NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers (Session 2026-27) – Free PDF, Tips & Complete Guide

Class 10 Maths Chapter 1 Real Numbers for new session
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If you are a Class 10 student stepping into the new academic session 2026-27, the very first chapter you will encounter in your Mathematics textbook is Real Numbers — and it is one of the most scoring chapters in the entire CBSE curriculum. Chapter 1 of Class 10 Maths builds your number sense from the ground up, teaching you how every integer, fraction and irrational number fits into a unified system. For session 2026-27, CBSE has maintained the revised and trimmed syllabus introduced in recent years, keeping only two focused exercises that test your ability to apply the Fundamental Theorem of Arithmetic and prove the irrationality of numbers such as √2, √3, and √5.

Tiwari Academy provides step-by-step, free NCERT Solutions for Class 10 Maths Chapter 1 in both Hindi and English medium, updated specifically for the 2026-27 board exam session. Whether you want to understand the logic behind prime factorisation, master proof-writing for irrational numbers or practise case-study and assertion-reason questions asked in recent CBSE board exams, this is your complete one-stop guide.

What’s New in Class 10 Maths Chapter 1 for Session 2026-27?

One of the most common searches students and parents make at the start of a new academic year is: “What has changed in Class 10 Maths Chapter 1 for the new session?” The answer matters because studying removed content wastes precious preparation time. Here is a clear, updated breakdown for the 2026-27 session based on the latest CBSE Syllabus:
Key Changes in Chapter 1 (Real Numbers) for 2026-27:

  1. Only 2 Exercises Remain — The chapter now contains Exercise 1.1 and Exercise 1.2 only for CBSE students. State Board students may still have the older four-exercise format.
  2. Euclid’s Division Lemma is removed — The old Exercise 1.1, which dealt with Euclid’s Division Algorithm for finding HCF step-by-step, has been dropped from the CBSE syllabus. Students following CBSE need not prepare this for board exams.
  3. Exercise 1.4 (Decimal Expansions) is also deleted — The topics of terminating and non-terminating recurring decimals, which formed Exercise 1.4 in the older format, are no longer assessed in board exams under the current and upcoming syllabus.
  4. Renumbering of Exercises — The old Exercise 1.2 (Fundamental Theorem of Arithmetic + HCF/LCM via prime factorisation) is now Exercise 1.1. The old Exercise 1.3 (Proofs of irrational numbers) is now Exercise 1.2.
  5. Retained Core Topics — The Fundamental Theorem of Arithmetic and proofs of irrationality of numbers like √2, √3, and √5 remain the backbone of this chapter.
  6. Board Exam Weightage — Chapter 1 carries approximately 6 marks in the CBSE Class 10 board exam (typically 1 question of 2 marks and 1 question of 4 marks, or via MCQ/assertion-reason format).
ExerciseOld NameStatus in 2026-27
Old Ex 1.1Euclid’s Division LemmaDELETED
Old Ex 1.2Fundamental Theorem of ArithmeticNew Exercise 1.1
Old Ex 1.3Proofs of IrrationalityNew Exercise 1.2
Old Ex 1.4Decimal ExpansionsDELETED
Important Note for 2026-27:
Students preparing for CBSE Board Exams should focus exclusively on Exercise 1.1 (HCF, LCM, prime factorisation) and Exercise 1.2 (irrational number proofs). State Board students should confirm their syllabus separately.

