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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6. a) 8, 12 and 20 are three given whole numbers. i) Write the factors of these numbers separately. Step 1: List the factors for each number. • Factors of 8: 1, 2, 4, 8 • Factors of 12: 1, 2, 3, 4, 6, 12 • Factors of 20: 1, 2, 4, 5, 10, 20 The factors are: 8: 1, 2, 4, 8, 12: 1, 2, 3, 4, 6, 12, 20: 1, 2, 4, 5, 10, 20. ii) What are the common factors of these numbers? Step 1: Identify the factors that appear in all three lists. The common factors of 8, 12, and 20 are 1, 2, 4. The common factors are 1, 2, 4. iii) Among these common factors, which one is the greatest number that divides 8, 12 and 20 without leaving a remainder? Step 1: From the common factors (1, 2, 4), identify the largest one. The greatest common factor is 4. The greatest number is 4. b) 48 and 60 are two given whole numbers. i) Express each number as the products of their prime factors. Step 1: Find the prime factorization for 48. 48 = 2 × 2 × 2 × 2 × 3 = 2^4 × 3^1 Step 2: Find the prime factorization for 60. 60 = 2 × 2 × 3 × 5 = 2^2 × 3^1 × 5^1 The prime factors are: 48: 2^4 × 3^1, 60: 2^2 × 3^1 × 5^1. ii) Find the product of the common prime factors of both numbers. Step 1: Identify the common prime factors and their lowest powers. Common prime factors are 2 and 3. The lowest power of 2 is 2^2, and the lowest power of 3 is 3^1. Step 2: Calculate the product. Product = 2^2 × 3^1 = 4 × 3 = 12 The product of the common prime factors is 12. iii) Is the product H.C.F. of the given numbers? Give reason. Step 1: Compare the product with the definition of H.C.F. Yes, the product (12) is the H.C.F. of 48 and 60. Reason: The H.C.F. is found by taking the product of the common prime factors, each raised to the lowest power that appears in the prime factorization of the given numbers. The product is Yes, because it is the product of the common prime factors raised to their lowest powers. c) 45, 60 and 75 are three given whole numbers. i) What do you mean by the H.C.F. of these numbers? The H.C.F. (Highest Common Factor) of these numbers is the largest positive integer that divides 45, 60, and 75 exactly, without leaving any remainder. ii) Find the greatest number that divides these three numbers leaving a remainder. Note: The question likely intends to ask for the greatest number that divides these three numbers without leaving a remainder, which is the H.C.F. I will proceed with this interpretation. Step 1: Find the prime factorization of each number. 45 = 3 × 3 × 5 = 3^2 × 5^1 60 = 2 × 2 × 3 × 5 = 2^2 × 3^1 × 5^1 75 = 3 × 5 × 5 = 3^1 × 5^2 Step 2: Identify the common prime factors and their lowest powers. The common prime factors are 3 and 5. The lowest power of 3 is 3^1, and the lowest power of 5 is 5^1. Step 3: Multiply the common prime factors with their lowest powers to find the H.C.F. H.C.F. = 3^1 × 5^1 = 3 × 5 = 15 The greatest number that divides these three numbers exactly is 15. That's 2 down. 3 left today — send the next one.