This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions for questions 3, 4, and 5: 3. Creative Section - A Let's write the sets of all possible factors and the sets of common factors of these pairs of numbers, then find their H.C.F. a) 6 and 9 Step 1: List factors for each number. • Factors of 6: \1, 2, 3, 6\ • Factors of 9: \1, 3, 9\ Step 2: Identify common factors. • Common factors: \1, 3\ Step 3: Find the H.C.F. The H.C.F. is the greatest common factor. H.C.F. of 6 and 9 is 3. b) 10 and 15 Step 1: List factors for each number. • Factors of 10: \1, 2, 5, 10\ • Factors of 15: \1, 3, 5, 15\ Step 2: Identify common factors. • Common factors: \1, 5\ Step 3: Find the H.C.F. H.C.F. of 10 and 15 is 5. c) 12 and 30 Step 1: List factors for each number. • Factors of 12: \1, 2, 3, 4, 6, 12\ • Factors of 30: \1, 2, 3, 5, 6, 10, 15, 30\ Step 2: Identify common factors. • Common factors: \1, 2, 3, 6\ Step 3: Find the H.C.F. H.C.F. of 12 and 30 is 6. d) 18 and 24 Step 1: List factors for each number. • Factors of 18: \1, 2, 3, 6, 9, 18\ • Factors of 24: \1, 2, 3, 4, 6, 8, 12, 24\ Step 2: Identify common factors. • Common factors: \1, 2, 3, 6\ Step 3: Find the H.C.F. H.C.F. of 18 and 24 is 6. e) 24 and 32 Step 1: List factors for each number. • Factors of 24: \1, 2, 3, 4, 6, 8, 12, 24\ • Factors of 32: \1, 2, 4, 8, 16, 32\ Step 2: Identify common factors. • Common factors: \1, 2, 4, 8\ Step 3: Find the H.C.F. H.C.F. of 24 and 32 is 8. f) 21 and 28 Step 1: List factors for each number. • Factors of 21: \1, 3, 7, 21\ • Factors of 28: \1, 2, 4, 7, 14, 28\ Step 2: Identify common factors. • Common factors: \1, 7\ Step 3: Find the H.C.F. H.C.F. of 21 and 28 is 7. 4. Let's find the H.C.F. of the following numbers by factorisation method. a) 9, 12 Step 1: Prime factorization. 9 = 3^2 12 = 2^2 × 3^1 Step 2: Identify common prime factors with lowest powers. The common prime factor is 3^1. H.C.F. of 9 and 12 is 3. b) 15, 18 Step 1: Prime factorization. 15 = 3^1 × 5^1 18 = 2^1 × 3^2 Step 2: Identify common prime factors with lowest powers. The common prime factor is 3^1. H.C.F. of 15 and 18 is 3. c) 12, 16 Step 1: Prime factorization. 12 = 2^2 × 3^1 16 = 2^4 Step 2: Identify common prime factors with lowest powers. The common prime factor is 2^2. H.C.F. of 12 and 16 is 2^2 = 4. d) 18, 24 Step 1: Prime factorization. 18 = 2^1 × 3^2 24 = 2^3 × 3^1 Step 2: Identify common prime factors with lowest powers. The common prime factors are 2^1 and 3^1. H.C.F. of 18 and 24 is 2 × 3 = 6. e) 36, 48 Step 1: Prime