CBSE Class 10 Mathematics (Standard) Chapter 1 Real Numbers Important Questions 2027
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23 questions from Real Numbers appeared in 6 of the 7 CBSE papers we analysed (2022–2026) — here are the topics that matter, the questions that repeat, and every previous-year question from this chapter.
Key facts
- Number Systems unit — 6 marks in the official syllabus
- Asked in 6 of 7 analysed papers (2022–2026)
- About 3.3 questions per paper (OR alternatives counted separately)
- 4 question patterns repeated across papers
Important topics in Real Numbers
| # | Topic | Asked in | Importance |
|---|---|---|---|
| 1 | HCF and LCM using prime factorisation | Board 2024Board 2025Board 2026Sample 2025Sample 2026 | ★★☆☆☆ |
| 2 | Proving irrationality (root 2, root 3, root 5 type) | Board 2023Board 2024Board 2025Board 2026Sample 2025 | ★★☆☆☆ |
| 3 | HCF-LCM relationship and word problems | Board 2023Board 2024Sample 2025Sample 2026 | ★★☆☆☆ |
| 4 | Fundamental Theorem of Arithmetic / prime factorisation | Board 2024Board 2025Board 2026Sample 2026 | ★★☆☆☆ |
Repeated questions from Real Numbers
1. Compute HCF/LCM of two numbers
3×Board 2025Board 2026Sample 20251/2 marks
The LCM of 960 and 240 is : (A) 960 (B) 240 (C) 60 (D) 15
2. Prove sqrt(p) is irrational
3×Board 2023Board 2026Sample 20253 marks
Prove that sqrt(5) is an irrational number.
3. Units digit of 5n
2×Board 2026Sample 20261 marks
For any natural number n, 5n ends with the digit : (A) 0 (B) 5 (C) 3 (D) 2
4. Minimum equal rooms/groups via HCF
2×Board 2024Sample 20263 marks
In a workshop, the number of teachers of English, Hindi and Science are 36, 60 and 84 respectively. Find the minimum number of rooms required, if in each room the same number of teachers are to be seated and all of them being of the same subject.
Real Numbers questions by paper
| Paper | Questions | Marks |
|---|---|---|
| Board 2022 | 0 | 0 |
| Board 2023 | 3 | 6 |
| Board 2024 | 4 | 8 |
| Board 2025 | 4 | 6 |
| Sample 2025 | 4 | 8 |
| Board 2026 | 4 | 6 |
| Sample 2026 | 4 | 6 |
Counts include both options of internal-choice (OR) questions.
Previous year questions: Real Numbers
Board 2026 4 questions
The LCM of 960 and 240 is : (A) 960 (B) 240 (C) 60 (D) 15
The natural number 1 is : (A) a prime number. (B) a composite number. (C) prime as well as composite. (D) neither prime nor composite.
For any natural number n, 5n ends with the digit : (A) 0 (B) 5 (C) 3 (D) 2
Prove that sqrt(5) is an irrational number.
Sample 2026 4 questions
If a = 22 x 3x, b = 22 x 3 x 5, c = 22 x 3 x 7 and LCM(a, b, c) = 3780, then x is equal to (A) 1 (B) 2 (C) 3 (D) 0
The sum of two numbers is 1215 and their HCF is 81, then the possible pairs of such numbers are (A) 2 (B) 3 (C) 4 (D) 5
Assertion (A): The number 5n cannot end with the digit 0, where n is a natural number. Reason (R): A number ends with 0, if its prime factorization contains both 2 and 5. [Options: (A) Both A and R true, R correct explanation (B) Both true, R not correct explanation (C) A true, R false (D) A false, R true]
In a workshop, the number of teachers of English, Hindi and Science are 36, 60 and 84 respectively. Find the minimum number of rooms required, if in each room the same number of teachers are to be seated and all of them being of the same subject.
Board 2025 4 questions
If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n - 7m is : (A) 0 (B) 28 (C) 98 (D) 198
If (-1)n + (-1)8 = 0, then n is : (A) any positive integer (B) any negative integer (C) any odd number (D) any even number
Which of the following is a rational number between sqrt(3) and sqrt(5) ? (A) 1.4142387954012 .... (B) 2.326 (with bar over 6, recurring) (C) pi (D) 1.857142
Prove that 1/sqrt(5) is an irrational number.
Sample 2025 4 questions
Assertion (A): HCF of any two consecutive even natural numbers is always 2. Reason (R): Even natural numbers are divisible by 2. [Choose: A) Both A and R true, R correct explanation B) Both true, R not correct explanation C) A true, R false D) A false, R true]
Find the H.C.F and L.C.M of 480 and 720 using the Prime factorisation method.
The H.C.F of 85 and 238 is expressible in the form 85m - 238. Find the value of m.
Prove that sqrt(3) is an irrational number.
Board 2024 4 questions
If two positive integers p and q can be expressed as p = 18 a2 b4 and q = 20 a3 b2, where a and b are prime numbers, then LCM(p, q) is: (a) 2 a2 b2 (b) 180 a2 b2 (c) 12 a2 b2 (d) 180 a3 b4
Prove that 5 - 2*sqrt(3) is an irrational number. It is given that sqrt(3) is an irrational number.
Show that the number 5 x 11 x 17 + 3 x 11 is a composite number.
In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48, 80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.
Board 2023 3 questions
Assertion (A): The perimeter of triangle ABC is a rational number. Reason (R): The sum of the squares of two rational numbers is always rational. [Figure: right triangle ABC right-angled at B with AB = 2 cm, BC = 3 cm] Select: (A) Both A and R true, R correct explanation (B) Both true, R not correct explanation (C) A true, R false (D) A false, R true
Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.
Prove that sqrt(5) is an irrational number.
Frequently asked questions
What are the most important topics in CBSE Class 10 Mathematics (Standard) Chapter 1 Real Numbers?
By how often and how recently they were asked: HCF and LCM using prime factorisation; Proving irrationality (root 2, root 3, root 5 type); HCF-LCM relationship and word problems.
How many marks does Real Numbers carry in the CBSE Class 10 Mathematics (Standard) exam?
Real Numbers is part of the Number Systems unit, which carries 6 marks in the official syllabus. In the analysed papers it had about 3.3 questions per paper.
Do questions from Real Numbers repeat in board exams?
Yes — 4 question patterns from this chapter appeared in two or more past board or official sample papers. They are listed on this page with the years they were asked.