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.
If A and B are two numbers, then find the relation between LCM and HCF.

Options

A. `a/b=LCM(A,B)HCF(A,B)`
B. `AxxB=LCM(A,B)xxHCF(A,B)`
C. `AxxLCM(A,B)=BxxHCF(A,B)`
D. None of these

Answer

To solve the problem regarding the relationship between the Least Common Multiple (LCM) and the Highest Common Factor (HCF) of two numbers A and B, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The relationship between LCM and HCF of two numbers A and B can be expressed as:
\[ \text{LCM}(A, B) \times \text{HCF}(A, B) = A \times B \]

2. **Assuming Values for A and B**:
Let's assume \( A = 12 \) and \( B = 16 \).

3. **Finding the Prime Factorization**:
- For \( A = 12 \):
- \( 12 = 2^2 \times 3^1 \)
- For \( B = 16 \):
- \( 16 = 2^4 \)

4. **Calculating the LCM**:
- The LCM is found by taking the highest power of each prime factor from both numbers.
- From \( A \) and \( B \):
- For \( 2 \): highest power is \( 2^4 \)
- For \( 3 \): highest power is \( 3^1 \)
- Therefore,
\[ \text{LCM}(12, 16) = 2^4 \times 3^1 = 16 \times 3 = 48 \]

5. **Calculating the HCF**:
- The HCF is found by taking the lowest power of each prime factor common to both numbers.
- From \( A \) and \( B \):
- For \( 2 \): lowest power is \( 2^2 \)
- Therefore,
\[ \text{HCF}(12, 16) = 2^2 = 4 \]

6. **Verifying the Relationship**:
- Now, we will verify the relationship:
\[ A \times B = 12 \times 16 = 192 \]
\[ \text{LCM}(A, B) \times \text{HCF}(A, B) = 48 \times 4 = 192 \]

7. **Conclusion**:
Since both calculations give us the same result (192), we can conclude that:
\[ \text{LCM}(A, B) \times \text{HCF}(A, B) = A \times B \]
This confirms the relationship between LCM and HCF.