Exercise 1.2 1. Write each of the numbers below as the product of its prime factors and then write down the HCF and LCM in each case. $$ \begin{array}{lll} ext { a) } 12 ext { and } 18 & ext { b) } 10 ext { and } 20 & ext { c) } 75 ext { and } 45 \ ext { d) } 24 ext { and } 60 & ext { e) } 8,4 ext { and } 12 & ext { f) } 6,10 ext { and } 15\end{array} $$ g) 6,8 and 12

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Exercise 1.2 1. Write each of the numbers below as the product of its prime factors and then write down the HCF and LCM in each case. $$ \begin{array}{lll}	ext { a) } 12 	ext { and } 18 & 	ext { b) } 10 	ext { and } 20 & 	ext { c) } 75 	ext { and } 45 \ 	ext { d) } 24 	ext { and } 60 & 	ext { e) } 8,4 	ext { and } 12 & 	ext { f) } 6,10 	ext { and } 15\end{array} $$ g) 6,8 and 12

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Certainly! Let's solve each part of Exercise 1.2 step by step. For each set of numbers, we'll:

1. **Factorize each number into its prime factors.**
2. **Determine the Highest Common Factor (HCF)**, also known as the Greatest Common Divisor (GCD).
3. **Determine the Lowest Common Multiple (LCM).**

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### **a) 12 and 18**

1. **Prime Factorization:**
   - $$12 = 2^2 	imes 3$$
   - $$18 = 2 	imes 3^2$$

2. **Highest Common Factor (HCF):**
   - Identify the common prime factors with the smallest exponents.
   - Common factors: $$2^1$$ and $$3^1$$
   - $$ 	ext{HCF} = 2 	imes 3 = 6 $$

3. **Lowest Common Multiple (LCM):**
   - Take all prime factors with the highest exponents.
   - Prime factors: $$2^2$$ and $$3^2$$
   - $$ 	ext{LCM} = 2^2 	imes 3^2 = 4 	imes 9 = 36 $$

---

### **b) 10 and 20**

1. **Prime Factorization:**
   - $$10 = 2 	imes 5$$
   - $$20 = 2^2 	imes 5$$

2. **Highest Common Factor (HCF):**
   - Common factors: $$2^1$$ and $$5^1$$
   - $$ 	ext{HCF} = 2 	imes 5 = 10 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$2^2$$ and $$5$$
   - $$ 	ext{LCM} = 2^2 	imes 5 = 4 	imes 5 = 20 $$

---

### **c) 75 and 45**

1. **Prime Factorization:**
   - $$75 = 3 	imes 5^2$$
   - $$45 = 3^2 	imes 5$$

2. **Highest Common Factor (HCF):**
   - Common factors: $$3^1$$ and $$5^1$$
   - $$ 	ext{HCF} = 3 	imes 5 = 15 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$3^2$$ and $$5^2$$
   - $$ 	ext{LCM} = 3^2 	imes 5^2 = 9 	imes 25 = 225 $$

---

### **d) 24 and 60**

1. **Prime Factorization:**
   - $$24 = 2^3 	imes 3$$
   - $$60 = 2^2 	imes 3 	imes 5$$

2. **Highest Common Factor (HCF):**
   - Common factors: $$2^2$$ and $$3^1$$
   - $$ 	ext{HCF} = 2^2 	imes 3 = 4 	imes 3 = 12 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$2^3$$, $$3$$, and $$5$$
   - $$ 	ext{LCM} = 2^3 	imes 3 	imes 5 = 8 	imes 3 	imes 5 = 120 $$

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### **e) 8, 4, and 12**

1. **Prime Factorization:**
   - $$8 = 2^3$$
   - $$4 = 2^2$$
   - $$12 = 2^2 	imes 3$$

2. **Highest Common Factor (HCF):**
   - Common factor among all three numbers: $$2^2$$
   - $$ 	ext{HCF} = 2^2 = 4 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$2^3$$ and $$3$$
   - $$ 	ext{LCM} = 2^3 	imes 3 = 8 	imes 3 = 24 $$

---

### **f) 6, 10, and 15**

1. **Prime Factorization:**
   - $$6 = 2 	imes 3$$
   - $$10 = 2 	imes 5$$
   - $$15 = 3 	imes 5$$

2. **Highest Common Factor (HCF):**
   - There are no common prime factors across all three numbers.
   - $$ 	ext{HCF} = 1 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$2$$, $$3$$, and $$5$$
   - $$ 	ext{LCM} = 2 	imes 3 	imes 5 = 30 $$

---

### **g) 6, 8, and 12**

1. **Prime Factorization:**
   - $$6 = 2 	imes 3$$
   - $$8 = 2^3$$
   - $$12 = 2^2 	imes 3$$

2. **Highest Common Factor (HCF):**
   - Common prime factor among all three numbers: $$2$$
   - $$ 	ext{HCF} = 2 $$

3. **Lowest Common Multiple (LCM):**
   - Prime factors: $$2^3$$ and $$3$$
   - $$ 	ext{LCM} = 2^3 	imes 3 = 8 	imes 3 = 24 $$

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### **Summary of Results**

| Part | Numbers          | Prime Factorization                               | HCF | LCM  |
|------|------------------|---------------------------------------------------|-----|------|
| a)   | 12 and 18        | $$12 = 2^2 	imes 3$$ $$18 = 2 	imes 3^2$$   | 6   | 36   |
| b)   | 10 and 20        | $$10 = 2 	imes 5$$ $$20 = 2^2 	imes 5$$     | 10  | 20   |
| c)   | 75 and 45        | $$75 = 3 	imes 5^2$$ $$45 = 3^2 	imes 5$$   | 15  | 225  |
| d)   | 24 and 60        | $$24 = 2^3 	imes 3$$ $$60 = 2^2 	imes 3 	imes 5$$ | 12  | 120  |
| e)   | 8, 4, and 12     | $$8 = 2^3$$ $$4 = 2^2$$ $$12 = 2^2 	imes 3$$ | 4    | 24   |
| f)   | 6, 10, and 15    | $$6 = 2 	imes 3$$ $$10 = 2 	imes 5$$ $$15 = 3 	imes 5$$ | 1    | 30   |
| g)   | 6, 8, and 12     | $$6 = 2 	imes 3$$ $$8 = 2^3$$ $$12 = 2^2 	imes 3$$ | 2    | 24   |

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Feel free to ask if you need further explanations or assistance with related topics!
Here are the HCF and LCM for each pair of numbers: - **a) 12 and 18** - HCF: 6 - LCM: 36 - **b) 10 and 20** - HCF: 10 - LCM: 20 - **c) 75 and 45** - HCF: 15 - LCM: 225 - **d) 24 and 60** - HCF: 12 - LCM: 120 - **e) 8, 4, and 12** - HCF: 4 - LCM: 24 - **f) 6, 10, and 15** - HCF: 1 - LCM: 30 - **g) 6, 8, and 12** - HCF: 2 - LCM: 24

Exercise 1.2

Write each of the numbers below as the product of its prime factors and then
write down the HCF and LCM in each case.

g) 6,8 and 12

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Exercise 1.2 Write each of the numbers below as the product of its prime factors and then write down the HCF and LCM in each case. g) 6,8 and 12