Monica, Veronica and Rachat begin to jog around a circular stadium. They complete their revolutions in 42 s, 56 s and 63 s, respectively. After how many seconds will they be together at the starting point?

Options

A. 366
B. 252
C. 504
D. Couldn't be determined

Answer

To determine after how many seconds Monica, Veronica, and Rachat will be together at the starting point, we need to find the Least Common Multiple (LCM) of their individual times to complete one revolution around the stadium. The times given are:

- Monica: 42 seconds
- Veronica: 56 seconds
- Rachat: 63 seconds

### Step 1: Prime Factorization
First, we will find the prime factorization of each time.

1. **42**:
- 42 = 2 × 3 × 7

2. **56**:
- 56 = 2^3 × 7

3. **63**:
- 63 = 3^2 × 7

### Step 2: Identify the Highest Power of Each Prime
Next, we will identify the highest power of each prime number that appears in the factorizations.

- For **2**: The highest power is \(2^3\) (from 56).
- For **3**: The highest power is \(3^2\) (from 63).
- For **7**: The highest power is \(7^1\) (appears in all).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors together:

\[ \text{LCM} = 2^3 \times 3^2 \times 7 \]

Calculating this step-by-step:

1. Calculate \(2^3 = 8\)
2. Calculate \(3^2 = 9\)
3. Multiply these results with 7:

\[ 8 \times 9 = 72 \]
\[ 72 \times 7 = 504 \]

### Final Result
Thus, the LCM of 42, 56, and 63 is 504 seconds. Therefore, Monica, Veronica, and Rachat will all be together at the starting point after **504 seconds**.