A, B, C begin to run in the same sirection from the same point in a circular stadium at the same time A completes one round in 252 seconds, B 308 seconds and C in 198 seconds. In how much time will they meet again at the initial point ?

Options

A. 26 min 18 sec
B. 42 min 36 sec
C. 45 min
D. 46 min 12 sec

Answer

To find out how much time A, B, and C will take to meet again at the initial point after starting to run in the circular stadium, we need to calculate the Least Common Multiple (LCM) of their individual times taken to complete one round.

### Step-by-Step Solution:

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

2. **Find the prime factorization of each time:**
- **For 252:**
- 252 = 2 × 126
- 126 = 2 × 63
- 63 = 3 × 21
- 21 = 3 × 7
- So, the prime factorization of 252 is: \( 2^2 \times 3^2 \times 7^1 \)

- **For 308:**
- 308 = 2 × 154
- 154 = 2 × 77
- 77 = 7 × 11
- So, the prime factorization of 308 is: \( 2^2 \times 7^1 \times 11^1 \)

- **For 198:**
- 198 = 2 × 99
- 99 = 3 × 33
- 33 = 3 × 11
- So, the prime factorization of 198 is: \( 2^1 \times 3^2 \times 11^1 \)

3. **Determine the LCM using the highest powers of each prime factor:**
- From the factorizations, we take the highest power of each prime:
- \( 2^2 \) (from 252 and 308)
- \( 3^2 \) (from 252 and 198)
- \( 7^1 \) (from 252 and 308)
- \( 11^1 \) (from 308 and 198)

Therefore, the LCM is:
\[ LCM = 2^2 \times 3^2 \times 7^1 \times 11^1 \]

4. **Calculate the LCM:**
- \( LCM = 4 \times 9 \times 7 \times 11 \)
- First, calculate \( 4 \times 9 = 36 \)
- Then, \( 36 \times 7 = 252 \)
- Finally, \( 252 \times 11 = 2772 \)

5. **Conclusion:**
- The LCM of 252, 308, and 198 is 2772 seconds.
- Therefore, A, B, and C will meet again at the initial point after 2772 seconds.

6. **Convert seconds into minutes and seconds:**
- 2772 seconds = 46 minutes and 12 seconds.

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