CBSE Class 10 Maths Real Numbers 2026-27: Euclid Division and Theorems

Q: What is Euclid's Division Lemma in simple words?

When one positive integer is divided by another positive integer, the remainder is always less than the divisor and the quotient is always unique.

Q: Is Euclid's Division Lemma still part of the CBSE Class 10 syllabus for 2026-27?

There are some variations in information in different sources: some mention only the Fundamental Theorem of Arithmetic now, others mention Euclid's Division Lemma. Always check the precise list of exercises with your school or the most recent CBSE syllabus PDF.

Q: What is the Fundamental Theorem of Arithmetic used for?

It is used to represent any number in an unique factorization of prime numbers so that HCF and LCM of any two numbers can be found quickly.

Q: How many marks does the Real Numbers chapter carry in board exams?

Typically contains 4 – 6 marks depending on the paper pattern for that year.

Q: How do I prove that √2 is irrational?

If you assume that √2 is a fraction p/q, then by squaring and simplifying the expression, you should be able to show that p and q are both even, which would contradict your original assumption — thus proving that √2 is not a fraction.

Q: Is HCF × LCM = Product of two numbers always true?

Yes, but just when you have only two numbers to work with, not three numbers.

CBSE Class 10 Real Numbers is the first chapter of the CBSE Class 10 Maths syllabus 2026-27, which is based on two important concepts: Euclid's Division Lemma (the rule used to find HCF) and Fundamental Theorem of Arithmetic (the rule used to find the unique prime factorisation of a number) and is typically worth 4-6 marks in the board exam. This chapter is small but the concepts learnt in this chapter form the basis for the subsequent chapters such as Polynomials and Pair of Linear Equations. This guide contains easy tables, smart tips and simple explanations to create effective Class 10 Maths notes for this chapter.

What Does the Real Numbers Chapter Teach You?

Real numbers are any number that you can imagine on a number line, including whole numbers, fractions, negative numbers, and numbers such as √2, which are not finite. This chapter does not present any new types of numbers in Class 10. Rather, it helps you learn about how these numbers play together, particularly:

Use division to find HCF (Highest Common Factor) of two numbers.
Prime Factorization – the ways each number can be expressed as a product of prime numbers
How to show that some numbers are not rational, such as √2 and √3

This chapter is a short but important chapter as the concepts of HCF and LCM are used again in Polynomials, Quadratic Equations and in later classes.

CBSE Class 10 Real Numbers Syllabus 2026-27 — Marks and Structure

Having a structure in mind when you are planning your revision will help you to do the job better.

Topic	What It Covers
Euclid's Division Lemma	Rule for dividing one number by another to get quotient and remainder
Euclid's Division Algorithm	Repeated use of the lemma to find HCF
Fundamental Theorem of Arithmetic	Every number is a unique product of primes
HCF and LCM using Prime Factorisation	Finding HCF/LCM without long division
Irrational Numbers	Proving numbers like √2 are irrational

The Real Numbers chapter is typically 4-6 marks in the Class 10 Maths board exam and the weightage may slightly change every year depending on the pattern of the paper. It is a good scoring chapter due to the limited concepts and repetitive questions that are asked in a predictable pattern each year.

Note: A few recent updates on the CBSE rationalisation of the chapter indicate that the chapter now has only two exercises for CBSE students, and the exercise on Euclid's Division Algorithm has been removed from the board assessment of some versions of the syllabus, while others still include Euclid's Division Algorithm, HCF, LCM and the Fundamental Theorem of Arithmetic in the chapter without major deletions at the concept level. Different schools may use slightly different exercise numbers so it is advisable that you always cross check the exact list of exercises with your school or the official CBSE syllabus PDF before deciding on your revision plan.

