Sets, Relations & Functions
Counting Elements in a Set
nta_pyq_2025_apr
Grade 11
Question:
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:
Step-by-Step Solution
Key Concept: For each denominator $n$ from 2 to 10, count the numerators $m < n$ with $\gcd(m,n)=1$ using Euler's totient function $\phi(n)$.
$n=2$: 1 fraction; $n=3$: 2; $n=4$: 2; $n=5$: 4; $n=6$: 2; $n=7$: 6; $n=8$: 4; $n=9$: 6; $n=10$: 4. Total $= 1+2+2+4+2+6+4+6+4 = 31$.
Correct Answer: 31