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

### Concept & Strategy For objects moving in a circle, the time required to meet again precisely at the starting point is determined by calculating the Least Common Multiple (LCM) of the individual lap times. $$\text{Meeting Time} = \text{LCM}(T_1, T_2, T_3)$$ ### Step-by-Step Solution **Given:** * Lap time for A = 252 seconds * Lap time for B = 308 seconds * Lap time for C = 198 seconds **Calculation:** 1. Perform prime factorization for each number to find the LCM: * $252 = 2 \times 126 = 2^2 \times 63 = 2^2 \times 3^2 \times 7$ * $308 = 2 \times 154 = 2^2 \times 77 = 2^2 \times 7 \times 11$ * $198 = 2 \times 99 = 2 \times 3^2 \times 11$ 2. Identify the highest power of each prime factor present ($2, 3, 7, 11$): * Highest power of 2 is $2^2$ * Highest power of 3 is $3^2$ * Highest power of 7 is $7^1$ * Highest power of 11 is $11^1$ 3. Multiply these together to find the LCM: $$\text{LCM} = 2^2 \times 3^2 \times 7 \times 11$$ $$\text{LCM} = 4 \times 9 \times 7 \times 11$$ $$\text{LCM} = 36 \times 77 = 2772 \text{ seconds}$$ 4. Convert the total seconds into minutes and seconds: $$\frac{2772}{60} = 46.2 \text{ minutes}$$ * The whole number is 46 minutes. * The remainder is $0.2 \text{ minutes}$. Convert back to seconds: $0.2 \times 60 = 12 \text{ seconds}$. * Total time = 46 minutes and 12 seconds. ### Exam Strategy & Shortcut Look closely at the numbers: 252, 308, 198. They are all multiples of 2. Even better, they all seem related to 11 and 9 rules. $198$ is obviously a multiple of $9$ and $11$. $308$ is $28 \times 11$. $252$ is a multiple of $9$. If you factor out common multipliers iteratively: $\text{LCM}(252, 308, 198) = 2 \times \text{LCM}(126, 154, 99)$ You can quickly build the factor tree without doing full primes for each if you recognize 11 and 7 as key factors, ultimately yielding 2772. Another shortcut: 2772 ends in a 2. When divided by 60, the remainder must end in a 2 ($2772 \pmod{10} = 2$). The only option ending in a "2" in the seconds column is option (d) 12 seconds. ### Common Pitfall During long multiplication like $36 \times 77$, simple arithmetic errors can cascade, leading to the wrong total seconds. Additionally, students sometimes divide by 60 and see $46.2$ on their scratchpad and mistakenly think it means "46 minutes and 20 seconds", rather than correctly doing $0.2 \times 60 = 12$ seconds. ### Final Answer **Therefore, the correct answer is 46 minutes 12 seconds.**