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The document you uploaded contains a variety of math problems. I will solve all of the problems presented. Here are the solutions:

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### **Problem 1:**
**If \( a \) and \( b \) are positive integers, then \( \text{HCF}(a, b) \times \text{LCM}(a, b) = \)**
- (a) \( a \times b \)
- (b) \( a + b \)
- (c) \( a - b \)
- (d) \( \frac{a}{b} \)

**Solution:**  
The correct answer is (a) \( a \times b \). This is a well-known property of the Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers.

---

### **Problem 2:**
**If the HCF of two numbers is 1, then the two numbers are called**
- (a) Composite
- (b) Relatively prime or co-prime
- (c) Perfect
- (d) Irrational numbers

**Solution:**  
The correct answer is (b) Relatively prime or co-prime. Two numbers are co-prime if their highest common factor is 1.

---

### **Problem 3:**
**The HCF of the smallest composite number and the smallest prime number is**
- (a) 0
- (b) 1
- (c) 2
- (d) 3

**Solution:**  
- The smallest composite number is 4.
- The smallest prime number is 2.
- The HCF of 4 and 2 is 2.

The correct answer is (c) 2.

---

### **Problem 4:**
**Given that \( \text{HCF}(1152, 1664) = 128 \), the LCM \( (1152, 1664) \) is**
- (a) 14976
- (b) 1664
- (c) 1152
- (d) None of these

**Solution:**  
Using the relationship:
\[
\text{HCF} \times \text{LCM} = \text{Product of the numbers}
\]
\[
128 \times \text{LCM} = 1152 \times 1664
\]
\[
\text{LCM} = \frac{1152 \times 1664}{128} = 14976
\]
The correct answer is (a) 14976.

---

### **Problem 5:**
**The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, then the other number is**
- (a) 23
- (b) 207
- (c) 1449
- (d) None of these

**Solution:**  
Let the other number be \( x \). Then,
\[
\text{HCF} \times \text{LCM} = \text{Product of the numbers}
\]
\[
23 \times 1449 = 161 \times x
\]
\[
x = \frac{23 \times 1449}{161} = 207
\]
The correct answer is (b) 207.

---

### **Problem 6:**
**A number when divided by 61 gives 27 as quotient and 32 as remainder. The number is**
- (a) 1679
- (b) 1664
- (c) 1449
- (d) None of these

**Solution:**  
The number can be found using:
\[
\text{Number} = 61 \times 27 + 32 = 1659 + 32 = 1691
\]
The correct answer is (a) 1679.

---

### **Problem 7:**
**The product of LCM and HCF of two numbers is equal to**
- (a) Sum of the numbers
- (b) Difference of the numbers
- (c) Product of the numbers
- (d) Quotients of the numbers

**Solution:**  
The product of the LCM and HCF of two numbers is equal to the product of the numbers themselves.  
The correct answer is (c) Product of the numbers.

---

### **Problem 8:**
**The LCM of two co-prime numbers is always**
- (a) Product of the numbers
- (b) Sum of the numbers
- (c) Difference of the numbers
- (d) None

**Solution:**  
For two co-prime numbers, the LCM is equal to the product of the numbers.  
The correct answer is (a) Product of the numbers.

---

I have solved the first 8 problems. Would you like me to continue with the rest, or provide more details on any specific problem?

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**8 Relative Questions:**
1. How can we use the relationship between HCF and LCM in problem-solving?
2. What is the significance of co-prime numbers in number theory?
3. How do we determine the HCF and LCM of large numbers efficiently?
4. Why is the property \( \text{HCF} \times \text{LCM} = \text{Product of the numbers} \) always true?
5. How can we apply the concept of HCF and LCM in real-life situations?
6. What are some examples of problems involving HCF, LCM, and divisibility?
7. How do prime numbers relate to the calculation of HCF and LCM?
8. What role does the Fundamental Theorem of Arithmetic play in finding HCF and LCM?

**Tip:** Always break down a large number into its prime factors to simplify the process of finding HCF and LCM.

Explore solutions to math problems involving Highest Common Factor (HCF) and Least Common Multiple (LCM). This detailed guide covers concepts like prime numbers, divisibility, and the Fundamental Theorem of Arithmetic. Suitable for Grades 7-9.

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