Euclid's Division Lemma — The Starting Point

The Division Lemma of Euclid is a simple yet powerful rule. It says:

If a and b are positive integers, then there exist unique integers q (quotient) and r (remainder) such that: a = bq + r, with 0 ≤ r < b

Simply put, when you divide one number by another you always have a remainder and the remainder is always less than the number you are dividing by. The Lemma is the proved mathematical statement and the Algorithm is the working procedure which repeatedly applies the Lemma to obtain the HCF of two numbers.

Example: If a = 17 and b = 5, then 17 = 5 × 3 + 2, so q = 3 and r = 2.

Euclid's Division Algorithm — Finding HCF Step by Step

The algorithm just keeps on repeating the lemma over and over until the remainder is zero. The property used to prove that this method finds the HCF of two numbers is that HCF of two numbers is HCF of the smaller number and the remainder; and the remainder in each step is always less than the divisor, so in each step the problem is reduced to a smaller problem till the remainder is 0.

Example: Find HCF of 210 and 55

Step	Division	Quotient (q)	Remainder (r)
1	210 = 55 × q + r	3	45
2	55 = 45 × q + r	1	10
3	45 = 10 × q + r	4	5
4	10 = 5 × q + r	2	0

The HCF of 210 and 55 is 5 (the last non-zero remainder) as at step 4 the remainder is 0.

The Fundamental Theorem of Arithmetic

This theorem says that each positive integer (n>1) is uniquely expressible as a product of prime numbers with the order of the factors not being considered. For instance, 12 can be written as 2 × 2 × 3 and no matter how you factor 12, you will always have the same prime numbers.

The concept is very useful as it helps to easily find HCF and LCM by prime factorisation without performing long division.

Example:

Number	Prime Factorisation
12	2 × 2 × 3
18	2 × 3 × 3
HCF (12, 18)	2 × 3 = 6
LCM (12, 18)	2 × 2 × 3 × 3 = 36

Important Theorems and Proofs You Must Know

In board exams, the examiner will require evidence of some statements. These are the proofs that you need to know from this chapter:

Theorem/Proof	Key Idea
√2 is irrational	Proof by contradiction using co-prime numbers
√3 is irrational	Same method as √2, using a different prime
HCF × LCM = Product of two numbers	Useful shortcut for finding LCM once HCF is known
Every composite number has a unique prime factorisation	Based on the Fundamental Theorem of Arithmetic

In order to show that √2 is irrational, you make the assumption that √2 = p/q where p and q are co-prime and q ≠ 0. If p is squared and equated to 2q squared, then p is even and q is even, but this is a contradiction to the assumption that p and q are co-prime, and thus it is impossible to write √2 as a fraction.

How to Prepare Strong Class 10 Maths Notes for This Chapter

Real Numbers' good notes should be formulaic, short and to the point as the chapter is short. Try this method:

One-page summary – Write the Euclid's Division Lemma formula, the Fundamental Theorem of Arithmetic statement and the HCF × LCM formula on a single page.
Step-by-step example bank – One worked example for each of HCF using division, HCF/LCM using prime factorisation, and irrationality proof.
Common mistake tracker – Many students do not remember the condition 0 ≤ r < b or do not recall HCF and LCM formulas – remember these separately.
Practice from a trusted question bank – Students in Class 10 benefit from using a question bank that has questions similar to the one they will be asked in the board exams, and they also benefit from the step marking provided in these resources, which helps them understand exactly how the irrationality proof and HCF steps will be written in the board exam.

Summary

In Real Numbers of CBSE Class 10 Mathematics, you will learn about Euclid's Division Lemma, Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic, which will help you to understand how numbers work. These concepts enable you to easily work out the HCF and LCM, and demonstrate that numbers such as √2 and √3 are irrational. The chapter carries only 4 to 6 marks but is the base of subsequent chapters of Maths; hence, it is important to have a good understanding of it. For this chapter, you can score all the marks in the board exam 2027 by making short notes on the formula of Class 10 Maths, practicing regularly HCF and Proof of Irrationality and practicing the board pattern questions in Oswaal Books Sample papers.