The persons start walking together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Options

A. 25 m 20 cm
B. 50 m 40 cm
C. 75 m 60 cm
D. 100m 80 cm

Answer

To find the minimum distance each person should walk so that they can cover the same distance in complete steps, we need to determine the least common multiple (LCM) of their step lengths. The step lengths given are 40 cm, 42 cm, and 45 cm.

### Step 1: Prime Factorization
First, we will perform the prime factorization of each step length.

- **40 cm:**
- 40 = 2 × 20
- 20 = 2 × 10
- 10 = 2 × 5
- So, 40 = \(2^3 \times 5^1\)

- **42 cm:**
- 42 = 2 × 21
- 21 = 3 × 7
- So, 42 = \(2^1 \times 3^1 \times 7^1\)

- **45 cm:**
- 45 = 3 × 15
- 15 = 3 × 5
- So, 45 = \(3^2 \times 5^1\)

### Step 2: Determine the LCM
Next, we will find the LCM by taking the highest power of each prime factor from the factorizations.

- For \(2\): The highest power is \(2^3\) (from 40).
- For \(3\): The highest power is \(3^2\) (from 45).
- For \(5\): The highest power is \(5^1\) (from both 40 and 45).
- For \(7\): The highest power is \(7^1\) (from 42).

Now, we can calculate the LCM:

\[ \text{LCM} = 2^3 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step by step:

1. \(2^3 = 8\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)
4. \(7^1 = 7\)

Now, multiply these together:

\[ 8 \times 9 = 72 \]
\[ 72 \times 5 = 360 \]
\[ 360 \times 7 = 2520 \]

So, the LCM of 40 cm, 42 cm, and 45 cm is 2520 cm.

### Step 3: Convert to Meters
Since the question asks for the distance in meters, we convert 2520 cm to meters:

\[ 2520 \text{ cm} = \frac{2520}{100} \text{ m} = 25.2 \text{ m} \]

### Conclusion
The minimum distance each person should walk so that they can cover the same distance in complete steps is **25.2 meters**.