Monica, Veronica and Rajat begin to jog around a circular stadium. They complete their revolutions in 24 s, 36 s and 45 s, respectively. After how many seconds they will be together at the starting points ?

Options

A. 360
B. 252
C. 504
D. None of these

Answer

To find out after how many seconds Monica, Veronica, and Rajat will be together at the starting point, we need to calculate the Least Common Multiple (LCM) of the times they take to complete one revolution: 24 seconds, 36 seconds, and 45 seconds.

### Step-by-Step Solution:

1. **Identify the numbers**: The times taken by Monica, Veronica, and Rajat are 24 seconds, 36 seconds, and 45 seconds.

2. **Prime Factorization**:
- **24**:
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- So, 24 = 2^3 × 3^1
- **36**:
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, 36 = 2^2 × 3^2
- **45**:
- 45 = 3 × 15
- 15 = 3 × 5
- So, 45 = 3^2 × 5^1

3. **Determine the LCM**:
- For LCM, we take the highest power of each prime factor from the factorizations:
- For 2: the highest power is 2^3 (from 24)
- For 3: the highest power is 3^2 (from 36 and 45)
- For 5: the highest power is 5^1 (from 45)
- Therefore, LCM = 2^3 × 3^2 × 5^1

4. **Calculate the LCM**:
- LCM = 8 (from 2^3) × 9 (from 3^2) × 5
- First, calculate 8 × 9 = 72
- Then, calculate 72 × 5 = 360

5. **Conclusion**:
- The LCM of 24, 36, and 45 is 360 seconds. Therefore, Monica, Veronica, and Rajat will all be together at the starting point after **360 seconds**.