Chapter at a Glance – Quick Overview Table

ParameterDetails
Chapter NameReal Numbers
Chapter Number1
Class10
SubjectMathematics
CBSE Session2026-27
Number of Exercises (CBSE)2
Total Board Exam MarksApprox. 6 Marks
Estimated Study Periods15 Periods
Medium AvailableHindi & English
Key TheoremsFundamental Theorem of Arithmetic
Core Skills TestedPrime Factorisation, HCF, LCM, Proof Writing
Question Types in Board ExamMCQ, Assertion-Reason, Short Answer, Case Study




Key Topics Covered in Class 10 Maths Chapter 1 (2026-27)

1. The Fundamental Theorem of Arithmetic

This is the pillar of the entire chapter. The theorem states that every positive integer greater than 1 can be expressed as a product of prime numbers in a unique way, regardless of the order in which the primes appear. For example:

  • 140 = 2² × 5 × 7
  • 156 = 2² × 3 × 13
  • 7429 = 17 × 19 × 23

This theorem is used to find the HCF and LCM of numbers. The key relationship to remember is:
HCF × LCM = Product of the two numbers (for any two positive integers)
You can practise more such problems with the Chapter 1 Important Questions section, which includes board-pattern problems with solutions.

2. HCF and LCM Using Prime Factorisation

Using the Fundamental Theorem of Arithmetic, you can systematically calculate HCF and LCM:

  • HCF = Product of the lowest powers of all common prime factors
  • LCM = Product of the highest powers of all prime factors involved

Example: Find HCF and LCM of 72 and 120.

  • 72 = 2³ × 3²
  • 120 = 2³ × 3 × 5
  • HCF = 2³ × 3 = 24
  • LCM = 2³ × 3² × 5 = 360
  • Verification: 24 × 360 = 8640 = 72 × 120

3. Proofs of Irrationality

The second exercise focuses entirely on proof-writing. You must be able to prove that numbers like √2, √3, √5, 3 + 2√5, 6 + √2, and 7√5 are irrational. The standard method used is proof by contradiction. All solved examples and practice problems for this topic are available in Exercise 1.2 Solutions on Tiwari Academy.
Standard Proof Template (Proof by Contradiction):

  • Assume the number is rational, i.e., it can be written as p/q where p and q are co-prime integers and q ≠ 0.
  • Through algebraic manipulation, show that this assumption leads to a known irrational number (like √2 or √5) being rational.
  • Since this contradicts the established fact that √2, √3, √5 are irrational, the assumption is false.
  • Conclude that the original number is irrational.




How to Prepare Class 10 Maths Chapter 1 for Session 2026-27

Preparing class 10 mathematics chapter 1 real numbers, smartly means knowing what to study, in what order and how deeply. Here is a day-by-day preparation plan aligned with the latest CBSE Syllabus for Class 10:

Focus AreaActivityRecommended Resource
1. Introduction & Number SystemRead the theory, understand rational vs irrational numbersNCERT Textbook + Tiwari Academy Notes
2. Prime FactorisationPractice finding prime factors of at least 10 numbersExercise 1.1, Q1–3
3. HCF & LCMSolve problems and verify HCF × LCM = product of numbersNCERT Exercise 1.1, Q2–4
4. Applications of HCF & LCMSolve word problems (circular path, bells, traffic lights)Board Questions from 2019–2026
5. Proofs of Irrationality (Basic)Learn and write proof for √2, √3, √5Exercise 1.2, Q1
6. Proofs of Irrationality (Advanced)Prove combined expressions like 3 + 2√5, 6 + √2, 7√5NCERT Exercise 1.2, Q2–3
7. Board Pattern QuestionsSolve Assertion-Reason, MCQ, Case Study questionsImportant Questions – Chapter 1
8. Full Chapter RevisionAttempt a timed practice testChapter 1 Practice Tests

Additional Preparation Tips:

  • Always keep a formula notebook where you write down theorems, definitions and key relationships like HCF × LCM = product of numbers.
  • Memorise the standard proof structure for irrational numbers — once you know the template, you can apply it to any number.
  • For case-study questions, practise reading the scenario carefully and identifying which concept is being tested before jumping to solve.
  • Solve at least 2 previous year CBSE Sample Papers for Class 10 focusing on Chapter 1 questions from 2019 to 2025.
  • Use Tiwari Academy’s free online MCQ tests for Chapter 1 to benchmark your readiness.

