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NCERT Class 10 Maths Chapter 1, Real Numbers, explained simply

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Real Numbers is the first chapter of the NCERT Class 10 Mathematics textbook. In the current book (Reprint 2026-27) it teaches two ideas: every composite number breaks into primes in exactly one way, apart from the order of the primes, and that fact can prove that numbers like √2 are irrational. This guide explains both in plain words, works through the question types the CBSE sample paper uses, and ends with a self-check and a way to test an AI's answer.

Keep the textbook open next to it. Our explanations and examples are our own; the textbook is the authority.

How much it counts in the board exam

CBSE's Class 10 Mathematics curriculum for 2026-27 puts this chapter in Unit I, Number Systems, worth 6 of the 80 theory marks. For this unit the curriculum lists two things: the Fundamental Theorem of Arithmetic, and proofs that √2, √3, √5 and numbers like 3 + 2√5 are irrational.

The 2026-27 sample paper (Mathematics Standard) asks the chapter in three places:

QuestionMarksWhat it asks
Q11The greatest number that divides two numbers and leaves given remainders (an HCF question)
Q212The longest ruler that measures a hall's length, breadth and height exactly (HCF), or how many more times three bells ring together before closing time (LCM)
Q263Given that √3 is irrational, prove that 2 + 3√3 is irrational

That is 6 marks, the whole unit. Every one of these question types is below.

1. The Fundamental Theorem of Arithmetic (section 1.2)

The theorem (Theorem 1.1 in the book) says: every composite number is a product of primes, and there is only one such product, apart from the order of the primes.

Two halves matter. Every composite number has a prime factorisation, and there is only one. You can find it with a factor tree, splitting until every branch ends in a prime:

2520 = 2 × 1260 = 2 × 2 × 630 = 2 × 2 × 2 × 315 = 2 × 2 × 2 × 3 × 105 = 2 × 2 × 2 × 3 × 3 × 35 = 2 × 2 × 2 × 3 × 3 × 5 × 7

In powers of primes: 2520 = 2³ × 3² × 5 × 7. Whichever way you split 2520 first, you end with these same primes.

The book adds a little history: the result probably appears first in Euclid's Elements, and Carl Friedrich Gauss was the first to prove it correctly.

Why "unique" is useful: can a power end in 0?

A number ends in the digit 0 only if it is divisible by 10, so its prime factorisation must contain both 2 and 5. The book uses this to show that 4ⁿ never ends in 0: 4ⁿ = 2²ⁿ, and uniqueness means no 5 can ever appear. Exercise 1.1 asks the same about 6ⁿ: 6ⁿ = 2ⁿ × 3ⁿ, still no 5, so 6ⁿ never ends in 0 either.

2. HCF and LCM by prime factorisation

Write both numbers as products of prime powers, then:

Worked example: 84 and 120.

The HCF × LCM rule, and its limit

For two positive integers, the book shows that their HCF times their LCM equals the product of the two numbers. Check it on our example: 12 × 840 = 10,080 and 84 × 120 = 10,080. So if you know the HCF, you get the LCM by one division: LCM = (a × b) ÷ HCF.

The rule does not work for three numbers. Take 4, 6 and 8: HCF = 2 and LCM = 24, so HCF × LCM = 48, but 4 × 6 × 8 = 192. The book makes the same point with its own three numbers.

3. The word problems: which one is HCF, which one is LCM?

The board paper rarely says "find the HCF". It hides it in a story. One test helps: is the answer no bigger than the smallest number, or at least as big as the largest?

Worked: the largest square tile. A floor is 4.2 m by 3.6 m. What is the side of the largest square tile that covers it exactly, with no cutting? Work in centimetres: 420 = 2² × 3 × 5 × 7 and 360 = 2³ × 3² × 5, so HCF = 2² × 3 × 5 = 60 cm.

Worked: buses leaving together. Two buses leave a depot every 12 and every 20 minutes. They leave together at 8:00 a.m. LCM(12, 20) = 60, so they next leave together 60 minutes later, at 9:00 a.m.

Worked: the "remainder" question. What is the greatest number that divides 70 and 125, leaving remainders 5 and 8? Take the remainders away first: 70 − 5 = 65 and 125 − 8 = 117. Now the number must divide 65 and 117 exactly. 65 = 5 × 13 and 117 = 3² × 13, so the answer is 13. Check: 70 = 13 × 5 + 5, and 125 = 13 × 9 + 8.

