Prove the following


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

equal to:

  1. (1) 5

  2. (2) 6

  3. (3) 8

  4. (4) 7

Correct Option: , 4


$\mathrm{A}=\{2,3,4,5, \ldots,, 30\}$

$(\mathrm{a}, \mathrm{b}) \simeq(\mathrm{c}, \mathrm{d}) \quad \Rightarrow \mathrm{ad}=\mathrm{bc}$

$(4,3) \simeq(c, d) \quad \Rightarrow \quad 4 d=3 c$

$\Rightarrow \frac{4}{3}=\frac{c}{d}$

$\frac{c}{d}=\frac{4}{3} \quad \& c, d \in\{2,3, \ldots \ldots, 30\}$

$(\mathrm{c}, \mathrm{d})=\{(4,3),(8,6),(12,9),(16,12),(20,$, $15),(24,18),(28,21)\}$

No. of ordered pair $=7$


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