How to Solve Class 10 Maths Chapter 1 Questions Easily

Many students find this chapter conceptually straightforward but lose marks due to errors in proof writing or wrong HCF/LCM calculations. Here are topic-wise shortcuts and strategies:

Solving HCF and LCM Problems Easily

  1. Always do prime factorisation first — Do not try to find HCF or LCM by guessing. Write out the full prime factorisation of each number using a factor tree or division method.
  2. Use a table format — If given three or more numbers, list each prime factor and its powers across the numbers. Then pick lowest powers for HCF and highest powers for LCM.
  3. Use the relationship formula as a shortcut — If HCF is given, you can find LCM directly using: LCM = (Product of two numbers) ÷ HCF. This saves significant time.
  4. For word problems, decide first: Will both events happen together? → Use LCM. or What is the largest common unit? → Use HCF.

Common Word Problem Keywords:

  • When will they meet again? / After how many minutes will they ring together? → LCM
  • What is the largest tile / largest group size? / Maximum number of equal groups → HCF

Solving Irrational Number Proofs Easily

The secret to writing clean, full-mark proofs is following the standard structure every single time. You can refer to fully worked-out proofs in Exercise 1.2 on Tiwari Academy to see exactly how each step should be written for board exams.
Step-by-step method for proving √p is irrational (where p is prime):

  1. Write: “Assume √p is a rational number.”
  2. Write: “Then √p = a/b, where a, b are integers, b ≠ 0, and a, b are co-prime.”
  3. Square both sides: p = a²/b², so a² = pb².
  4. Conclude: a² is divisible by p, therefore a is divisible by p.
  5. Let a = pk for some integer k.
  6. Substitute: p²k² = pb², which gives b² = pk².
  7. Conclude: b² is divisible by p, therefore b is divisible by p.
  8. Contradiction: “Both a and b are divisible by p, which contradicts our assumption that a and b are co-prime.”
  9. Final statement: “Therefore, our assumption is wrong and √p is irrational.”
Pro Tip: In board exams, you will lose marks if you skip any step in the proof. The contradiction step and final conclusion must always be explicitly stated.

Solving Assertion-Reason Questions Easily

For these question types, check both statements separately first:

  • Is the Assertion (A) true or false?
  • Is the Reason (R) true or false?
  • Then check: Does R correctly explain A?

Most common trap: Both A and R are individually true, but R does not explain A → Answer is (B). You can practise a full set of Assertion-Reason questions at the Chapter 1 Important Questions page.

How to Practice Class 10 Maths Chapter 1 Effectively

  1. Exercise-first approach: Complete all NCERT Exercise 1.1 and Exercise 1.2 problems with proper step-by-step solutions. Do not just read the answers — write them out.
  2. Board paper mining: Go through CBSE Sample Papers for Class 10 from 2019 to 2025. Chapter 1 questions appear every year. Categorise them by type: Proof questions (3–5 marks), HCF/LCM numericals (2–3 marks), MCQs (1 mark), and Case Study / Assertion-Reason (2–4 marks).
  3. Timed practice tests: Tiwari Academy offers six practice test papers for Chapter 1. Set a timer and attempt each test in exam conditions. Access all tests through the Chapter 1 Assignments and Practice Tests page.
  4. Flashcard method for theorems: Write the theorem name on one side and the statement + proof steps on the other. Quiz yourself daily until you can write proofs without referring to notes.
  5. Peer discussion: Post your doubts question-by-question on the Tiwari Academy Discussion Forum and get answers from the community and subject experts.
  6. Revision schedule: Revisit Chapter 1 once every 10 days during board exam preparation. Short but regular revision is more effective than long cramming sessions. Once Chapter 1 is solid, move ahead to Chapter 2 Polynomials.

Exercise-wise Breakdown of NCERT Class 10 Maths Chapter 1

Exercise 1.1 – Fundamental Theorem of Arithmetic

This exercise has 7 questions covering prime factorisation, HCF and LCM with verification, three-number HCF/LCM, given-HCF problems, the 6ⁿ digit check, composite number proof and the circular path LCM word problem. Get all solutions at Exercise 1.1 on Tiwari Academy.

