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

Options

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

Answer

To find out after what time A, B, and C will meet again at the starting point, we need to calculate the Least Common Multiple (LCM) of the times taken by each runner to complete one round. Here are the steps to solve the problem:

### 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: Prime Factorization of Each Time
- **For A (252 seconds)**:
- 252 = 2 × 126
- 126 = 2 × 63
- 63 = 3 × 21
- 21 = 3 × 7
- So, the prime factorization of 252 is:
\[ 252 = 2^2 \times 3^2 \times 7 \]

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

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

### Step 3: Determine the LCM
To find the LCM, we take the highest power of each prime factor from the factorizations:

- For 2: the highest power is \(2^2\) (from A and B)
- For 3: the highest power is \(3^2\) (from A and C)
- For 7: the highest power is \(7^1\) (from A and B)
- For 11: the highest power is \(11^1\) (from B and C)

Now, we can calculate the LCM:
\[ \text{LCM} = 2^2 \times 3^2 \times 7^1 \times 11^1 \]

Calculating this step by step:
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(7^1 = 7\)
- \(11^1 = 11\)

Now multiply these together:
\[ 4 \times 9 = 36 \]
\[ 36 \times 7 = 252 \]
\[ 252 \times 11 = 2772 \]

Thus, the LCM is 2772 seconds.

### Step 4: Convert LCM to Minutes and Seconds
To convert 2772 seconds into minutes:
- Divide 2772 by 60:
\[ 2772 \div 60 = 46 \text{ remainder } 12 \]
So, 2772 seconds is equal to 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**.

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