Let A =\ 1,2,3, ldots ldots ., 10\ and B = \ m/n : m, n in A, m < n, \ and \ \ \ gcd (m, n) = 1 \ . Then n(B) is equal to:

Mathematics · Sets and Relations

JEE Main 2025 — 22 January, Morning Shift — Question 17

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

Option A:
31
Correct
Option B:
36
Option C:
37
Option D:
29

Answer: A

Step-by-step solution

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

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

n(B)

n = 2 \qquad \left\{ \frac{1}{2} \right\}

n = 3 \qquad \left\{ \frac{1}{3}, \frac{2}{3} \right\}

n = 4 \qquad \left\{ \frac{1}{4}, \frac{3}{4} \right\}

n = 5 \qquad \left\{ \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \right\}

n = 6 \qquad \left\{ \frac{1}{6}, \frac{5}{6} \right\}

n = 7 \qquad \left\{ \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} \right\}

n = 8 \qquad \left\{ \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} \right\}

n = 9 \qquad \left\{ \frac{1}{9}, \frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}, \frac{8}{9} \right\}

n = 10 \qquad \left\{ \frac{1}{10}, \frac{3}{10}, \frac{7}{10}, \frac{9}{10} \right\}

n(B) = 31

Answer key and solution verified before publishing.

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Exam: JEE Main 2025
Paper: 22 January, Morning Shift
Subject: Mathematics
Chapter: Sets and Relations
Topic: Sets

The number of non-empty equivalence relations on the set \1,2,3\ is :

Answer: A

Step-by-step solution

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