CBSE Class 10 Maths — Real Numbers

Chapter Overview & Weightage

Real Numbers is the first chapter of CBSE Class 10 Maths, and it sets the foundation for algebra and number theory throughout the year. The chapter covers the Euclid’s Division Lemma, the Fundamental Theorem of Arithmetic, and proofs of irrationality.

In CBSE Class 10 board exams, Real Numbers carries approximately 6–8 marks. Typical distribution: 1 MCQ (1 mark) + 1 short answer on HCF/LCM (2 marks) + 1 long answer on irrational number proof (3 marks). This is a scoring chapter — most students who prepare it well score full marks.

Exam Year	Marks	Topics Covered
2024	7	HCF by Euclid, LCM, irrational proof
2023	6	Prime factorisation, LCM, √2 irrational
2022	8	Euclid’s algorithm, HCF×LCM property, decimal expansions
2021	6	Fundamental theorem, irrational number

Key Concepts You Must Know

1. Euclid’s Division Lemma: For any two positive integers $a$ and $b$ , there exist unique integers $q$ and $r$ such that:

a = bq + r, \quad 0 \leq r < b

This lemma is the basis for the Euclid’s Division Algorithm for finding HCF.

2. Euclid’s Division Algorithm for HCF: Apply the lemma repeatedly until the remainder becomes 0. The last non-zero remainder is the HCF.

3. Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes in a unique way (except for the order of the prime factors).

4. HCF and LCM relationship: $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$ (for two numbers)

5. Decimal Expansions:

If $p/q$ (lowest terms) has $q = 2^m \times 5^n$ , then it is a terminating decimal
Otherwise, it is a non-terminating repeating decimal

6. Irrational Numbers: Cannot be expressed as $p/q$ where $p, q$ are integers and $q \neq 0$ .

Important Formulas

a = bq + r \quad (0 \leq r < b)

\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b

Note: This property holds for exactly two numbers. Do NOT apply it to three numbers.

$\frac{p}{q}$ (in lowest terms) terminates iff $q = 2^m \times 5^n$ for non-negative integers $m, n$ .

Solved Previous Year Questions

PYQ 1 — Find HCF of 420 and 130 using Euclid’s Algorithm

Solution:

420 = 130 \times 3 + 30

130 = 30 \times 4 + 10

30 = 10 \times 3 + 0

Last non-zero remainder = 10. HCF(420, 130) = 10.

Using this: LCM = $\frac{420 \times 130}{10} = 5460$ .

PYQ 2 — Prove $\sqrt{2}$ is irrational (CBSE 2023)

Solution (Proof by contradiction):

Assume $\sqrt{2}$ is rational. Then $\sqrt{2} = \frac{p}{q}$ where $p, q$ are integers, $q \neq 0$ , and $\gcd(p, q) = 1$ .

Squaring: $2 = \frac{p^2}{q^2} \implies p^2 = 2q^2$

So $p^2$ is even $\implies$ $p$ is even. Let $p = 2k$ .

Then $4k^2 = 2q^2 \implies q^2 = 2k^2 \implies q$ is even.

But if both $p$ and $q$ are even, they have a common factor 2, contradicting $\gcd(p, q) = 1$ .

This contradiction proves $\sqrt{2}$ is irrational. $\square$

PYQ 3 — Decimal Expansion

Determine whether $\frac{13}{125}$ has a terminating or non-terminating decimal expansion.

Solution: $125 = 5^3$ . Since $q = 5^3 = 2^0 \times 5^3$ (of the form $2^m \times 5^n$ with $m = 0, n = 3$ ), the decimal expansion terminates.

$\frac{13}{125} = \frac{13 \times 2^3}{5^3 \times 2^3} = \frac{104}{1000} = 0.104$

Difficulty Distribution

Difficulty	Example Topics	Approximate %
Easy	HCF/LCM by prime factorisation, terminating decimals	50%
Medium	Euclid’s algorithm, HCF × LCM property	30%
Hard	Irrational number proofs, three-number LCM problems	20%

Expert Strategy

Start with prime factorisation. For most HCF/LCM problems, prime factorisation is faster and more reliable than Euclid’s algorithm for numbers up to 4 digits.

Irrational proofs follow a template. The “assume rational, derive contradiction via common factor” approach works for $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ , and $3 + 2\sqrt{5}$ . Practise the template until it becomes automatic.

For proofs of the form “prove $a + b\sqrt{p}$ is irrational,” the approach is: assume it’s rational → rearrange to isolate $\sqrt{p}$ → note that LHS is rational and RHS ( $\sqrt{p}$ ) is irrational → contradiction. This covers all Class 10 irrational proof variants.

For HCF × LCM = product of two numbers: This is only for two numbers. For three numbers, use prime factorisation separately.

Decimal expansion: Simplify the fraction first (find lowest terms), then check the denominator. If even a single prime factor other than 2 or 5 appears, it’s non-terminating repeating.

Common Traps

Trap 1: Applying HCF × LCM = a × b to three numbers. The formula only works for two numbers. For three numbers $a$ , $b$ , $c$ : use prime factorisation for both HCF and LCM. Some students write HCF(a,b,c) × LCM(a,b,c) = a × b × c — this is wrong.

Trap 2: Not simplifying the fraction before testing terminating decimal. $\frac{12}{15}$ seems non-terminating (15 = 3 × 5), but $\frac{12}{15} = \frac{4}{5}$ , and $5 = 5^1$ , which IS terminating. Always reduce to lowest terms first.

Trap 3: Stopping Euclid’s algorithm one step early. Continue until the remainder is exactly 0. The last non-zero remainder is the HCF. Students sometimes stop when they see a “small” remainder and report that as HCF.

Trap 4: In irrational proofs, not showing that “p² even implies p even.” CBSE marks this step. The logic: if $p$ is odd, $p^2$ is odd; contrapositive: if $p^2$ is even, $p$ is even. This step must be explicitly stated, not assumed.