Class 10 - Real Numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/category/class-10-real-numbers/ Mathematics is like music - Enjoy it!!! Sat, 03 May 2025 12:01:05 +0000 en-US hourly 1 https://wordpress.org/?v=7.1 https://mathemafia.com/wp-content/uploads/2020/08/cropped-M__Logo__SM-1-32x32.png Class 10 - Real Numbers Archives – Mathemafia http://sh016.global.temp.domains/~mathemaf/category/class-10-real-numbers/ 32 32 Exercise 1.2 (NCERT) – Real Numbers, Class 10 – All Solutions https://mathemafia.com/exercise-1-2-ncert-real-numbers-class-10-all-solutions/ Wed, 30 Apr 2025 07:32:47 +0000 https://mathemafia.com/?p=13359 Complete solution of Exercise 1.2, Chapter 1, Real Numbers of NCERT book of Mathematics for Class 10. The exercise is about proving that the square roots of prime numbers are irrational and some more questions around this concept.

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Exercise 1.2 (NCERT) - Real Numbers, Class 10

Chapter 1 Exercise 1.2 Class 10

Below are the Quick links for all questions of Exercise 1.2, Real Numbers, Class 10.

Click a link to view the solution of corresponding question.

Below are the solutions of every question of exercise 1.2 of chapter 1, Real Numbers, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.  

 

Question 1

Prove that √5 is irrational.

We prove such questions by contradiction.

Let us assume that \( \begin{aligned} \sqrt{5}\end{aligned}\) is a rational number.

This is a contradiction, becasue

Hence our assumption is wrong.

This implies √5 is irrational.

Question 2

Prove that 3 +2 √5 is irrational.

Question 3

Prove that the following are irrationals :

(i) 1/√2

(ii) 7 √5

(iii) 6 + √2

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]]> Exercise 1.1 (NCERT) – Real Numbers, Class 10 – All Solutions https://mathemafia.com/exercise-1-1-ncert-real-numbers-class-10-all-solutions/ Sun, 02 Mar 2025 15:11:59 +0000 https://mathemafia.com/?p=13289 Complete solution of Exercise 1.1, Chapter 1, Real Numbers of NCERT book of Mathematics for Class 10. The chapter is primarily about prime factorisation, LCM and HCF of the numbers. Easily readable sections and embedded YouTube tutorials will help students understand the solutions quickly.

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Exercise 1.1 (NCERT) - Real Numbers, Class 10

Chapter 1 Exercise 1.1 Class 10

Below are the Quick links for all questions of Exercise 1.1, Real Numbers, Class 10.

Click a link to view the solution of corresponding question.

Below are the solutions of every question of exercise 1.1 of chapter 1, Real Numbers, NCERT of class 10. YouTube video tutorial of every question is also embedded along with the written solution.  

Question 1

Express each number as a product of its prime factors:

You must learn the Ladder method to solve this question. Watch the video embedded below.

(i) 140

140 = 2 x 2 x 5 x 7 = 22 x 5 x 7

(ii) 156

156 = 2 x 2 x 3 x 13 = 22 x 3 x 13

(iii) 3825

3825 = 3 x 3 x 5 x 5 x 7 = 32 x 52 x 17

(iv) 5005

5005 = 5 x 7 x 11 x 13

(v) 7429

7429 = 17 x 19 x 23

Question 2

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91

26=2×1391=7×13𝐿𝐶𝑀(26,91)𝐻𝐶𝐹(26,91)=21×70×131=20×71×131=21×71×131=182=20×70×131=13

Verification: LCM × HCF = product of the two numbers

LCM (26, 91) x HCF (26, 91) = 182 x 13 = 2366 … (1)

26 x 91 = 2,366 … (2)

From (1) and (2) 

LCM (26, 91) x HCF (26, 91) = 26 x 91

(ii) 510 and 92

510=2×3×5×1792=2×2×23𝐿𝐶𝑀(510,92)𝐻𝐶𝐹(510,92)=21×31×51×171×230=22×30×50×170×231=22×31×51×171×231=23460=21×30×50×170×230=2

Verification: LCM × HCF = product of the two numbers

LCM (510, 92) x HCF (510, 92) = 23460 x 2 = 46920 … (1)

