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 while jogging around a circular stadium, we need to determine the least common multiple (LCM) of the times they take to complete one revolution. The times are as follows:

- Monica: 24 seconds
- Veronica: 36 seconds
- Rajat: 45 seconds

### Step-by-Step Solution:

**Step 1: Find the prime factorization of each time.**

- **24 seconds:**
\( 24 = 2^3 \times 3^1 \)

- **36 seconds:**
\( 36 = 2^2 \times 3^2 \)

- **45 seconds:**
\( 45 = 3^2 \times 5^1 \)

**Step 2: Identify the highest power of each prime factor.**

- For the prime factor 2: The highest power is \( 2^3 \) (from 24).
- For the prime factor 3: The highest power is \( 3^2 \) (from both 36 and 45).
- For the prime factor 5: The highest power is \( 5^1 \) (from 45).

**Step 3: Multiply these highest powers together to find the LCM.**

\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \( 2^3 = 8 \)
2. Calculate \( 3^2 = 9 \)
3. Calculate \( 5^1 = 5 \)

Now multiply these results:

\[ \text{LCM} = 8 \times 9 \times 5 \]

First, multiply \( 8 \times 9 = 72 \).

Then multiply \( 72 \times 5 = 360 \).

**Step 4: Conclusion**

The least common multiple of 24, 36, and 45 is 360 seconds. Therefore, Monica, Veronica, and Rajat will all be together at the starting point after 360 seconds.

### Final Answer:
They will be together at the starting point after **360 seconds**.

---