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

**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**: \[ \text{LCM} = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 = 900 \]