A, B, and C start together around a circular track. A takes 252 seconds per lap, B takes 308 seconds, and C takes 198 seconds. After how much time will they all be back together at the starting point?

Given data

• A's lap time = 252 s
• B's lap time = 308 s
• C's lap time = 198 s

Concept / Approach

• They meet together at the start after the least common multiple (LCM) of their lap times.
• Use prime factorization to compute LCM.

Step-by-step calculation (prime factors)252 = 2^2 × 3^2 × 7308 = 2^2 × 7 × 11198 = 2 × 3^2 × 11LCM = 2^2 × 3^2 × 7 × 11 = 4 × 9 × 7 × 11 = 2772 s

Verification2772 ÷ 252 = 11; 2772 ÷ 308 = 9; 2772 ÷ 198 = 14 — all integers, so 2772 s is a common multiple and the least by construction.Convert to minutes: 2772 s = 46 min 12 s.

Common pitfalls

• Adding or averaging lap times (not valid). The meeting time is the LCM, not the sum/average.
• Missing a prime power when forming the LCM.

Final Answer2772 seconds (i.e., 46 minutes 12 seconds)