510 x 92 = 42960 … (2)

From (1) and (2) 

LCM (510, 92) x HCF (510, 92) = 510 x 92

(iii) 336 and 54

336=24×31×7154=21×33𝐿𝐶𝑀(336,54)𝐻𝐶𝐹(336,54)=24×31×71=21×33×70=24×33×71=3024=21×31×70=6

Verification: LCM × HCF = product of the two numbers

LCM (336, 54) x HCF (336, 54) = 3024 x 6 = 18144 … (1)

336 x 54 = 18144 … (2)

From (1) and (2) 

LCM (336, 54) x HCF (336, 54) = 336 x 54

Watch the video embedded below in case you have any doubts about finding LCM and HCF.

Question 3

Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21

12=22×315=3×521=3×7𝐿𝐶𝑀(12,15,21)𝐻𝐶𝐹(12,15,21)=22×31×50×70=20×31×51×70=20×31×50×71=22×31×51×71=420=20×31×50×70=3

(ii) 17, 23 and 29

17=1723=2329=29𝐿𝐶𝑀(17,23,29)𝐻𝐶𝐹(17,23,29)=171×230×290=170×231×290=170×230×291=171×231×291=11339=170×230×290=1

(iii) 8, 9 and 25

8=239=3225=52𝐿𝐶𝑀(8,9,25)𝐻𝐶𝐹(8,9,25)=23×30×50=20×32×50=20×30×52=23×32×52=1800=20×30×50=1

Question 4

Given that HCF (306, 657) = 9, find LCM (306, 657).

Let a = 306, b = 657

Now, we know that 

𝐿𝐶𝑀(𝑎,𝑏)×𝐻𝐶𝐹(𝑎,𝑏)⇒𝐿𝐶𝑀(306,657)×𝐻𝐶𝐹(306,657)⇒𝐿𝐶𝑀(306,657)×9⇒𝐿𝐶𝑀(306,657)=𝑎×𝑏=306×657=306×657=306×6579=34×657=22338

Question 5

Check whether 6n can end with the digit 0 for any natural number n.

Any number that ends with 0, is divisible by 10. 

For example, 20, 30, 50 etc. 

This means the number ending with 0 is always divisible by 2 and 5. 

Let us now check if both 2 and 5 are factors of 6n for some natural n or not.

6𝑛=(2×3)𝑛=2𝑛×3𝑛

We can see that for any natural n, only 2 and 3 are the prime factors of 6n.

So, it can never end with 0 because 5 is not a factor.

Question 6

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

A number is a composite number if it can be expressed as product of at least 2 prime numbers. 

7 × 11 × 13 + 13

7×11×13+13=13(7×11+1)=13(77+1)=13×78=13×2×3×13

We can see that 7 × 11 × 13 + 13 can be written as product of 4 prime numbers. 

Hence it is a composite number. 

7 × 6 × 5 × 4 × 3 × 2 × 1 + 5

7×6×5×4×3×2×1+5=5(7×6×4×3×2×1+1)=5(1008+1)=5×1009

We can see that 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 can be written as product of 2 prime numbers. 

Hence it is a composite number. 

Question 7

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 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?

Time taken by Sonia to complete 1 round = 18 minutes

Time taken by Ravi to complete 1 round = 12 minutes

This means after every 18 minutes, Sonia reaches the Starting Point and after every 12 minutes, Ravi reaches the Starting Point.

Or we can say that:

Sonia reaches the Starting Point at multiple of 18

Ravi reaches the Starting Point at multiple of 12

So we have to find the LCM (because both will need at starting point at multiple of 18 and 12 respectively)

12 = 2 x 2 x 3

18 = 2 x 3 x 3

LCM (12, 18) = 2 x 2 x 3 x 3 = 36

Both will meet at starting point after 36 minutes.

\(\begin{aligned}  \end{aligned}\)

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]]> Class 10 | Chapter 1 | All Solutions | Real Numbers https://mathemafia.com/class-10-chapter-01-all-solutions-real-numbers/ Fri, 17 Dec 2021 07:36:57 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7363 This playlist covers all the questions, concepts and examples of chapter 1 (Real Numbers) of class 10, NCERT book. Click a thumbnail to watch the tutorial.  Watch complete chapter on YouTube

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This playlist covers all the questions, concepts and examples of chapter 1 (Real Numbers) of class 10, NCERT book. Click a thumbnail to watch the tutorial. 

