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# NCERT solutions for Class 10 Mathematics chapter 1 - Real Numbers

## Chapter 1: Real Numbers

Ex. 1.1Ex. 1.2Ex. 1.3Ex. 1.4

#### Chapter 1: Real Numbers Exercise 1.1 solutions [Page 7]

Ex. 1.1 | Q 1.1 | Page 7

Using Euclid's division algorithm, find the H.C.F. of 135 and 225

Ex. 1.1 | Q 1.2 | Page 7

Using Euclid's division algorithm, find the H.C.F. of 196 and 38220

Ex. 1.1 | Q 1.3 | Page 7

Using Euclid's division algorithm, find the H.C.F. of (iii) 867 and 255

Ex. 1.1 | Q 2 | Page 7

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

Ex. 1.1 | Q 3 | Page 7

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?

Ex. 1.1 | Q 4 | Page 7

Use Euclid's Division Algorithm to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Ex. 1.1 | Q 5 | Page 7

Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m

#### Chapter 1: Real Numbers Exercise 1.2 solutions [Page 11]

Ex. 1.2 | Q 1.1 | Page 11

Express the number as a product of its prime factor:

140

Ex. 1.2 | Q 1.2 | Page 11

Express the number as a product of its prime factor:

156

Ex. 1.2 | Q 1.3 | Page 11

Express the number as a product of its prime factor:

3825

Ex. 1.2 | Q 1.4 | Page 11

Express the number as a product of its prime factor:

5005

Ex. 1.2 | Q 1.5 | Page 11

Express the number as a product of its prime factor:

7429

Ex. 1.2 | Q 2.1 | Page 11

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

26 and 91

Ex. 1.2 | Q 2.2 | Page 11

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

510 and 92

Ex. 1.2 | Q 2.3 | Page 11

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

336 and 54

Ex. 1.2 | Q 3.1 | Page 11

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

12, 15 and 21

Ex. 1.2 | Q 3.2 | Page 11

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

17, 23 and 29

Ex. 1.2 | Q 3.3 | Page 11

Find the LCM and HCF of the following integers by applying the prime factorisation method
8, 9 and 25

Ex. 1.2 | Q 4 | Page 11

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

Ex. 1.2 | Q 5 | Page 11

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

Ex. 1.2 | Q 6 | Page 11

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

Ex. 1.2 | Q 7 | Page 11

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?

#### Chapter 1: Real Numbers Exercise 1.3 solutions [Page 14]

Ex. 1.3 | Q 1 | Page 14

Prove that sqrt5 is irrational.

Ex. 1.3 | Q 2 | Page 14

Prove that 3 + 2sqrt5 is irrational

Ex. 1.3 | Q 3.1 | Page 14

Prove that the following is irrational

1/sqrt2

Ex. 1.3 | Q 3.2 | Page 14

Prove that the following is irrational

 7sqrt5

Ex. 1.3 | Q 3.3 | Page 14

Prove that the following are irrationals
 6+sqrt2

#### Chapter 1: Real Numbers Exercise 1.4 solutions [Pages 17 - 18]

Ex. 1.4 | Q 1.01 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

13/3125

Ex. 1.4 | Q 1.02 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

17/8

Ex. 1.4 | Q 1.03 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

64/455

Ex. 1.4 | Q 1.04 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

15/1600

Ex. 1.4 | Q 1.05 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

29/343

Ex. 1.4 | Q 1.06 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

23/(2^3xx5^2)

Ex. 1.4 | Q 1.07 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

129/(2^2xx5^7xx7^5)

Ex. 1.4 | Q 1.08 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

6/15

Ex. 1.4 | Q 1.09 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

35/50

Ex. 1.4 | Q 1.1 | Page 17

State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

77/210

Ex. 1.4 | Q 2.01 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

13/3125

Ex. 1.4 | Q 2.02 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

17/8

Ex. 1.4 | Q 2.03 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

64/455

Ex. 1.4 | Q 2.04 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

15/1600

Ex. 1.4 | Q 2.05 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

29/343

Ex. 1.4 | Q 2.06 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

23/(2^3xx5^2)

Ex. 1.4 | Q 2.07 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

129/(2^2xx5^7xx7^5)

Ex. 1.4 | Q 2.08 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

6/15

Ex. 1.4 | Q 2.09 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

35/50

Ex. 1.4 | Q 2.1 | Page 18

Write down the decimal expansion of the following number  which have terminating decimal expansion.

77/210

Ex. 1.4 | Q 3 | Page 18

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?

(1) 43.123456789

(2) 0.120120012000120000. . .

(3) 43.bar(123456789)

## Chapter 1: Real Numbers

Ex. 1.1Ex. 1.2Ex. 1.3Ex. 1.4

## NCERT solutions for Class 10 Mathematics chapter 1 - Real Numbers

NCERT solutions for Class 10 Maths chapter 1 (Real Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Textbook for Class 10 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com are providing such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Mathematics chapter 1 Real Numbers are Introduction of Real Numbers, Real Numbers Examples and Solutions, Euclidâ€™s Division Lemma, Fundamental Theorem of Arithmetic, Fundamental Theorem of Arithmetic Motivating Through Examples, Proofs of Irrationality, Revisiting Irrational Numbers, Revisiting Rational Numbers and Their Decimal Expansions.

Using NCERT Class 10 solutions Real Numbers exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer NCERT Textbook Solutions to score more in exam.

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