(2^3 \times 5^1 \times 7^1\)
- LCM = \(8 \times 5 \times 7\)
- First, calculate \(8 \times 5 = 40\)
- Then, calculate \(40 \times 7 = 280\)

6. **Convert the LCM from minutes to hours and minutes:**
- The LCM is 280 minutes.
- To convert minutes to hours, divide by 60:
- \(280 \div 60 = 4\) hours with a remainder of \(40\) minutes.

7. **Final Answer:**
- They will meet for the first time after **4 hours and 40 minutes**.