YOUR STUDY SPACE · CBSE

Real Numbers — Class 10 Mathematics

Mathematics · Chapter 1

Download revision card →

CBSE Class 10 community

View all

All available subjects

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Mathematics
Science
+ 10 more

~7 pages per kitPage 1 free

Get all 12 kits · ₹299

Mathematics (Standard and Basic)

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

70+ formulas and theorems
Board-paper chapter counts
24 traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Social Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

History
Economics
+ 2 more

~7 pages per kitPage 1 free

Get all 4 kits · ₹69

English

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

First Flight
Footprints without Feet

~7 pages per kitPage 1 free

Get both kits · ₹69

Science

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

TrendingStart now for December pre-boards. Reuse it for boards.

Equations and formulas
Book tables as charts
Traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi A · Kshitij 2 + Kritika 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

15 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Computer Applications 2027

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

HTML & CSS examples
Protocol tables
Cyber ethics

Only 7 pagesPage 1 free

Get the kit · ₹69

Hindi B · Sparsh 2 + Sanchayan 2

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

17 पाठ
व्याकरण
लेखन प्रारूप

Only 7 pagesPage 1 free

Get the kit · ₹69

Sanskrit · Shemushi 2 + Vyakaranavithi + Abhyaswaan Bhav

~50days leftPre-boards · school dates varyDec 2026 · school-specific · tentative

Start now for December pre-boards. Reuse it for boards.

9 पाठ
सन्धि-समास
प्रत्यय

Only 7 pagesPage 1 free

Get the kit · ₹69

Browse all Class 10 kits →

Chapter summary

This chapter builds the foundation of number theory for Class 10. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of primes, which lets us find the HCF and LCM of numbers through prime factorisation. A key relationship is that for any two positive integers, equals the product of the numbers. The chapter also revisits irrational numbers, proving results such as the irrationality of using the method of contradiction. Together these ideas sharpen the understanding of how integers and primes underpin the rest of arithmetic and algebra.

Fundamental Theorem of ArithmeticHCF and LCM by prime factorisationIrrational numbersProofs of irrationality

Key ideas & terms

Fundamental Theorem of Arithmetic: Every composite number has a unique prime factorisation.
HCF: The highest common factor (greatest common divisor) of given numbers.
LCM: The lowest common multiple of given numbers.
Irrational number: A number that cannot be written as a ratio of two integers.

Extra questions & answers

State the Fundamental Theorem of Arithmetic.

Every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.

Prove that is irrational.

Assume in lowest terms; squaring gives , so is even, say ; then , so is also even — contradicting that is in lowest terms. Hence is irrational.

How do you find the HCF and LCM of two numbers by prime factorisation?

Express each number as a product of primes; the HCF is the product of the smallest powers of common primes, and the LCM is the product of the greatest powers of all primes involved.

What is the relationship between the HCF, LCM and product of two numbers?

For any two positive integers and , .

Find the HCF and LCM of 6 and 20 by prime factorisation.

and , so and (and ).

Define a rational and an irrational number.

A rational number can be written as where and are integers and ; an irrational number cannot be written in this form, e.g. and .

Prove that is irrational (outline).

Assume in lowest terms; then , so 3 divides , say ; then , so 3 divides too — contradicting that is in lowest terms. Hence is irrational.

When does a rational number have a terminating decimal expansion?

When, in its lowest form , the denominator has only 2 and/or 5 as its prime factors .

Why can the number never end with the digit 0 for any natural number ?

A number ending in 0 must have both 2 and 5 as factors; has no factor of 5, so it can never end in 0.

State the Fundamental Theorem of Arithmetic.

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the factors.

EXERCISE 1.1

1. Express each number as a product of its prime factors:
(i) (ii) (iii) (iv) (v)

(i) .

(ii) .

(iii) .

(iv) .

(v) .

2. Find the LCM and HCF of the following pairs of integers and verify that product of the two numbers.
(i) and (ii) and (iii) and

(i) and .
Therefore,

Verification:

(ii) and .
Therefore,

Verification:

(iii) and .
Therefore,

Verification:

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) and (ii) and (iii) and

(i) , , and .
The common prime factor with the smallest power is . Hence,

Taking the greatest power of every prime factor,

(ii) , , and are distinct prime numbers. Hence,

(iii) , , and . They have no common prime factor. Hence,

4. Given that , find .

For two positive integers,

Therefore,

Thus, .

5. Check whether can end with the digit for any natural number .

A number ending with the digit must be divisible by , so its prime factorisation must contain both and . However,

contains only the prime factors and , and it has no factor . Therefore, cannot end with the digit for any natural number .

6. Explain why and are composite numbers.

For the first number,

It is a product of two integers greater than , so it is composite.

For the second number,

It is also a product of two integers greater than , so it is composite.

7. There is a circular path around a sports field. Sonia takes minutes to drive one round of the field, while Ravi takes minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

They will meet again at the starting point after a time equal to the LCM of minutes and minutes.

Therefore,

Hence, they will meet again at the starting point after minutes.

EXERCISE 1.2

1. Prove that is irrational.

Assume, contrary to what is required, that is rational. Then it can be written in lowest terms as

where and are coprime integers and .

Squaring both sides gives

so

Thus, . Since is prime, . Therefore, let for some integer .

Substituting this value gives

Hence, , and therefore .

Thus, both and are divisible by , contradicting the assumption that they are coprime. Therefore, is irrational.

2. Prove that is irrational.

Assume, contrary to what is required, that is rational. Let

where is rational. Rearranging,

Since , , and are rational, is rational. This would mean that is rational, which contradicts the fact that is irrational. Therefore, is irrational.

3. Prove that the following are irrationals :
(i) (ii) (iii)

(i) Assume that is rational and equal to a non-zero rational number . Then

which gives

Since is a non-zero rational number, is rational. This contradicts the fact that is irrational. Hence, is irrational.

(ii) Assume that is rational and equal to . Then

so

Since and are rational, is rational. This contradicts the fact that is irrational. Hence, is irrational.

(iii) Assume that is rational and equal to . Then

so

Since and are rational, is rational. This contradicts the fact that is irrational. Hence, is irrational.

Real Numbers — Class 10 Mathematics

CBSE Class 10 community

Chapter summary

Key ideas & terms

Extra questions & answers

State the Fundamental Theorem of Arithmetic.

Prove that is irrational.

How do you find the HCF and LCM of two numbers by prime factorisation?

What is the relationship between the HCF, LCM and product of two numbers?

Find the HCF and LCM of 6 and 20 by prime factorisation.

Define a rational and an irrational number.

Prove that is irrational (outline).

When does a rational number have a terminating decimal expansion?

Why can the number never end with the digit 0 for any natural number ?

State the Fundamental Theorem of Arithmetic.

EXERCISE 1.1

1. Express each number as a product of its prime factors:(i) (ii) (iii) (iv) (v)

2. Find the LCM and HCF of the following pairs of integers and verify that product of the two numbers.(i) and (ii) and (iii) and

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.(i) and (ii) and (iii) and

4. Given that , find .

5. Check whether can end with the digit for any natural number .

6. Explain why and are composite numbers.

EXERCISE 1.2

1. Prove that is irrational.

2. Prove that is irrational.

3. Prove that the following are irrationals :(i) (ii) (iii)

1. Express each number as a product of its prime factors:
(i) (ii) (iii) (iv) (v)

2. Find the LCM and HCF of the following pairs of integers and verify that product of the two numbers.
(i) and (ii) and (iii) and

3. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i) and (ii) and (iii) and

3. Prove that the following are irrationals :
(i) (ii) (iii)