Let $A=\{1,2,3, \ldots ., 10\}$ and $$ B=\left\{\frac{m}{n}: m, n \in A, m

Question

Let $A=\{1,2,3, \ldots ., 10\}$ and $$ B=\left\{\frac{m}{n}: m, n \in A, m<n 	ext { and } \operatorname{gcd}(m, n)=1ight\} $$ Then $n(B)$ is equal to :\

Professor PC Thomas Classes · Answer

(b): We have, $A=\{1,2,3, \ldots . ., 10\}$ $$ B=\left\{\frac{m}{n}: m, n \in A, m<n 	ext { and } \operatorname{gcd}(m, n)=1ight\} $$ For $m=1, n=2,3, \ldots \ldots . ., 10 \Rightarrow 9$ cases $$ \begin{array}{ll} m=2, n=3,5,7,9, & \Rightarrow 4 	ext { cases } \ m=3, n=4,5,7,8,10 & \Rightarrow 5 	ext { cases } \ m=4, n=5,7,9 & \Rightarrow 3 	ext { cases } \ m=5, n=6,7,8,9 & \Rightarrow 4 	ext { cases } \ m=6, n=7 & \Rightarrow 1 	ext { case } \ m=7, n=8,9,10 & \Rightarrow 3 	ext { cases } \ m=8, n=9 & \Rightarrow 1 	ext { case } \ m=9, n=10 & \Rightarrow 1 	ext { case } \end{array} $$ $  	herefore \quad n(B)=31 $