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 the same time. After what 
time will they meet again at the 
starting point?

To find the time after which A, B, and C will meet again at the starting point, we need to determine the least common multiple (LCM) of the times they take to complete one round.

The given times are:
- A: 252 seconds
- B: 308 seconds
- C: 198 seconds

### Steps:
1. **Prime factorization of each time**:
   - 252 = \( 2^2 \times 3^2 \times 7 \)
   - 308 = \( 2^2 \times 7 \times 11 \)
   - 198 = \( 2 \times 3^2 \times 11 \)

2. **LCM calculation**:
   The LCM is calculated by taking the highest powers of all prime factors that appear in any of the factorizations.

   - \( 2^2 \) (from 252 and 308)
   - \( 3^2 \) (from 252 and 198)
   - \( 7 \) (from 252 and 308)
   - \( 11 \) (from 308 and 198)

   Therefore, the LCM is:
   \[
   LCM = 2^2 \times 3^2 \times 7 \times 11 = 4 \times 9 \times 7 \times 11
   \]
   Let's calculate this.

\[
LCM = 4 \times 9 = 36
\]
\[
36 \times 7 = 252
\]
\[
252 \times 11 = 2772
\]

Thus, the least common multiple is **2772 seconds**.

### Final Answer:
A, B, and C will meet again at the starting point after **2772 seconds**.

Would you like a detailed breakdown of the LCM calculation? Here are 5 related questions:

1. How do you find the LCM of more than two numbers?
2. Why is the LCM useful in problems involving circular motion or periodic events?
3. Can you explain how prime factorization helps in finding the LCM?
4. What are the real-world applications of LCM in scheduling problems?
5. If one of the runners had a different starting time, how would that affect the meeting point?

**Tip**: Always break down a problem involving periodic events by finding common multiples!

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 the same time. After what time will they meet again at the starting point?

Learn how to calculate the least common multiple (LCM) for runners in circular motion, where A, B, and C complete laps in 252, 308, and 198 seconds respectively. This step-by-step solution uses prime factorization and applies the LCM theorem to determine when they meet again at the starting point.

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