Let $$A = \{2, 3, 4, 5, \ldots, 30\}$$ and '$$\sim$$' be an equivalence relation on $$A \times A$$, defined by $$(a, b) \sim (c, d)$$, if and only if $$ad = bc$$. Then the number of ordered pairs which satisfy this equivalence relation with ordered pair $$(4, 3)$$ is equal to:

We have $$A = \{2, 3, 4, 5, \ldots, 30\}$$ and the equivalence relation $$(a, b) \sim (c, d)$$ if and only if $$ad = bc$$. We need to find the number of ordered pairs $$(a, b)$$ equivalent to $$(4, 3)$$.

The condition $$(a, b) \sim (4, 3)$$ means $$3a = 4b$$, i.e., $$\frac{a}{b} = \frac{4}{3}$$. So $$a = 4k$$ and $$b = 3k$$ for some positive integer $$k$$, with both $$a, b \in A$$.

We need $$2 \leq 4k \leq 30$$ and $$2 \leq 3k \leq 30$$. From the first: $$k \leq 7$$ (and $$k \geq 1$$). From the second: $$k \leq 10$$ (and $$k \geq 1$$). So $$1 \leq k \leq 7$$.

The valid pairs are: $$(4$$, $$3)$$, $$(8$$, $$6)$$, $$(12$$, $$9)$$, $$(16$$, $$12)$$, $$(20$$, $$15)$$, $$(24$$, $$18)$$, $$(28$$, $$21)$$. All values lie in $$A$$, giving us 7 ordered pairs.