The HCF of two numbers is 4 and the two factors of LCM are 5 and 7 . Find the smaller of the two numbers .
A.10.
B.14
C. 20 .
D.28

Options

A. C
B. B
C. D
D. A

Answer

To solve the problem, we need to find the smaller of two numbers given that their HCF (Highest Common Factor) is 4 and the two factors of their LCM (Least Common Multiple) are 5 and 7.

### Step-by-Step Solution:

1. **Understanding HCF and LCM**:
- The HCF of two numbers is the largest number that divides both of them. In this case, the HCF is given as 4.
- The LCM of two numbers is the smallest number that is a multiple of both. The factors of the LCM are given as 5 and 7.

2. **Finding the LCM**:
- The LCM can be calculated using the formula:
\[ \text{LCM} = \text{HCF} \times \left(\frac{\text{Number 1}}{\text{HCF}} \times \frac{\text{Number 2}}{\text{HCF}}\right) \]
- Since the factors of the LCM are 5 and 7, we can calculate the LCM as:
\[ \text{LCM} = 5 \times 7 = 35 \]

3. **Using the relationship between HCF and LCM**:
- We know that:
\[ \text{HCF} \times \text{LCM} = \text{Number 1} \times \text{Number 2} \]
- Substituting the known values:
\[ 4 \times 35 = \text{Number 1} \times \text{Number 2} \]
- This simplifies to:
\[ 140 = \text{Number 1} \times \text{Number 2} \]

4. **Finding the two numbers**:
- Let’s denote the two numbers as \( a \) and \( b \).
- Since the HCF is 4, we can express the numbers as:
\[ a = 4m \quad \text{and} \quad b = 4n \]
- Where \( m \) and \( n \) are coprime integers (i.e., their HCF is 1).
- Substituting these into the product equation:
\[ 4m \times 4n = 140 \]
\[ 16mn = 140 \]
\[ mn = \frac{140}{16} = 8.75 \]
- Since \( m \) and \( n \) must be integers, we need to find pairs of integers that multiply to give us a product of 8.75. However, since we are looking for integer multiples of 4, we can instead check possible pairs of \( m \) and \( n \) that fit the HCF of 4.

5. **Finding pairs of coprime integers**:
- The pairs of coprime integers that can be formed with the factors of 35 (which are 5 and 7) are:
- \( (5, 1) \)
- \( (7, 1) \)
- Therefore, we can find the numbers:
- If \( m = 5 \) and \( n = 1 \):
\[ a = 4 \times 5 = 20 \quad \text{and} \quad b = 4 \times 1 = 4 \]
- If \( m = 7 \) and \( n = 1 \):
\[ a = 4 \times 7 = 28 \quad \text{and} \quad b = 4 \times 1 = 4 \]

6. **Conclusion**:
- The two numbers are 20 and 4 or 28 and 4. The smaller of the two numbers is:
\[ \text{Smaller number} = 20 \]

### Final Answer:
The smaller of the two numbers is **20**.