Puneet , Arun and Manish start together for a morning walk . Their steps measure 60 cm , 84 cm and 108 cm, What minimum distance they should walk , so that all can cover the same distance in complete steps ?

Options

A. 3780 cm
B. 1260 cm
C. 3720 cm
D. 1380 cm

Answer

To find the minimum distance that Puneet, Arun, and Manish should walk so that all can cover the same distance in complete steps, we need to determine the least common multiple (LCM) of their step lengths. The step lengths are:

- Puneet: 60 cm
- Arun: 84 cm
- Manish: 108 cm

### Step 1: Find the prime factorization of each step length.

1. **Puneet's step (60 cm)**:
- 60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹

2. **Arun's step (84 cm)**:
- 84 = 2 × 2 × 3 × 7 = 2² × 3¹ × 7¹

3. **Manish's step (108 cm)**:
- 108 = 2 × 2 × 3 × 3 × 3 = 2² × 3³

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

- For the prime number 2: The highest power is 2² (from all three).
- For the prime number 3: The highest power is 3³ (from Manish).
- For the prime number 5: The highest power is 5¹ (from Puneet).
- For the prime number 7: The highest power is 7¹ (from Arun).

### Step 3: Calculate the LCM using the highest powers.

The LCM is calculated as follows:
\[ \text{LCM} = 2² × 3³ × 5¹ × 7¹ \]

Calculating this step by step:
1. Calculate \(2² = 4\)
2. Calculate \(3³ = 27\)
3. Calculate \(5¹ = 5\)
4. Calculate \(7¹ = 7\)

Now multiply these together:
- First, multiply \(4 × 27 = 108\)
- Next, multiply \(108 × 5 = 540\)
- Finally, multiply \(540 × 7 = 3780\)

Thus, the LCM of 60, 84, and 108 is **3780 cm**.

### Step 4: Conclusion

The minimum distance they should walk so that all can cover the same distance in complete steps is **3780 cm**.

---