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.
A largest positive integer that divides given two positive integers is called:

Options

A. HCF
B. LCM
C. co-prime
D. None of these

Answer

To solve the question, we need to identify the largest positive integer that divides the given step lengths of Naveeka, Arjun, and Vedant, which are 240 cm, 90 cm, and 120 cm respectively. This integer is known as the Highest Common Factor (HCF) or Greatest Common Divisor (GCD).

### Step-by-Step Solution:

1. **Identify the Numbers**:
The step lengths of Naveeka, Arjun, and Vedant are:
- Naveeka: 240 cm
- Arjun: 90 cm
- Vedant: 120 cm

2. **Find the Prime Factorization of Each Number**:
- For 240:
- 240 = 2 × 120
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, 240 = 2^4 × 3^1 × 5^1

- For 90:
- 90 = 2 × 45
- 45 = 3 × 15
- 15 = 3 × 5
- Therefore, 90 = 2^1 × 3^2 × 5^1

- For 120:
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- Therefore, 120 = 2^3 × 3^1 × 5^1

3. **Identify the Common Factors**:
Now we will identify the common prime factors from the factorizations:
- 240: 2^4, 3^1, 5^1
- 90: 2^1, 3^2, 5^1
- 120: 2^3, 3^1, 5^1

The common prime factors are:
- For 2: the minimum power is 2^1 (from 90)
- For 3: the minimum power is 3^1 (from both 240 and 120)
- For 5: the minimum power is 5^1 (common in all)

4. **Calculate the HCF**:
Now, we multiply the common factors with their minimum powers:
HCF = 2^1 × 3^1 × 5^1 = 2 × 3 × 5 = 30

5. **Conclusion**:
The largest positive integer that divides the step lengths of Naveeka, Arjun, and Vedant is **30 cm**.