Exercise 1.9 1 Find the HCF of these numbers by using prime factors a 48 and 108

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To find the Highest Common Factor (HCF) of the numbers **48** and **108** using the **prime factorization** method, follow these steps:

### **Step 1: Prime Factorization of Each Number**

**a) Prime Factors of 48**

1. **Divide by 2** (the smallest prime number):
   $$
   48 \div 2 = 24
   $$
2. **Divide by 2** again:
   $$
   24 \div 2 = 12
   $$
3. **Divide by 2** once more:
   $$
   12 \div 2 = 6
   $$
4. **Divide by 2** for the last time:
   $$
   6 \div 2 = 3
   $$
5. **Now, 3** is a prime number, so we stop here.

**Prime Factorization of 48:**
$$
48 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1
$$

---

**b) Prime Factors of 108**

1. **Divide by 2**:
   $$
   108 \div 2 = 54
   $$
2. **Divide by 2** again:
   $$
   54 \div 2 = 27
   $$
3. **Divide by 3**:
   $$
   27 \div 3 = 9
   $$
4. **Divide by 3** once more:
   $$
   9 \div 3 = 3
   $$
5. **Divide by 3** for the last time:
   $$
   3 \div 3 = 1
   $$

**Prime Factorization of 108:**
$$
108 = 2 \times 2 \times 3 \times 3 \times 3 = 2^2 \times 3^3
$$

---

### **Step 2: Identify Common Prime Factors**

From the prime factorizations:

- **48:** $$ 2^4 \times 3^1 $$
- **108:** $$ 2^2 \times 3^3 $$

**Common Prime Factors:**
- **2** and **3**

### **Step 3: Determine the Minimum Exponents for Common Factors**

- For **2**:
  - Minimum exponent is $$ \min(4, 2) = 2 $$
- For **3**:
  - Minimum exponent is $$ \min(1, 3) = 1 $$

### **Step 4: Calculate the HCF**

Multiply the common prime factors raised to their minimum exponents:
$$
\text{HCF} = 2^2 \times 3^1 = 4 \times 3 = 12
$$

### **Conclusion**

The **HCF of 48 and 108** is **12**.
The HCF of 48 and 108 is 12.

Exercise 1.9
1 Find the HCF of these numbers by using prime factors
a 48 and 108

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