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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a) Factors of 24 and 32 Step 1: Find the prime factorization of 24. 24 = 2 × 12 = 2 × 2 × 6 = 2 × 2 × 2 × 3 = 2^3 × 3 Step 2: Find the prime factorization of 32. 32 = 2 × 16 = 2 × 2 × 8 = 2 × 2 × 2 × 4 = 2 × 2 × 2 × 2 × 2 = 2^5 Step 3: Identify the common prime factors with the lowest power. The common prime factor is 2. The lowest power of 2 is 2^3. Step 4: Calculate the Highest Common Factor (HCF). HCF(24, 32) = 2^3 = 8 The HCF of 24 and 32 is 8. b) Factors of 64 and 32 Step 1: Find the prime factorization of 64. 64 = 2 × 32 = 2 × 2^5 = 2^6 Step 2: Find the prime factorization of 32. 32 = 2^5 Step 3: Identify the common prime factors with the lowest power. The common prime factor is 2. The lowest power of 2 is 2^5. Step 4: Calculate the HCF. HCF(64, 32) = 2^5 = 32 The HCF of 64 and 32 is 32. c) Factors of 48 and 36 Step 1: Find the prime factorization of 48. 48 = 2 × 24 = 2 × 2^3 × 3 = 2^4 × 3 Step 2: Find the prime factorization of 36. 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3 = 2^2 × 3^2 Step 3: Identify the common prime factors with the lowest power. Common prime factors are 2 and 3. The lowest power of 2 is 2^2. The lowest power of 3 is 3^1. Step 4: Calculate the HCF. HCF(48, 36) = 2^2 × 3^1 = 4 × 3 = 12 The HCF of 48 and 36 is 12. d) Factors of 72 and 32 Step 1: Find the prime factorization of 72. 72 = 2 × 36 = 2 × 2^2 × 3^2 = 2^3 × 3^2 Step 2: Find the prime factorization of 32. 32 = 2^5 Step 3: Identify the common prime factors with the lowest power. The common prime factor is 2. The lowest power of 2 is 2^3. Step 4: Calculate the HCF. HCF(72, 32) = 2^3 = 8 The HCF of 72 and 32 is 8. Problem solving 1. Factorise 358. Step 1: Divide 358 by the smallest prime number, 2. 358 ÷ 2 = 179 Step 2: Determine if 179 is a prime number. We check for divisibility by prime numbers up to sqrt(179) ≈ 13.37. Primes to check: 3, 5, 7, 11, 13. 179 is not divisible by 3 (sum of digits 1+7+9=17, not divisible by 3). 179 is not divisible by 5 (does not end in 0 or 5). 179 ÷ 7 = 25 with a remainder of 4. 179 ÷ 11 = 16 with a remainder of 3. 179 ÷ 13 = 13 with a remainder of 10. Since 179 is not divisible by any prime number up to 13, it is a prime number. Step 3: Write the prime factorization. 358 = 2 × 179 The prime factorization of 358 is 2 × 179. 2. What is the sum of the highest common factor of 100 and 150 together with the highest common factor of 200 and 250? Step 1: Find the HCF of 100 and 150. Prime factorization of 100: 100 = 2^2 × 5^2 Prime factorization of 150: 150 = 2 × 3 × 5^2 HCF(100, 150) = 2^1 × 5^2 = 2 × 25 = 50 Step 2: Find the HCF of 200 and 250. Prime factorization of 200: 200 = 2^3 × 5^2 Prime factorization of 250: 250 = 2 × 5^3 HCF(200, 250) = 2^1 × 5^2 = 2 × 25 = 50 Step 3: Calculate the sum of the two HCFs. Sum = 50 + 50 = 100 The sum is 100. Send me the next one 📸