To solve this problem, we need to find the smaller of the two numbers, given the Highest Common Factor (HCF) and the other factors of the Least Common Multiple (LCM).

Let's begin step-by-step:

Step 1: Understand the given information.

• The HCF of the two numbers = 4.
• The two other factors of the LCM = 5 and 7.

Step 2: Derive the LCM from the given factors.

Since the LCM is formed by taking the highest power of all prime factors involved in both numbers, and we are given the two other factors (5 and 7), we can write:

• LCM = 4 (since it is part of both numbers) × 5 × 7 = 140.

Step 3: Identify the condition for two numbers based on their HCF and LCM.

Let the two numbers be \(a\) and \(b\).

• We know that \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\).
• Substituting the given values: \(4 \times 140 = a \times b\).
• Therefore, \(a \times b = 560\).

Step 4: Use the HCF to represent the numbers.

Given that the HCF of the numbers is 4, we can write:

• \(a = 4 \times m\)
• \(b = 4 \times n\)
• where \(\text{HCF}(m, n) = 1\) (m and n are coprime).

Step 5: Solve using coprime numbers m and n.

Substituting in the product \(a \times b = 560\):

• \(16 \times m \times n = 560\)
• Simplifying, we get \(m \times n = \frac{560}{16} = 35\).
• The pairs of coprime numbers (m, n) such that their product is 35 are (5, 7) or (7, 5).

Thus, the numbers can be:

• \(a = 4 \times 5 = 20\)
• \(b = 4 \times 7 = 28\)

Conclusion: The options given were 10, 14, 20, and 28. Therefore, the smaller of the two numbers is 20. The correct answer is option C: 20.