Neeraj , Manoj and Geeta start running around a circular stadium and complete one round in 10 seconds , 6 seconds and 14 seconds respectively . In how much time will they meet again at the starting point ?

Options

A. 3 minutes 30 seconds
B. 2 minutes 28 seconds
C. 4 minutes 45 seconds
D. 1 minute 40 seconds

Explanation

To solve the problem of when Neeraj, Manoj, and Geeta will meet again at the starting point after starting to run around a circular stadium, we need to find the least common multiple (LCM) of their individual times to complete one round. Here’s how we can do it step by step:

### Step-by-Step Solution:

1. **Identify the times taken by each runner:**
- Neeraj takes 10 seconds to complete one round.
- Manoj takes 6 seconds to complete one round.
- Geeta takes 14 seconds to complete one round.

2. **List the times:**
- The times are: 10 seconds, 6 seconds, and 14 seconds.

3. **Find the LCM of the three times:**
- To find the LCM, we can use the prime factorization method:
- 10 = 2 × 5
- 6 = 2 × 3
- 14 = 2 × 7
- Now, take the highest power of each prime number that appears in the factorizations:
- For 2: highest power is \(2^1\)
- For 3: highest power is \(3^1\)
- For 5: highest power is \(5^1\)
- For 7: highest power is \(7^1\)

4. **Calculate the LCM:**
- LCM = \(2^1 \times 3^1 \times 5^1 \times 7^1 = 2 \times 3 \times 5 \times 7\)
- Calculate step by step:
- \(2 \times 3 = 6\)
- \(6 \times 5 = 30\)
- \(30 \times 7 = 210\)
- Therefore, the LCM of 10, 6, and 14 is 210 seconds.

5. **Convert LCM to minutes and seconds:**
- Since 1 minute = 60 seconds,
- 210 seconds = 3 minutes and 30 seconds (because 3 minutes = 180 seconds, and 210 - 180 = 30 seconds).

### Final Answer:
Neeraj, Manoj, and Geeta will meet again at the starting point after **3 minutes and 30 seconds**.