Ononot — Let U = \{1, 2, 3, \dots, 12\}. Let A be the set of divisors

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Set Theoryhard

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Let $U = \{1, 2, 3, \dots, 12\}$ . Let $A$ be the set of divisors of 12 in $U$ , and $B$ be the set of divisors of 18 in $U$ . We define a set of ordered pairs $S = \{ (x, y) \in A \times B : \text{gcd}(x, y) \in A \cap B \}$ . What is the cardinality of $S$ ?