Four runners started running in a circular path at the same time from one point. They complete one circle in 200 seconds, 300 seconds, 360 seconds and 450 seconds respectively. Accordingly. When will they meet at starting point ?

Options

A. 1800 second
B. 3600 second
C. 2400 second
D. 4800 second

Answer

To find out when the four runners will meet at the starting point after starting together, we need to calculate the Least Common Multiple (LCM) of the times taken by each runner to complete one circle. The times given are:

- Runner 1: 200 seconds
- Runner 2: 300 seconds
- Runner 3: 360 seconds
- Runner 4: 450 seconds

### Step 1: Prime Factorization of Each Time
First, we will perform the prime factorization of each of the times:

1. **200 seconds**:
- \(200 = 2^3 \times 5^2\)

2. **300 seconds**:
- \(300 = 2^2 \times 3^1 \times 5^2\)

3. **360 seconds**:
- \(360 = 2^3 \times 3^2 \times 5^1\)

4. **450 seconds**:
- \(450 = 2^1 \times 3^2 \times 5^2\)

### Step 2: Identify the Highest Powers of Each Prime Factor
Next, we will identify the highest powers of each prime factor from the factorizations:

- 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).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors:

\[ \text{LCM} = 2^3 \times 3^2 \times 5^2 \]

Calculating this step-by-step:

1. Calculate \(2^3 = 8\)
2. Calculate \(3^2 = 9\)
3. Calculate \(5^2 = 25\)

Now, multiply these results together:

\[ \text{LCM} = 8 \times 9 \times 25 \]

Calculating \(8 \times 9 = 72\)

Then, calculate \(72 \times 25\):

\[ 72 \times 25 = 1800 \]

### Final Answer
Thus, the runners will meet at the starting point after **1800 seconds**.

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