Question TypeMarksExample
Prime factorisation1–2 MarksFind prime factors of 3825
HCF and LCM with verification3 MarksVerify HCF × LCM = product for 336 and 54
Given HCF, find LCM2 MarksHCF(306, 657) = 9, find LCM
Ending digit / composite proof2-3 MarksCan 6ⁿ end with digit 0?
Word problem (LCM application)3 MarksCircular path / traffic light problems

Exercise 1.2 – Proofs of Irrationality

This exercise has 3 questions: prove √5 is irrational (Q1), prove 3 + 2√5 is irrational (Q2) and prove 1/√2, 7√5, and 6 + √2 are irrational (Q3). All three questions appear regularly in board exams for 2–3 marks each. Get step-by-step proofs for all three at Exercise 1.2 Solutions.

Most Important Board Exam Questions from Chapter 1 (2019–2026)

Based on the pattern of CBSE Board Exams, the following question types from Chapter 1 appear most frequently:

  1. Prove that √2 is irrational. (CBSE 2023, 2025)
  2. Prove that √3 is irrational. (CBSE 2020)
  3. Prove that 4 – √5 is irrational. (CBSE 2026)
  4. Prove that 3 + 2√5 is irrational, given √5 is irrational. (CBSE 2019)
  5. Show that 6ⁿ cannot end with digit 0 for any natural number n. (CBSE 2023)
  6. Find HCF and LCM of 72 and 120. (CBSE 2023)
  7. Find HCF and LCM of 96 and 120 using prime factorisation. (CBSE 2023)
  8. Traffic lights at three crossings change after 48s, 72s, and 138s — when do they change together? (CBSE 2020)
  9. Given HCF (306, 657) = 9, find LCM (306, 657). (Classic NCERT + frequently asked)
  10. If a = x³y² and b = xy³, find LCM(a, b). (CBSE 2019)

For the full set of board questions with solutions, visit the Chapter 1 Board Questions section on Tiwari Academy.

Common Mistakes to Avoid in Class 10 Maths Chapter 1

  • Incomplete proofs: Skipping the contradiction step or the final conclusion costs 1 full mark. Always write “This contradicts our assumption” and “Therefore, [number] is irrational.”
  • Wrong HCF/LCM identification: Confusing HCF (lowest common prime powers) with LCM (highest common prime powers). Double-check by verifying: HCF × LCM = product of the two numbers.
  • Not writing co-prime assumption: In proofs, always state that a and b are co-prime at the very beginning. Without this, the proof is logically incomplete.
  • Applying HCF where LCM is needed: In word problems, identify whether the question asks for the “largest” or “together.” Largest usually means HCF; together usually means LCM.
  • Attempting deleted topics: For the 2026-27 CBSE board exam, Euclid’s Division Algorithm and Decimal Expansions are not in the syllabus. You can confirm the current syllabus at CBSE Official website or the Tiwari Academy Syllabus page.
  • Not verifying HCF × LCM: In Q2 of Exercise 1.1, the question specifically asks for verification. Many students skip this step and lose marks.



Frequently Asked Questions (FAQ) – Class 10 Maths Chapter 1 Real Numbers

How many exercises are there in Class 10 Maths Chapter 1 for the session 2026-27?

For the session 2026-27, Class 10 Maths Chapter 1 Real Numbers contains only 2 exercises for CBSE students – Exercise 1.1 and Exercise 1.2. The older exercises 1.1 (Euclid’s Division Lemma) and 1.4 (Decimal Expansions) have been removed from the CBSE syllabus. State Board students may still follow the older four-exercise format available on Tiwari Academy.

What is the Fundamental Theorem of Arithmetic in Class 10?