4. Proving a number is irrational (section 1.3)

A number is irrational if it is not a fraction of two integers: there is no way to write it as p/q, where p and q are integers and q ≠ 0. In Class 9 you met √2 and √3 on the number line; this chapter proves they are irrational.

The proof needs one fact first (Theorem 1.2 in the book): if a prime p divides a², then p divides a. It follows from the unique factorisation: the primes in a² are exactly the primes in a, each used twice.

The proof that √2 is irrational, step by step

The method is proof by contradiction: assume the opposite, and show it leads to something impossible.

  1. Suppose √2 is rational. Then √2 = a/b, where a and b are integers, b ≠ 0, and a and b have no common factor other than 1 (if they had one, we would cancel it first).
  2. Squaring, 2 = a²/b², so a² = 2b². So 2 divides a², and by Theorem 1.2, 2 divides a. Write a = 2c.
  3. Then (2c)² = 2b², so 4c² = 2b², which gives b² = 2c². So 2 divides b², and so 2 divides b.
  4. Now 2 divides both a and b. That contradicts step 1, where a and b had no common factor.
  5. So the assumption was wrong, and √2 is irrational.

The same five steps prove √3 and √5 are irrational: replace 2 with 3 or 5.

Rational plus or times irrational (the 3-mark question)

The sample paper's Q26 gives you that √3 is irrational and asks you to prove 2 + 3√3 is irrational. You do not repeat the long proof. You get the √ alone and show it would come out rational. Here is the same method on a different number, 4 − 5√2:

  1. Suppose 4 − 5√2 is rational: 4 − 5√2 = a/b, with a and b integers and b ≠ 0.
  2. Rearrange: 5√2 = 4 − a/b, so √2 = (4b − a)/(5b).
  3. a and b are integers, so (4b − a)/(5b) is rational. Then √2 would be rational.
  4. That contradicts the fact that √2 is irrational. So 4 − 5√2 is irrational.

How the marks are given. In CBSE's marking scheme for Q26, the 3 marks are split one per step:

Write all three steps, even when the last one feels obvious.

5. Three things that trip students up

  1. Using HCF × LCM = product for three numbers. It holds only for two.
  2. Forgetting to subtract the remainders before finding the HCF in "leaves remainders" questions. Always check your answer by dividing back, and check it is bigger than both remainders.
  3. Skipping the words in a proof. "a and b have no common factor other than 1", "b ≠ 0" and "this contradicts..." are what the marks are for. A proof that only shows the algebra loses marks.

6. Self-check: 6 questions with answers

Try them on paper first. Answers are below.

  1. Write 1260 as a product of prime powers.
  2. Find the HCF and LCM of 36 and 48, and check the HCF × LCM rule.
  3. Two numbers have HCF 8 and product 960. What is their LCM?
  4. Is there a natural number n for which 15ⁿ ends in 0? Why?
  5. Two bells ring every 9 and every 15 minutes. They ring together at noon. When do they next ring together?
  6. Given that √5 is irrational, prove that 2 + √5 is irrational.

Answers: 1) 1260 = 2² × 3² × 5 × 7. 2) 36 = 2² × 3² and 48 = 2⁴ × 3, so HCF = 2² × 3 = 12 and LCM = 2⁴ × 3² = 144. Check: 12 × 144 = 1728 = 36 × 48. 3) LCM = 960 ÷ 8 = 120. 4) No. 15ⁿ = 3ⁿ × 5ⁿ has no factor 2, so it is never divisible by 10. (It always ends in 5.) 5) LCM(9, 15) = 45, so at 12:45 p.m. 6) Suppose 2 + √5 = a/b with integers a, b and b ≠ 0. Then √5 = a/b − 2 = (a − 2b)/b, which is rational. This contradicts the fact that √5 is irrational, so 2 + √5 is irrational.

7. How to check an AI's answer for this chapter

Chatbots are quick at this chapter and still make mistakes. These checks take a minute:

More on using AI well: how to use ChatGPT for homework without cheating yourself.

Practise this chapter

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Sources

All checked on 20 October 2026.

The theorems, the section numbers and the question types are from the documents above. All explanations, worked examples and self-check questions on this page are our own; every number was checked by calculation.

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