Find the greatest common divisor of each of these pairs of numbers by the method of intersection of sets of divisors. (a) 135 and 125 (b) 51 and 21 (c) 162 and 108 (a) The greatest common divisor of 135 and 125 is

To find the greatest common divisor (GCD) of the numbers using the method of intersection of sets of divisors, we will follow these steps: 1. **Find the divisors of each number.** 2. **List the divisors in sets.** 3. **Identify the common divisors.** 4. **Determine the greatest common divisor from the common divisors.** Let's start with part (a): finding the GCD of 135 and 125. ### Step 1: Find the divisors of 135 and 125 **Divisors of 135:** - The prime factorization of 135 is \(3^3 \times 5^1\). - The divisors can be calculated as follows: - \(1, 3, 5, 9, 15, 27, 45, 135\) **Divisors of 125:** - The prime factorization of 125 is \(5^3\). - The divisors can be calculated as follows: - \(1, 5, 25, 125\) ### Step 2: List the divisors in sets - Set of divisors of 135: \(D_{135} = \{1, 3, 5, 9, 15, 27, 45, 135\}\) - Set of divisors of 125: \(D_{125} = \{1, 5, 25, 125\}\) ### Step 3: Identify the common divisors Now, we find the intersection of the two sets: - Common divisors: \(D_{135} \cap D_{125} = \{1, 5\}\) ### Step 4: Determine the greatest common divisor The greatest common divisor is the largest number in the intersection: - GCD(135, 125) = \(5\) Thus, the greatest common divisor of 135 and 125 is \(5\). Now, let's proceed to part (b): finding the GCD of 51 and 21. ### Step 1: Find the divisors of 51 and 21 **Divisors of 51:** - The prime factorization of 51 is \(3^1 \times 17^1\). - The divisors can be calculated as follows: - \(1, 3, 17, 51\) **Divisors of 21:** - The prime factorization of 21 is \(3^1 \times 7^1\). - The divisors can be calculated as follows: - \(1, 3, 7, 21\) ### Step 2: List the divisors in sets - Set of divisors of 51: \(D_{51} = \{1, 3, 17, 51\}\) - Set of divisors of 21: \(D_{21} = \{1, 3, 7, 21\}\) ### Step 3: Identify the common divisors Now, we find the intersection of the two sets: - Common divisors: \(D_{51} \cap D_{21} = \{1, 3\}\) ### Step 4: Determine the greatest common divisor The greatest common divisor is the largest number in the intersection: - GCD(51, 21) = \(3\) Thus, the greatest common divisor of 51 and 21 is \(3\). Now, let's proceed to part (c): finding the GCD of 162 and 108. ### Step 1: Find the divisors of 162 and 108 **Divisors of 162:** - The prime factorization of 162 is \(2^1 \times 3^4\). - The divisors can be calculated as follows: - \(1, 2, 3, 6, 9, 18, 27, 54, 81, 162\) **Divisors of 108:** - The prime factorization of 108 is \(2^2 \times 3^3\). - The divisors can be calculated as follows: - \(1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\) ### Step 2: List the divisors in sets - Set of divisors of 162: \(D_{162} = \{1, 2, 3, 6, 9, 18, 27, 54, 81, 162\}\) - Set of divisors of 108: \(D_{108} = \{1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\}\) ### Step 3: Identify the common divisors Now, we find the intersection of the two sets: - Common divisors: \(D_{162} \cap D_{108} = \{1, 2, 3, 6, 9, 18, 27, 54\}\) ### Step 4: Determine the greatest common divisor The greatest common divisor is the largest number in the intersection: - GCD(162, 108) = \(54\) Thus, the greatest common divisor of 162 and 108 is \(54\). ### Summary of Results - GCD(135, 125) = \(5\) - GCD(51, 21) = \(3\) - GCD(162, 108) = \(54\)