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

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Sets Relations and Functions

Let

A = {1, 2, 3, \dots, 10}

and

B = {\frac{m}{n} : m, n \in A, m < n and gcd (m, n) = 1}

. Then

n (B)

is equal to :

Answer: (A)

For a fixed denominator

n

(

2 \leq n \leq 10

), the numerators

m

with

1 \leq m < n

and

g cd (m, n) = 1

number exactly

φ (n)

, Euler's totient. Since each such fraction

m / n

is already in lowest terms, fractions with different

(m, n)

pairs are automatically distinct values, so

n (B) = n = 2 \sum 10 φ (n) .

Computing:

φ (2) = 1, φ (3) = 2, φ (4) = 2, φ (5) = 4, φ (6) = 2, φ (7) = 6, φ (8) = 4, φ (9) = 6, φ (10) = 4.

Summing:

1 + 2 + 2 + 4 + 2 + 6 + 4 + 6 + 4 = 31.

JEE Main · 2025 · 22 Jan 2025 · Shift 1 · Single correct option