Question

Three runners, Harshad, Aniket, and Rajesh, running around a circular track can complete one revolution in 12, 18 and 24 minutes respectively. After how many minutes will they meet at the starting point?

• A. \(70 \)
• B. \(73 \)
• C. \(75 \)
• D. \(72 \)
• E. \(76 \)

Hint:

For circular motion or repeating events: - Use LCM to find when all events coincide again

Answer 1

The Correct Option is D

Concept: When multiple events repeat at different time intervals, the time at which they all occur together again is given by the Least Common Multiple (LCM).
• Required time = LCM of their individual times
Step 1: Find LCM of 12, 18, and 24.
Prime factorization: \[ 12 = 2^2 \times 3 \] \[ 18 = 2 \times 3^2 \] \[ 24 = 2^3 \times 3 \]

Step 2: Take highest powers of all primes.
\[ \text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72 \]

Step 3: Interpretation.
They will all meet at the starting point after: \[ 72 \text{ minutes} \]