Watch complete chapter on YouTube

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]]> Class 10 | Chapter 1 | Real Numbers | Exercise 1.4 https://mathemafia.com/class-10-chapter-1-real-numbers-exercise-1-4/ Sun, 20 Jun 2021 03:02:02 +0000 http://sh016.global.temp.domains/~mathemaf/?p=7013 Question 1: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: Question 2: Write down the decimal expansions of those rational numbers in Question 1…

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Question 1: Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

class 10 1.4 q1

Question 2: Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Question 3: The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p/q, what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

(iii) 43.123456789123456789…

Class 10 | Exercise 1.3
Class 10 | Chapter 2 | Theorems & Concepts

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– Rishi’s Parents (USA)

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Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

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Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

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]]> Class 10 | Chapter 1 | Real Numbers | Exercise 1.3 https://mathemafia.com/class-10-chapter-1-real-numbers-exercise-1-3/ Thu, 10 Jun 2021 03:54:56 +0000 http://sh016.global.temp.domains/~mathemaf/?p=6998 Question 1: Prove that √5 is irrational. Question 2: Prove that 3 + √5 is irrational. Question 3: Prove that the following are irrationals:(i) 1/√2 (ii) 7√5 (iii) 6 + √2 Class 10 | Exercise 1.2 Class 10 | Exercise…

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Question 1: Prove that √5 is irrational.

Question 2: Prove that 3 + √5 is irrational.

Question 3: Prove that the following are irrationals:
(i) 1/√2 (ii) 7√5 (iii) 6 + √2

Class 10 | Exercise 1.2
Class 10 | Exercise 1.4

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

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Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

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]]> Class 10 | Chapter 1 | Real Numbers | Exercise 1.2 https://mathemafia.com/class-10-chapter-1-real-numbers-exercise-1-2/ Mon, 19 Apr 2021 10:06:13 +0000 http://sh016.global.temp.domains/~mathemaf/?p=6593 Question 1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429 Question 2: Find the LCM and HCF of the following pairs of integers and verify that LCM ×…

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Question 1: Express each number as a product of its prime factors: (i) 140 (ii) 156 (iii) 3825 (iv) 5005 (v) 7429

Question 2: Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

(i) 26 and 91 (ii) 510 and 92 (iii) 336 and 54

Question 3: Find the LCM and HCF of the following integers by applying the prime factorisation method.

(i) 12, 15 and 21 (ii) 17, 23 and 29 (iii) 8, 9 and 25

Question 4: Given that HCF (306, 657) = 9, find LCM (306, 657).

Question 5: Check whether 6n can end with the digit 0 for any natural number n.

Question 6: Explain why 7×11×13 +13 and 7×6×5×4×3×2×1+5 are composite numbers.

Question 7: There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 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?

Class 10 | Exercise 1.1
Class 10 | Exercise 1.3

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Saiba

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

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Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

Super Cool Stuff

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]]> Class 10 | Chapter 1 | Real Numbers | Exercise 1.1 https://mathemafia.com/class-10-chapter-1-real-numbers-exercise-1-1/ Wed, 17 Mar 2021 03:51:12 +0000 http://sh016.global.temp.domains/~mathemaf/?p=6417 Question 1: Use Euclid’s division algorithm to find the HCF of :(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255 Question 2: Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5,…

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Question 1: Use Euclid’s division algorithm to find the HCF of :
(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Question 2: Show that any positive odd integer is of the form 6q+1, or 6q+3, or 6q+5, where q is some integer.

Question 3: An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Question 4: Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Question 5: Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.

Class 10 | Chapter 1 | All Examples
Class 10 | Exercise 1.2

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

Super Cool Stuff

The post Class 10 | Chapter 1 | Real Numbers | Exercise 1.1 appeared first on Mathemafia.

]]> Class 10 | Chapter 1 | Real Numbers | Examples https://mathemafia.com/class-10-chapter-1-real-numbers-examples/ Fri, 05 Mar 2021 05:18:58 +0000 http://sh016.global.temp.domains/~mathemaf/?p=6387 Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576. Example 2: Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1,…

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Example 1 : Use Euclid’s algorithm to find the HCF of 4052 and 12576.