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed as a product of prime numbers in exactly one way, up to the order of the factors. For example, 12 = 2² × 3. This unique factorisation is the foundation for finding HCF and LCM in Chapter 1.

How to prove that √2 is irrational in Class 10 Maths?

To prove √2 is irrational, assume √2 = p/q where p and q are co-prime and q ≠ 0. Squaring gives p² = 2q², so p is even. Writing p = 2k gives q² = 2k², so q is also even. This contradicts that p and q are co-prime. Therefore, √2 is irrational. The full proof with each step is available at Exercise 1.2 Solutions.

Is Euclid’s Division Lemma in Class 10 Maths syllabus for 2026-27?

No. Euclid’s Division Lemma has been removed from the CBSE Class 10 Maths syllabus for 2026-27. It will not appear in the CBSE board exam. Verify this on the updated CBSE Syllabus page. However, students of certain State Boards may still need to study it — the solutions for the older four-exercise format are available separately on Tiwari Academy.

How many marks does Chapter 1 Real Numbers carry in the Class 10 board exam?

Chapter 1 carries approximately 6 marks in the CBSE Class 10 Mathematics board exam. For complete exam-oriented preparation, practise with CBSE Sample Papers for Class 10 to understand the exact question distribution.

What is the difference between HCF and LCM in Class 10?

HCF is the greatest number that divides two or more numbers exactly, found using the lowest powers of common prime factors. LCM is the smallest number divisible by all given numbers, found using the highest powers of all prime factors. The key relationship: HCF × LCM = Product of the two numbers.

How to solve irrational number proof questions easily in board exams?

Use proof by contradiction: assume the number is rational, express it as p/q in lowest terms, manipulate to show a known irrational would have to be rational, state the contradiction and conclude. Practise this with all three questions in Exercise 1.2 before the exam.

Which questions from Chapter 1 are most important for the 2026-27 board exam?

The most important questions are: (1) Prove √2, √3, or √5 is irrational, (2) Show 6ⁿ cannot end with digit 0, (3) Find HCF and LCM using prime factorisation with verification, (4) Given HCF, find LCM and (5) Real-life LCM word problems. For a complete curated set, visit Chapter 1 Important Questions.

Can 5ⁿ end with the digit 0 for any natural number n?

No. For a number to end with 0, it must be divisible by 10 = 2 × 5. The prime factorisation of 5ⁿ contains only 5 as a factor, not 2. Since 5ⁿ is never divisible by 2, it can never end with digit 0.

Where can I get free NCERT Solutions for Class 10 Maths Chapter 1?

You can get completely free, step-by-step NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers at Tiwari Academy. Solutions are available for both CBSE and State Board formats, in Hindi and English medium, along with practice tests, board exam questions, case studies, MCQ tests and downloadable assignments — all at no cost.

Why Choose Tiwari Academy for Class 10 Maths Chapter 1?

  • 100% Free — No subscription, no hidden charges, no registration required.
  • Updated for 2026-27 — Content revised every session to reflect CBSE syllabus changes.
  • Bilingual — Solutions in both Hindi Medium and English Medium.
  • Board Exam Focused — Includes previous year questions (2019–2026), case study questions, assertion-reason questions and MCQs.
  • Discussion Forum — Post doubts at the Tiwari Academy Discussion Forum for expert help.
  • Full NCERT Coverage — Covers all Class 10 subjects — Maths, Science, English, Social Science, Hindi and Sanskrit.
  • App Available — Download the Tiwari Academy app on Android and iOS for on-the-go studying.

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CONCLUSION

Class 10 Maths Chapter 1 Real Numbers for session 2026-27 is a compact, focused, and highly scorable chapter. With only two exercises and a well-defined set of theorems and proof techniques, students who prepare it systematically can score full marks in board exams. Start with Tiwari Academy’s free NCERT Solutions for Chapter 1, practise the six chapter test papers, solve previous year board questions, and once you’ve mastered this chapter, continue your preparation with Chapter 2 Polynomials.