Understanding Runners Meeting on Circular Track

The problem asks us to find the time when three runners, A, B, and C, will meet again at the starting point of a circular stadium. They start at the same time and run in the same direction, completing rounds in 252 seconds, 308 seconds, and 198 seconds, respectively.

For the runners to meet again at the starting point, the time elapsed must be a multiple of each runner's individual round time. Therefore, we need to find the Least Common Multiple (LCM) of their individual timings (252, 308, and 198 seconds).

Calculating Meeting Time using LCM

First, find the prime factorization of each runner's time:

• Runner A's time: $252$ seconds
• Runner B's time: $308$ seconds
• Runner C's time: $198$ seconds

Now, let's find the prime factors:

• $252 = 2 \times 126 = 2 \times 2 \times 63 = 2^2 \times 3 \times 21 = 2^2 \times 3^2 \times 7$
• $308 = 2 \times 154 = 2 \times 2 \times 77 = 2^2 \times 7 \times 11$
• $198 = 2 \times 99 = 2 \times 9 \times 11 = 2 \times 3^2 \times 11$

To find the LCM, we take the highest power of each prime factor present in any of the factorizations:

The prime factors involved are 2, 3, 7, and 11.

• Highest power of 2: $2^2$
• Highest power of 3: $3^2$
• Highest power of 7: $7^1$
• Highest power of 11: $11^1$

So, the LCM is calculated as:

$LCM(252, 308, 198) = 2^2 \times 3^2 \times 7 \times 11$

$LCM = 4 \times 9 \times 7 \times 11$

$LCM = 36 \times 77$

$LCM = 2772$ seconds.

Converting Seconds to Minutes and Seconds

The runners will meet again at the starting point after 2772 seconds. To convert this into minutes and seconds, we divide by 60 (since there are 60 seconds in a minute):

Total seconds = $2772$

Number of minutes = $\lfloor \frac{2772}{60} \rfloor$

Number of minutes = $46$ minutes.

Remaining seconds = $2772 \mod 60$

Remaining seconds = $2772 - (46 \times 60)$

Remaining seconds = $2772 - 2760$

Remaining seconds = $12$ seconds.

Therefore, they will meet again at the starting point after 46 minutes and 12 seconds.