Question

Let $A=\{1,2,3,\dots ,40\}$ and $R$ be an equivalence relation on $A\times A$ defined by $(a,b)R(c,d)$ if and only if $ad=bc.$ If $n$ is the number of elements in the equivalence class $\left[\right(1,3\left)\right]$ and $m$ is the number of elements in the equivalence class $\left[\right(1,4\left)\right],$ then the value of $m+n$ is

Answer 1

$R$ is defined as $(a,b)R(c,d)$ iff $ad=bc$
Here, $(a,b)R(1,3)$ iff $3a=b\times 1$
$\Rightarrow \frac{b}{a}=3$
$\therefore \left[\right(1,3\left)\right]=\left\{\right(1,3),(2,6),\dots ,(13,39\left)\right\}$
$n=13$

And $(a,b)R(1,4)$ iff $4a=b\times 1$
$\Rightarrow \frac{b}{a}=4$
$\therefore \left[\right(1,4\left)\right]=\left\{\right(1,4),(2,8),\dots ,(10,40\left)\right\}$
$m=10$
$\therefore m+n=13+10=23$