The LCM and the HCF of the two numbers are 620 and 4 respectively. How many such pairs are possible?


We are given that the **LCM** (Least Common Multiple) and the **HCF** (Highest Common Factor) of two numbers are \( 620 \) and \( 4 \) respectively.

### Step-by-step solution:

1. **Relationship between LCM and HCF**:  
   For any two numbers \( a \) and \( b \), the product of the LCM and HCF of the numbers is equal to the product of the numbers themselves:
   \[
   \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b
   \]
   Using the values from the problem:
   \[
   620 \times 4 = a \times b
   \]
   \[
   a \times b = 2480
   \]

2. **Expressing \( a \) and \( b \) in terms of the HCF**:  
   Let \( a = 4x \) and \( b = 4y \), where \( x \) and \( y \) are co-prime integers (i.e., their HCF is 1). This is because the HCF of \( a \) and \( b \) is 4. Substituting into the product equation:
   \[
   (4x) \times (4y) = 2480
   \]
   \[
   16xy = 2480
   \]
   \[
   xy = \frac{2480}{16} = 155
   \]

3. **Finding pairs of \( x \) and \( y \)**:  
   Now, \( x \) and \( y \) are co-prime factors of 155. The prime factorization of 155 is:
   \[
   155 = 5 \times 31
   \]
   The co-prime pairs \( (x, y) \) that multiply to 155 are:
   \[
   (1, 155), (5, 31)
   \]

4. **Counting the pairs**:  
   For each co-prime pair \( (x, y) \), we can assign \( a = 4x \) and \( b = 4y \) or \( a = 4y \) and \( b = 4x \). Hence, each pair can be arranged in two ways. Since there are 2 pairs of \( x \) and \( y \), we have:
   \[
   2 \times 2 = 4
   \]
   Thus, there are **4 such pairs** of numbers \( a \) and \( b \).

### Answer:
There are **4 pairs** of numbers that satisfy the given conditions.

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Would you like a detailed explanation on any part of this solution? Here are some related questions you might find useful:

1. How do you find the LCM and HCF of two numbers manually?
2. What is the prime factorization method, and how is it useful in LCM/HCF problems?
3. Why do we use the relationship \( \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \)?
4. How can you find co-prime pairs of a number?
5. How would the process change if the LCM and HCF values were different?

**Tip**: When dealing with LCM and HCF problems, always break the numbers into their prime factors—it simplifies the process and reveals hidden relationships!

The LCM and the HCF of the two numbers are 620 and 4 respectively. How many such pairs are possible?

Explore how to determine the number of pairs of numbers that have an LCM of 620 and an HCF of 4. This problem involves the key concepts of LCM, HCF, and prime factorization. Suitable for students in Grades 8-10.

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