The set A contains all natural numbers from 1 to 10, inclusive. The set B consists of fractions (m)/(n) where both m and n are elements of A, m is strictly less than n, and the greatest common divisor (GCD) of m and n is 1, meaning m and n are coprime.

The set A contains all natural numbers from 1 to 10, inclusive. The set B consists of fractions (m)/(n) where both m and n are elements of A, m is strictly less than n, and the greatest common divisor (GCD) of m and n is 1, meaning m and n are coprime.

We systematically list all possible values for n from 2 to 10. For each n, we find all m A such that m < n and (m, n) = 1. These pairs (m, n) form the fractions in set B.

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

Answer: 31

Step-by-step solution

Step 1: Understand the Sets

The set $A$ contains all natural numbers from 1 to 10, inclusive. The set $B$ consists of fractions $\frac{m}{n}$ where both $m$ and $n$ are elements of $A$ , $m$ is strictly less than $n$ , and the greatest common divisor (GCD) of $m$ and $n$ is 1, meaning $m$ and $n$ are coprime.

Step 2: List Possible Denominators (n)

Since $m < n$ and $m \in A$ , the smallest possible value for $m$ is 1. Therefore, $n$ must be at least 2. Also, $n \in A$ , so $n$ can range from 2 to 10.

Step 3: Find Coprime Pairs (m, n) for each n

We systematically list all possible values for $n$ from 2 to 10. For each $n$ , we find all $m \in A$ such that $m < n$ and $\gcd(m, n) = 1$ . These pairs $(m, n)$ form the fractions in set $B$ .

Step 4: Calculate the Total Number of Elements in B

The total number of elements in set $B$ , denoted as $n(B)$ , is the sum of the counts of valid $m$ values for each $n$ . Summing the counts from the table gives us the final answer.

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Let A={1,2,3,…,10}A = \{1, 2, 3, \ldots, 10\}A={1,2,3,…,10} and B={mn:m,n∈A,m<n and gcd⁡(m,n)=1}B = \left\{\frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1\right\}B={nm​:m,n∈A,m<n and gcd(m,n)=1}. Then n(B)n(B)n(B) is equal to: