Question

Let $A=\{2,3,4,5,\dots ,30\}$ and ${}^{\prime }{\cong }^{\prime }$ be an equivalence relation on $A\times A,$ defined by $(a,b)\cong (c,d),$ 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 :

• A. $7$
• B. $5$
• C. $6$
• D. $8$

Answer 1

The correct option is A $7$

$(a,b)R(c,d)$ iff $ad=bc$
$(a,b)R(4,3)\Rightarrow 3a=4b$
$\Rightarrow a=\frac{4b}{3}$
$b$ must be multiple of $3.$
$b$ can be $3,6,9,\dots ,30$
Also, $a$ must be less than or equal to $30.$
So $(a,b)$ can be $(4,3),(8,6),(12,9),(16,12),(20,15),(24,18),(28,21)$
$\Rightarrow 7$ ordered pairs