Let \( G=\{n \mid n \) divides 90\( \} \) and \( D=\{n \mid n \) divides 130\( \} \). In listed form, \( G=\{1,2,3,5,6,9,10,15,18,30,45,90\} \). (a) Find the listed form of the set \( D \). (b) Find \( G \cap D \). (c) Which element of \( G \cap D \) is largest? A. \( \{1,2,5,10,13,26,65,130\} \) B. \( 1,2,5,10,13,26,65,130 \) (a) Find the listed form of the set \( D \). C. \( \{1,2,3,4,5,6,7,8,9,130\} \) D. \( \{1,2,3,5,6,9,10,15,18,30,45,90\} \)

To find the set \( D \), we need the divisors of 130. The prime factorization of 130 is \( 2 \times 5 \times 13 \). The divisors of 130 are found by considering all combinations of these factors, leading to the following divisors: \( D = \{1, 2, 5, 10, 13, 26, 65, 130\} \). Next, let's find the intersection \( G \cap D \), which consists of elements that belong to both set \( G \) and set \( D \): - From \( G \): \( \{1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90\} \) - From \( D \): \( \{1, 2, 5, 10, 13, 26, 65, 130\} \) The common elements are: \( G \cap D = \{1, 2, 5, 10\} \). Now, to determine the largest element in \( G \cap D \), we can simply look at the intersection we've established. The largest element of \( G \cap D \) is \( 10 \). So the response to your queries are: (a) \( D = \{1, 2, 5, 10, 13, 26, 65, 130\} \). (b) \( G \cap D = \{1, 2, 5, 10\} \). (c) The largest element of \( G \cap D \) is \( 10 \). Lastly, looking at your options, the set \( D \) matches option A when expressed in set notation but is also the same as option B when expressed as a list. The final answer for \( D \) in a listed form is \( \{1, 2, 5, 10, 13, 26, 65, 130\} \).