Case Study-2 : Read the above paragraph and answer the following questions.
In a morning walk, Naveeka, Arjun and Vedant step off together, their steps measuring 240 cm, 90 cm, 120 cm respectively. They want to go for a juice shop for a health issue, which is situated near by them.
Find the comman distance covered by all of them to reach the juice shop:

Options

A. 20
B. 30
C. 35
D. 40

Answer

To find the common distance covered by Naveeka, Arjun, and Vedant to reach the juice shop, we need to determine the highest common factor (HCF) of their step lengths. Here are the steps to solve the problem:

### Step 1: Identify the step lengths
- Naveeka's step length = 240 cm
- Arjun's step length = 90 cm
- Vedant's step length = 120 cm

### Step 2: Find the prime factorization of each step length
1. **For Naveeka (240 cm)**:
- 240 = 2 × 120
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 240 is:
\[ 240 = 2^4 \times 3^1 \times 5^1 \]

2. **For Arjun (90 cm)**:
- 90 = 2 × 45
- 45 = 3 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 90 is:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

3. **For Vedant (120 cm)**:
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

### Step 3: Identify the common prime factors
- The common prime factors from the factorizations are:
- For 2: The minimum power is \(2^1\) (from Arjun).
- For 3: The minimum power is \(3^1\) (from Naveeka and Vedant).
- For 5: The minimum power is \(5^1\) (common in all).

### Step 4: Calculate the HCF
The HCF can be calculated by multiplying the common prime factors with their minimum powers:
\[ HCF = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30 \]

### Conclusion
The common distance covered by all three to reach the juice shop is **30 cm**.

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