Step 1 :Given the universal set \(U = \{1, 2, 3, 4, 5, 6\}\), and the sets \(A = \{2, 3, 5, 6\}\), \(B = \{1, 3, 5\}\), and \(C = \{1, 2, 3, 4, 5\}\).
Step 2 :First, we find the union of sets A and B, denoted as \(A \cup B\). This is the set of elements which are in A, or B, or in both. So, \(A \cup B = \{1, 2, 3, 5, 6\}\).
Step 3 :Next, we find the complement of the union of A and B, denoted as \((A \cup B)^{\prime}\). This is the set of elements in the universal set U that are not in \(A \cup B\). So, \((A \cup B)^{\prime} = \{4\}\).
Step 4 :Finally, we find the intersection of \((A \cup B)^{\prime}\) and C, denoted as \((A \cup B)^{\prime} \cap C\). This is the set of elements which are in both \((A \cup B)^{\prime}\) and C. So, \((A \cup B)^{\prime} \cap C = \{4\}\).
Step 5 :Final Answer: \((A \cup B)^{\prime} \cap C = \boxed{4}\)