QJEEM25Mafa06: JEE Main 2025 Mathematics - Class 11 JEEM25Mafa06 Solution: Let and and . Then is equal to :

To find the number of elements in set $B$ , we consider pairs $\left(\frac{m}{n}\right)$ where $m, n \in A$ with m < n and $\text{gcd}(m, n) = 1$ .

Here's the breakdown for each possible $m$ :

For $m = 1$ :

Possible values for $n$ are $2, 3, 4, 5, 6, 7, 8, 9, 10$ .

Total pairs: $9$ .

For $m = 2$ :

Possible values for $n$ are $3, 5, 7, 9$ (since these have $\text{gcd}(2, n) = 1$ ).

Total pairs: $4$ .

For $m = 3$ :

Possible values for $n$ are $4, 5, 7, 8, 10$ .

Total pairs: $5$ .

For $m = 4$ :

Possible values for $n$ are $5, 7, 9$ .

Total pairs: $3$ .

For $m = 5$ :

Possible values for $n$ are $6, 7, 8, 9$ .

Total pairs: $4$ .

For $m = 6$ :

Possible value for $n$ is $7$ .

Total pairs: $1$ .

For $m = 7$ :

Possible values for $n$ are $8, 9, 10$ .

Total pairs: $3$ .

For $m = 8$ :

Possible value for $n$ is $9$ .

Total pairs: $1$ .

For $m = 9$ :

Possible value for $n$ is $10$ .

Total pairs: $1$ .

Adding all these up, the total number of elements in set $B$ is:

$9 + 4 + 5 + 3 + 4 + 1 + 3 + 1 + 1 = 31$