Q3 Use prime factors to find (i) the HCF and (ii) the LCM of each of the following pairs of numbers. a) 60 and 75 HCF-is b) 54 and 96 c) 108 and 144 d) 200 and 240 e) 168 and 196 f) 150 and 180

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**a) For 60 and 75**

1. Write the prime factorizations:
   $$
   60 = 2^2 \times 3 \times 5, \quad 75 = 3 \times 5^2
   $$

2. **HCF** is found by taking the lowest power of the common primes:
   $$
   \text{HCF} = 3^1 \times 5^1 = 15
   $$

3. **LCM** is found by taking the highest power of each prime that appears:
   $$
   \text{LCM} = 2^2 \times 3^1 \times 5^2 = 4 \times 3 \times 25 = 300
   $$

---

**b) For 54 and 96**

1. Write the prime factorizations:
   $$
   54 = 2 \times 3^3, \quad 96 = 2^5 \times 3
   $$

2. **HCF** (lowest power of common factors):
   $$
   \text{HCF} = 2^1 \times 3^1 = 6
   $$

3. **LCM** (highest power of each prime):
   $$
   \text{LCM} = 2^5 \times 3^3 = 32 \times 27 = 864
   $$

---

**c) For 108 and 144**

1. Write the prime factorizations:
   $$
   108 = 2^2 \times 3^3, \quad 144 = 2^4 \times 3^2
   $$

2. **HCF**:
   $$
   \text{HCF} = 2^2 \times 3^2 = 4 \times 9 = 36
   $$

3. **LCM**:
   $$
   \text{LCM} = 2^4 \times 3^3 = 16 \times 27 = 432
   $$

---

**d) For 200 and 240**

1. Write the prime factorizations:
   $$
   200 = 2^3 \times 5^2, \quad 240 = 2^4 \times 3 \times 5
   $$

2. **HCF**:
   $$
   \text{HCF} = 2^3 \times 5^1 = 8 \times 5 = 40
   $$

3. **LCM**:
   $$
   \text{LCM} = 2^4 \times 3^1 \times 5^2 = 16 \times 3 \times 25 = 1200
   $$

---

**e) For 168 and 196**

1. Write the prime factorizations:
   $$
   168 = 2^3 \times 3 \times 7, \quad 196 = 2^2 \times 7^2
   $$

2. **HCF**:
   $$
   \text{HCF} = 2^2 \times 7^1 = 4 \times 7 = 28
   $$

3. **LCM**:
   $$
   \text{LCM} = 2^3 \times 3^1 \times 7^2 = 8 \times 3 \times 49 = 1176
   $$

---

**f) For 150 and 180**

1. Write the prime factorizations:
   $$
   150 = 2 \times 3 \times 5^2, \quad 180 = 2^2 \times 3^2 \times 5
   $$

2. **HCF**:
   $$
   \text{HCF} = 2^1 \times 3^1 \times 5^1 = 30
   $$

3. **LCM**:
   $$
   \text{LCM} = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900
   $$
**a) 60 and 75** - **HCF:** 15 - **LCM:** 300 **b) 54 and 96** - **HCF:** 6 - **LCM:** 864 **c) 108 and 144** - **HCF:** 36 - **LCM:** 432 **d) 200 and 240** - **HCF:** 40 - **LCM:** 1200 **e) 168 and 196** - **HCF:** 28 - **LCM:** 1176 **f) 150 and 180** - **HCF:** 30 - **LCM:** 900

Q3 Use prime factors to find (i) the HCF and (ii) the LCM of each of the following pairs of numbers. a) 60 and 75 HCF-is b) 54 and 96 c) 108 and 144 d) 200 and 240 e) 168 and 196 f) 150 and 180

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