A,B and C start at the same time in the same direction to run around a circular stadium. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at same point After what time will they again at the starting point ?

Options

A. 46 minutes 12 seconds
B. 45 minutes
C. 42 minutes 36 second
D. 26 minutes 18 seconds

Answer

To find out after what time A, B, and C will meet again at the starting point after running around a circular stadium, we need to calculate the least common multiple (LCM) of their respective times to complete one round.

### Step-by-Step Solution:

1. **Identify the times taken by A, B, and C:**
- A completes a round in 252 seconds.
- B completes a round in 308 seconds.
- C completes a round in 198 seconds.

2. **Find the prime factorization of each time:**
- For **252**:
- \( 252 = 2^2 \times 3^2 \times 7 \)
- For **308**:
- \( 308 = 2^2 \times 7 \times 11 \)
- For **198**:
- \( 198 = 2 \times 3^2 \times 11 \)

3. **Determine the LCM:**
- To find the LCM, take the highest power of each prime factor present in the factorizations:
- For \( 2 \): highest power is \( 2^2 \)
- For \( 3 \): highest power is \( 3^2 \)
- For \( 7 \): highest power is \( 7^1 \)
- For \( 11 \): highest power is \( 11^1 \)
- Therefore, the LCM is:
\[ LCM = 2^2 \times 3^2 \times 7 \times 11 \]

4. **Calculate the LCM:**
- Calculate step-by-step:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- Now multiply: \( 4 \times 9 = 36 \)
- Next, multiply by \( 7 \): \( 36 \times 7 = 252 \)
- Finally, multiply by \( 11 \): \( 252 \times 11 = 2772 \)

5. **Convert the LCM to minutes and seconds:**
- \( 2772 \) seconds can be converted to minutes:
- \( 2772 \div 60 = 46 \) minutes with a remainder of \( 12 \) seconds.
- Thus, \( 2772 \) seconds is \( 46 \) minutes and \( 12 \) seconds.

### Final Answer:
A, B, and C will all meet again at the starting point after **46 minutes and 12 seconds**.