Example 2: Show that every positive even integer is of the form 2q, and that every positive odd integer is of the form 2q + 1, where q is some integer.

Example 3: Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Example 4: A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of that can be placed in each stack for this purpose?

Example 5: Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.

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

Example 7: Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.

Example 8: Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Example 9: Prove that √ 3 is irrational.

Example 10: Show that 5 – √ 3 is irrational.

Example 11: Show that 3√2 is irrational.

Class 10 | Chapter 1 | Theorems & Concepts
Class 10 | Exercise 1.1

Testimonials

Saiba

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Sunil Sir has been an excellent math teacher for Saiba. His teaching methods are highly informative, engaging, and motivational which have greatly strengthened Saiba’s understanding of maths concepts. He consistently provides support beyond the classroom and has played an important role in improving Saiba’s problem-solving skills and preparation for math competitions. His dedication and professionalism make him an outstanding educator. Saiba is thankful for his continued support and positive influence.

 

– Saiba’s Parents (USA)

Amazing Math Coach

Sunil is an amazing math coach! My son Rishi loves to learn from him. He makes learning fun and engages Rishi. Sunil is also very patient with kids and understands my son’s learning methods and teaches accordingly with positive reinforcement. Very thankful to him!

 

– Rishi’s Parents (USA)

Rishi (USA Student)

Unique, Patient and Good Mentor

Sunil sir has a unique way of explaining complex concepts, making them simple and understandable. You are very patient and always encourages me to ask queries as many times until the concept is clearer. You are a great person besides being a good mentor and always in a light mood and never gets frustrated for anything. My grades have immensely improved and I am much more confident in maths now.

 

Also, your videos on Mathemafia youtube channel are very informative.

 

– Lakshay (USA)

WhatsApp for 1-1 Classes

❇️ ❇️ M️aster List – All Maths Formulas, Class 10 ❇️ ❇️

🇮🇳 हर भारतीय के दिल की आवाज़… आइए, इस स्वतंत्रता दिवस मिलकर गाएँ – “भारत माता की जय“।

Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

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]]> Class 10 | Chapter 1 | Real Numbers | Theorems & Concepts https://mathemafia.com/class-10-chapter-1-real-numbers-conceptual-tutorials/ Fri, 05 Mar 2021 04:57:32 +0000 http://sh016.global.temp.domains/~mathemaf/?p=6372 Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. Euclid’s Division Algorithm : This algorithm is used to find…

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Theorem 1.1 (Euclid’s Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.

Euclid’s Division Algorithm : This algorithm is used to find the h.c.f (Highest Common Factor) or g.c.d (Greatest Common Divisor)of two positive integers. It is based on Euclid’s Division Lemma that has been stated and explained above. 

  • Step 1 : Apply Euclid’s division lemma, to a and b to find integers (more precisely whole numbers) q and r such that a = bq + r, 0 ≤ r < b.
  • Step 2 : If r=0, b is the g.c.d of a and b. If r≠0, apply the division lemma to b and r.
  • Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required GCD (HCF).

Theorem 1.3: Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

Theorem 1.4: √2 is irrational.

Theorem – Good to know: If a non -ve integer is not a perfect square, then its square root is irrational.

Theorem 1.5: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers.

Theorem 1.6: Let x = p/q be a rational number, such that the prime factorisation q of q is of the form 2n5m, where n, m are non-negative integers. Then x has a decimal expansion which terminates.

Theorem 1.7: Let x = p/q , where p and q are coprimes, be a rational number, such that the prime factorisation of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).

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Class 10 – Mathematics – NCERT Solutions

  1. Real Numbers
  2. Polynomials
  3. Pair of Linear Equation in Two Variables
  1. Quadratic Equations
  2. Arithmetic Progressions
  3. Triangles
  1. Coordinate Geometry
  2. Introduction to Trigonometry
  3. Some Applications of Trigonometry
    • Exercise 9.1
  4. Circles
  1. Areas Related to Circles
  2. Surface Areas and Volumes
  3. Statistics
    • Exercise 13.1
    • Exercise 13.2
    • Exercise 13.3
  4. Probability

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