(d) Devemine the HCF of the (1) 36 and 54. (3) 120 and 154 . (5) 132 and 864 . (7) 480 and 432 . (10)

Certainly! Let's determine the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), for each pair of numbers you've provided. The HCF of two numbers is the largest number that divides both of them without leaving a remainder. We'll use two methods to find the HCF: 1. **Prime Factorization**: Breaking down each number into its prime factors and then multiplying the common factors. 2. **Euclidean Algorithm**: A systematic method that involves dividing and finding remainders until a remainder of zero is achieved. --- ### (1) HCF of 36 and 54 **Prime Factorization Method:** - **36**: \(2^2 \times 3^2\) - **54**: \(2^1 \times 3^3\) **Common Factors**: \(2^1 \times 3^2 = 2 \times 9 = 18\) **HCF = 18** **Euclidean Algorithm Method:** 1. \(54 ÷ 36 = 1\) with a remainder of \(18\). 2. \(36 ÷ 18 = 2\) with a remainder of \(0\). Since the remainder is now \(0\), the HCF is **18**. --- ### (3) HCF of 120 and 154 **Prime Factorization Method:** - **120**: \(2^3 \times 3^1 \times 5^1\) - **154**: \(2^1 \times 7^1 \times 11^1\) **Common Factors**: \(2^1 = 2\) **HCF = 2** **Euclidean Algorithm Method:** 1. \(154 ÷ 120 = 1\) with a remainder of \(34\). 2. \(120 ÷ 34 = 3\) with a remainder of \(18\). 3. \(34 ÷ 18 = 1\) with a remainder of \(16\). 4. \(18 ÷ 16 = 1\) with a remainder of \(2\). 5. \(16 ÷ 2 = 8\) with a remainder of \(0\). Since the remainder is now \(0\), the HCF is **2**. --- ### (5) HCF of 132 and 864 **Prime Factorization Method:** - **132**: \(2^2 \times 3^1 \times 11^1\) - **864**: \(2^5 \times 3^3\) **Common Factors**: \(2^2 \times 3^1 = 4 \times 3 = 12\) **HCF = 12** **Euclidean Algorithm Method:** 1. \(864 ÷ 132 = 6\) with a remainder of \(72\). 2. \(132 ÷ 72 = 1\) with a remainder of \(60\). 3. \(72 ÷ 60 = 1\) with a remainder of \(12\). 4. \(60 ÷ 12 = 5\) with a remainder of \(0\). Since the remainder is now \(0\), the HCF is **12**. --- ### (7) HCF of 480 and 432 **Prime Factorization Method:** - **480**: \(2^5 \times 3^1 \times 5^1\) - **432**: \(2^4 \times 3^3\) **Common Factors**: \(2^4 \times 3^1 = 16 \times 3 = 48\) **HCF = 48** **Euclidean Algorithm Method:** 1. \(480 ÷ 432 = 1\) with a remainder of \(48\). 2. \(432 ÷ 48 = 9\) with a remainder of \(0\). Since the remainder is now \(0\), the HCF is **48**. --- ### (10) [No numbers provided] It seems like part (10) was mentioned, but the corresponding pair of numbers isn't provided. Could you please provide the numbers for part (10) so I can assist you in finding their HCF? --- **Summary of HCFs:** 1. **HCF of 36 and 54:** 18 3. **HCF of 120 and 154:** 2 5. **HCF of 132 and 864:** 12 7. **HCF of 480 and 432:** 48 Feel free to ask if you have further questions or need additional assistance!