Let A={1,2,3,…,10} and B={nm:m,n∈A,m<n and gcd(m,n)=1}. Then n(B) is equal to:
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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 nm 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∈A, the smallest possible value for m is 1. Therefore, n must be at least 2. Also, n∈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∈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.