Three persons A, B and C can complete one round of circular track in 8, 15 and 20 minutes respectively. If they start together at 8 A.M from a certain point, at which time will they be together again at the starting point?

Options

A. `10 AM`
B. `12 "Noon"`
C. `11 AM`
D. `9 AM`

Answer

To find out when persons A, B, and C will be together again at the starting point after starting together at 8 A.M., we need to determine the least common multiple (LCM) of the times it takes each person to complete one round of the circular track.

### Step-by-Step Solution:

1. **Identify the times taken by A, B, and C:**
- A takes 8 minutes to complete one round.
- B takes 15 minutes to complete one round.
- C takes 20 minutes to complete one round.

2. **Find the LCM of the times:**
- To find the LCM of 8, 15, and 20, we can use the prime factorization method:
- 8 = 2^3
- 15 = 3^1 × 5^1
- 20 = 2^2 × 5^1

- Now, we take the highest power of each prime number:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^1 (from 15)
- For 5: the highest power is 5^1 (from both 15 and 20)

- Therefore, the LCM is:
\[ LCM = 2^3 × 3^1 × 5^1 = 8 × 3 × 5 = 120 \]
- So, the LCM of 8, 15, and 20 is 120 minutes.

3. **Convert the LCM into hours:**
- Since 120 minutes is equal to 2 hours (because 120 ÷ 60 = 2).

4. **Calculate the time they will meet again:**
- They start at 8 A.M., and after 2 hours, they will meet again.
- Therefore, 8 A.M. + 2 hours = 10 A.M.

### Final Answer:
They will be together again at the starting point at **10 A.M.**.

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