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

Options

A. 15 minutes 15 seconds
B. 42 minutes 30 seconds
C. 42 minutes
D. 46 minutes and 12 seconds

Explanation

To find out after what time A, B, and C will meet again 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 round.

**Step 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.

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

- For B (308):
- 308 = 2 × 154
- 154 = 2 × 77
- 77 = 7 × 11
- So, 308 = 2^2 × 7^1 × 11^1

- For C (198):
- 198 = 2 × 99
- 99 = 3 × 33
- 33 = 3 × 11
- So, 198 = 2^1 × 3^2 × 11^1

**Step 3: Determine the LCM using the highest power of each prime factor.**
- From the factorizations:
- The highest power of 2 is 2^2 (from A and B).
- The highest power of 3 is 3^2 (from A and C).
- The highest power of 7 is 7^1 (from A and B).
- The highest power of 11 is 11^1 (from B and C).

Thus, the LCM is:
\[ \text{LCM} = 2^2 × 3^2 × 7^1 × 11^1 \]

**Step 4: Calculate the LCM.**
- Calculate each part:
- 2^2 = 4
- 3^2 = 9
- 7^1 = 7
- 11^1 = 11

Now multiply these together:
\[ \text{LCM} = 4 × 9 × 7 × 11 \]

Calculating step-by-step:
- 4 × 9 = 36
- 36 × 7 = 252
- 252 × 11 = 2772

So, the LCM is 2772 seconds.

**Step 5: Conclusion.**
A, B, and C will meet again at the starting point after 2772 seconds.

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