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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You're on a roll — here are the solutions for questions 4 (continued) and 5: 4. Let's find the H.C.F. of the following numbers by factorisation method. e) 36, 48 Step 1: Find the prime factorization of each number. 36 = 2 × 2 × 3 × 3 = 2^2 × 3^2 48 = 2 × 2 × 2 × 2 × 3 = 2^4 × 3^1 Step 2: Identify the common prime factors and their lowest powers. The common prime factors are 2^2 and 3^1. Step 3: Multiply these common prime factors. H.C.F. = 2^2 × 3^1 = 4 × 3 = 12 H.C.F. of 36 and 48 is 12. f) 30, 75, 90 Step 1: Find the prime factorization of each number. 30 = 2 × 3 × 5 = 2^1 × 3^1 × 5^1 75 = 3 × 5 × 5 = 3^1 × 5^2 90 = 2 × 3 × 3 × 5 = 2^1 × 3^2 × 5^1 Step 2: Identify the common prime factors and their lowest powers. The common prime factors are 3^1 and 5^1. Step 3: Multiply these common prime factors. H.C.F. = 3^1 × 5^1 = 3 × 5 = 15 H.C.F. of 30, 75, and 90 is 15. g) 32, 64, 80 Step 1: Find the prime factorization of each number. 32 = 2 × 2 × 2 × 2 × 2 = 2^5 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6 80 = 2 × 2 × 2 × 2 × 5 = 2^4 × 5^1 Step 2: Identify the common prime factors and their lowest powers. The common prime factor is 2^4. Step 3: Calculate the value. H.C.F. = 2^4 = 16 H.C.F. of 32, 64, and 80 is 16. h) 96, 144, 216 Step 1: Find the prime factorization of each number. 96 = 2^5 × 3^1 144 = 2^4 × 3^2 216 = 2^3 × 3^3 Step 2: Identify the common prime factors and their lowest powers. The common prime factors are 2^3 and 3^1. Step 3: Multiply these common prime factors. H.C.F. = 2^3 × 3^1 = 8 × 3 = 24 H.C.F. of 96, 144, and 216 is 24. 5. Let's find the H.C.F. of the following numbers by division method. a) 18, 24 Step 1: Divide the larger number (24) by the smaller number (18). 24 = 18 × 1 + 6 Step 2: Divide the previous divisor (18) by the remainder (6). 18 = 6 × 3 + 0 The last non-zero remainder is the H.C.F. H.C.F. of 18 and 24 is 6. b) 28, 32 Step 1: Divide 32 by 28. 32 = 28 × 1 + 4 Step 2: Divide 28 by 4. 28 = 4 × 7 + 0 H.C.F. of 28 and 32 is 4. c) 36, 42 Step 1: Divide 42 by 36. 42 = 36 × 1 + 6 Step 2: Divide 36 by 6. 36 = 6 × 6 + 0 H.C.F. of 36 and 42 is 6. d) 52, 65 Step 1: Divide 65 by 52. 65 = 52 × 1 + 13 Step 2: Divide 52 by 13. 52 = 13 × 4 + 0 H.C.F. of 52 and 65 is 13. e) 35, 56, 70 Step 1: Find the H.C.F. of the first two numbers, 35 and 56. Divide 56 by 35: 56 = 35 × 1 + 21 Divide 35 by 21: 35 = 21 × 1 + 14 Divide 21 by 14: 21 = 14 × 1 + 7 Divide 14 by 7: 14 = 7 × 2 + 0 The H.C.F. of 35 and 56 is 7. Step 2: Find the H.C.F. of the result (7) and the third number (70). Divide 70 by 7: 70 = 7 × 10 + 0 H.C.F. of 35, 56, and 70 is 7. f) 30, 45, 60 Step 1: Find the H.C.F. of the first two numbers, 30 and 45. Divide 45 by 30: 45 = 30 × 1 + 15 Divide 30 by 15: 30 = 15 × 2 + 0 The H.C.F. of 30 and 45 is 15. Step 2: Find the H.C.F. of the result (15) and the third number (60). Divide 60 by 15: 60 = 15 × 4 + 0 H.C.F. of 30, 45, and 60 is 15. g) 156, 221, 390 Step 1: Find the H.C.F. of the first two numbers, 156 and 221. Divide 221 by 156: 221 = 156 × 1 + 65 Divide 156 by 65: 156 = 65 × 2 + 26 Divide 65 by 26: 65 = 26 × 2 + 13 Divide 26 by 13: 26 = 13 × 2 + 0 The H.C.F. of 156 and 221 is 1