9. Give the greatest common factor of each of the following pairs of numbers. - Option 1: Do this by placing the two quantities in a ratio and scaling the ratio until it is a simplest ratio of two integers. Then, determine the greatest common factor based on your scaling process. - Option 2: Do this by listing all of the common factors of both quantities and finding the greatest factor they share in common. (a) 32 and 60 (b) 56 and 42 (c) 7 and 50 (d) 24 and 48 (e) 8 and 9

Let's solve the problem using both options for each pair of numbers. ### Option 1: Scaling the Ratio To find the greatest common factor (GCF) using the ratio method, we will express the two numbers as a ratio and scale it down to its simplest form. The GCF can then be determined from the scaling process. ### Option 2: Listing Common Factors We will also list the factors of each number and find the greatest common factor. Let's start with each pair of numbers: ### (a) 32 and 60 **Option 1: Scaling the Ratio** 1. Write the ratio: \( \frac{32}{60} \). 2. Simplify the ratio by dividing both numbers by their GCF. The GCF of 32 and 60 can be found by scaling down: - \( 32 = 2^5 \) - \( 60 = 2^2 \times 3 \times 5 \) - The common factor is \( 2^2 = 4 \). 3. Scale down: - \( \frac{32 \div 4}{60 \div 4} = \frac{8}{15} \). **Option 2: Listing Common Factors** - Factors of 32: 1, 2, 4, 8, 16, 32 - Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 - Common factors: 1, 2, 4 - GCF: 4 ### (b) 56 and 42 **Option 1: Scaling the Ratio** 1. Write the ratio: \( \frac{56}{42} \). 2. Simplify the ratio: - \( 56 = 2^3 \times 7 \) - \( 42 = 2 \times 3 \times 7 \) - The common factor is \( 2^1 \times 7 = 14 \). 3. Scale down: - \( \frac{56 \div 14}{42 \div 14} = \frac{4}{3} \). **Option 2: Listing Common Factors** - Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56 - Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 - Common factors: 1, 2, 7, 14 - GCF: 14 ### (c) 7 and 50 **Option 1: Scaling the Ratio** 1. Write the ratio: \( \frac{7}{50} \). 2. Since 7 is a prime number and does not divide 50, the GCF is 1. 3. The simplest ratio is \( \frac{7}{50} \). **Option 2: Listing Common Factors** - Factors of 7: 1, 7 - Factors of 50: 1, 2, 5, 10, 25, 50 - Common factor: 1 - GCF: 1 ### (d) 24 and 48 **Option 1: Scaling the Ratio** 1. Write the ratio: \( \frac{24}{48} \). 2. Simplify the ratio: - \( 24 = 2^3 \times 3 \) - \( 48 = 2^4 \times 3 \) - The common factor is \( 2^3 \times 3 = 24 \). 3. Scale down: - \( \frac{24 \div 24}{48 \div 24} = \frac{1}{2} \). **Option 2: Listing Common Factors** - Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 - Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 - Common factors: 1, 2, 3, 4, 6, 8, 12, 24 - GCF: 24 ### (e) 8 and 9 **Option 1: Scaling the Ratio** 1. Write the ratio: \( \frac{8}{9} \). 2. Since 8 and 9 have no common factors, the GCF is 1. 3. The simplest ratio is \( \frac{8}{9} \). **Option 2: Listing Common Factors** - Factors of 8: 1, 2, 4, 8 - Factors of 9: 1, 3, 9 - Common factor: 1 - GCF: 1 ### Summary of GCFs - (a) GCF of 32 and 60: 4 - (b) GCF of 56 and 42: 14 - (c) GCF of 7 and 50: 1 - (d) GCF of 24 and 48: 24 - (e) GCF of 8 and 9: 1