Four runners started running simultaneously from a point on a circular track. They took 200 seconds, 300 seconds , 360 seconds and 450 seconds to complete one round . After how much time do they meet at the starting point for the first time ?

Options

A. 1800 seconds
B. 3600 seconds
C. 2400 seconds
D. 2400 seconds

Answer

To find out when the four runners will meet at the starting point for the first time, we need to calculate the least common multiple (LCM) of the times taken by each runner to complete one round. The times are 200 seconds, 300 seconds, 360 seconds, and 450 seconds.

### Step-by-step Solution:

1. **List the times taken by each runner:**
- Runner 1: 200 seconds
- Runner 2: 300 seconds
- Runner 3: 360 seconds
- Runner 4: 450 seconds

2. **Find the prime factorization of each time:**
- **200:**
- \( 200 = 2^3 \times 5^2 \)
- **300:**
- \( 300 = 2^2 \times 3^1 \times 5^2 \)
- **360:**
- \( 360 = 2^3 \times 3^2 \times 5^1 \)
- **450:**
- \( 450 = 2^1 \times 3^2 \times 5^2 \)

3. **Identify the highest power of each prime factor:**
- For \(2\): The highest power is \(2^3\) (from 200 and 360).
- For \(3\): The highest power is \(3^2\) (from 360 and 450).
- For \(5\): The highest power is \(5^2\) (from 200, 300, and 450).

4. **Calculate the LCM using the highest powers:**
\[ \text{LCM} = 2^3 \times 3^2 \times 5^2 \]
- Calculate each part:
- \(2^3 = 8\)
- \(3^2 = 9\)
- \(5^2 = 25\)

5. **Multiply these values together:**
\[ \text{LCM} = 8 \times 9 \times 25 \]
- First calculate \(8 \times 9 = 72\)
- Then calculate \(72 \times 25 = 1800\)

6. **Conclusion:**
The runners will meet at the starting point for the first time after **1800 seconds**.