- Find the ccm and LCF of: a) \( 12 ; 20 \) b) \( 25 ; 3625 ; 35 \) c) \( 36 ; 54 ; 72 \) d) \( 63 ; 105 ; 147 \) d) \( 693 ; 1617 ; 2541 \)

Sure, let's determine the Highest Common Factor (HCF) and Least Common Multiple (LCM) for each set of numbers provided. --- ### **a) \(12\) and \(20\)** **Prime Factorization:** - \(12 = 2^2 \times 3\) - \(20 = 2^2 \times 5\) **HCF:** - Multiply the lowest powers of common prime factors. - Common prime factor: \(2^2\) - **HCF = \(4\)** **LCM:** - Multiply the highest powers of all prime factors. - \(2^2 \times 3 \times 5 = 60\) - **LCM = \(60\)** --- ### **b) \(25\), \(3625\), and \(35\)** **Prime Factorization:** - \(25 = 5^2\) - \(35 = 5 \times 7\) - \(3625 = 5^3 \times 29\) **HCF:** - Common prime factor: \(5\) - Minimum exponent: \(5^1\) - **HCF = \(5\)** **LCM:** - \(5^3 \times 7 \times 29 = 125 \times 7 \times 29 = 25,375\) - **LCM = \(25,375\)** --- ### **c) \(36\), \(54\), and \(72\)** **Prime Factorization:** - \(36 = 2^2 \times 3^2\) - \(54 = 2 \times 3^3\) - \(72 = 2^3 \times 3^2\) **HCF:** - Common prime factors: \(2\) and \(3\) - Minimum exponents: \(2^1 \times 3^2 = 18\) - **HCF = \(18\)** **LCM:** - \(2^3 \times 3^3 = 8 \times 27 = 216\) - **LCM = \(216\)** --- ### **d) \(63\), \(105\), and \(147\)** **Prime Factorization:** - \(63 = 3^2 \times 7\) - \(105 = 3 \times 5 \times 7\) - \(147 = 3 \times 7^2\) **HCF:** - Common prime factors: \(3\) and \(7\) - Minimum exponents: \(3^1 \times 7^1 = 21\) - **HCF = \(21\)** **LCM:** - \(3^2 \times 5 \times 7^2 = 9 \times 5 \times 49 = 2,205\) - **LCM = \(2,205\)** --- ### **e) \(693\), \(1617\), and \(2541\)** **Prime Factorization:** - \(693 = 3^2 \times 7 \times 11\) - \(1617 = 3 \times 7^2 \times 11\) - \(2541 = 3 \times 7 \times 11^2\) **HCF:** - Common prime factors: \(3\), \(7\), and \(11\) - Minimum exponents: \(3^1 \times 7^1 \times 11^1 = 231\) - **HCF = \(231\)** **LCM:** - \(3^2 \times 7^2 \times 11^2 = 9 \times 49 \times 121 = 53,361\) - **LCM = \(53,361\)** --- **Summary:** | **Set of Numbers** | **HCF** | **LCM** | |-------------------------|---------|----------| | **a) 12, 20** | 4 | 60 | | **b) 25, 3625, 35** | 5 | 25,375 | | **c) 36, 54, 72** | 18 | 216 | | **d) 63, 105, 147** | 21 | 2,205 | | **e) 693, 1617, 2541** | 231 | 53,361 | If you have any further questions or need additional assistance, feel free to ask!