Three athletes can complete one round around a circular field in 16 minutes, 24 minutes and 36 minutes respectively. If they start together from the same point and run in the same direction, after how much time will they all meet again at the starting point for the first time?

Introduction / Context:
This question involves finding when three athletes running at constant speeds around a circular track will all meet again at the starting point at the same time. This is a standard concept in aptitude exams and is solved using the least common multiple of their individual times to complete one round. The circular nature of the track means that every time an athlete completes an integral number of rounds, they return to the starting point.

Given Data / Assumptions:

• First athlete completes one round in 16 minutes.
• Second athlete completes one round in 24 minutes.
• Third athlete completes one round in 36 minutes.
• All start together from the same point at the same instant and run in the same direction.
• We need the earliest time when all three are again together at the starting point.

Concept / Approach:
Each athlete will be at the starting point after times equal to multiples of their individual round times. We want the smallest positive time that is a multiple of 16, 24, and 36 simultaneously. This is exactly the least common multiple of the three times. Once we find the least common multiple in minutes, we can convert it into hours and minutes for a clear and understandable answer.

Step-by-Step Solution:
Step 1: Express the three times: 16 minutes, 24 minutes, and 36 minutes. Step 2: Find the least common multiple of 16 and 24. Prime factors are 16 = 2^4 and 24 = 2^3 times 3, so least common multiple is 2^4 times 3 = 48. Step 3: Now find the least common multiple of 48 and 36. Prime factors of 36 are 2^2 times 3^2. Step 4: Least common multiple of 48 and 36 uses the highest powers of primes: 2^4 and 3^2, giving 16 times 9 = 144. Step 5: Thus the least common multiple of 16, 24, and 36 is 144 minutes. Step 6: Convert 144 minutes into hours and minutes: 144 divided by 60 equals 2 hours and 24 minutes.

Verification / Alternative check:
We can check by seeing how many rounds each runner completes in 144 minutes. The first athlete completes 144 divided by 16 = 9 rounds. The second completes 144 divided by 24 = 6 rounds. The third completes 144 divided by 36 = 4 rounds. Since each completes an integer number of rounds, all are at the starting point at 144 minutes. Any smaller time would not be a multiple of all three individual times, so 144 minutes is indeed the earliest such time.

Why Other Options Are Wrong:

• 2 hours 44 minutes corresponds to 164 minutes, which is a larger common point but not the earliest.
• 1 hour 24 minutes equals 84 minutes, which is not a multiple of all three times.
• 1 hour 44 minutes equals 104 minutes, which again is not a multiple of 16, 24, and 36.

Common Pitfalls:
Some learners mistakenly compute the greatest common divisor instead of the least common multiple. Others may forget to check all three times or may not factor numbers correctly. Carefully using prime factorization and taking the highest powers of each prime ensures the correct least common multiple and a reliable answer.

Final Answer:
The three athletes will meet again at the starting point after 2 hours 24 minutes.