A, B and C start at the same time in the same direction to run around a…

Concept: When two or more bodies start together and repeat their motion in fixed cycle times, they return together to the starting point after a time equal to the Least Common Multiple (LCM) of their individual cycle times — the smallest time that is simultaneously divisible by every period.

Application:

1. Write each lap time as a product of prime powers: 252 = 22 × 32 × 7 seconds; 308 = 22 × 7 × 11 seconds; 198 = 2 × 32 × 11 seconds.

2. For each prime that appears, take the highest power seen across the three factorizations: 22, 32, 7, and 11.

3. Multiply these highest powers to get the LCM: 22 × 32 × 7 × 11 = 4 × 9 × 7 × 11 = 2772 seconds.

4. Convert 2772 seconds into minutes and seconds: 2772 ÷ 60 = 46 remainder 12, so 2772 seconds = 46 minutes 12 seconds.

Cross-check: 2772 ÷ 252 = 11, 2772 ÷ 308 = 9, and 2772 ÷ 198 = 14 — each division gives a whole number, confirming A, B and C each complete an exact whole number of laps at this instant, so 46 minutes 12 seconds is indeed the first time all three are together again at the starting point.