NCERT solutions class 10 maths chapter 1 â€“ Real Numbers are provided here. These solutions are prepared by our expert faculties to help students in their board exam preparations. Our expert faculties solve and provide the NCERT solutions for maths so that it should help students to solve the problems easily. They also focus on preparing the solutions in such a way that it is easy to understand for the students. They provide a detailed and stepwise explanation of each answer to the questions given in the exercises in the NCERT book.

In NCERT Solutions for Class 10 Maths â€“ the First Chapter is Real numbers chapter, where students are introduced to a lot of important concepts which will be useful for those who wish to pursue mathematics as a subject in their class 11. Based on these solutions, students can prepare for their upcoming Board Exams. These solutions are helpful as the syllabus covered here follows NCERT guidelines.

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### Class 10 Maths Chapter 1 Exercise 1.1 Page: 7

**1. Use Euclidâ€™s division algorithm to find the HCF of:****(i) 135 and 225****(ii) 196 and 38220****(iii) 867 and 225**

**Solutions: **

**(iii) 867 and 225 **

As we know, 867 is greater than 225. Let us apply now Euclidâ€™s division algorithm on 867, to get,

867 = 225 Ã— 3 + 102

Remainder 102 â‰ 0, therefore taking 225 as divisor and applying the division lemma method, we get,

225 = 102 Ã— 2 + 51

Again, 51 â‰ 0. Now 102 is the new divisor, so repeating the same step we get,

102 = 51 Ã— 2 + 0

The remainder is now zero, so our procedure stops here. Since, in the last step, the divisor is 51, therefore, HCF (867,225) = HCF(225,102) = HCF(102,51) = 51.

* Hence, the HCF of 867 and 225 is 51. *

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

**Sol: **Let a be any positive integer and b = 6. Then, by Euclidâ€™s algorithm, a = 6q + r, for some integer q â‰¥ 0, and r = 0, 1, 2, 3, 4, 5, because 0â‰¤r<6.

Now substituting the value of r, we get,

If r = 0, then a = 6q

Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2. A positive integer can be either even or odd Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.

**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?**

**Solution: **Given,

Number of army contingent members = 616

Number of army band members = 32

If the two groups have to march in the same column, we have to find out the highest common factor between the two groups. HCF(616, 32), gives the maximum number of columns in which they can march.

By Using Euclidâ€™s algorithm to find their HCF, we get,

Since, 616>32, therefore,

616 = 32 Ã— 19 + 8

Since, 8 â‰ 0, therefore, taking 32 as new divisor, we have,

32 = 8 Ã— 4 + 0

Now we have got remainder as 0, therefore, HCF (616, 32) = 8.

Hence, the maximum number of columns in which they can march is 8.

**4. Use Euclidâ€™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.**

**Solutions: **Let x be any positive integer and y = 3.

**5. Use Euclidâ€™s division lemma to show that the cube of any positive integer is of the form 9m, 9m+1 or 9m+8.**

**Solution: **Let x be any positive integer and y = 3.

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### Class 10 Maths Chapter 1 Exercise 1.2 Page: 11

**1. Express each number as a product of its prime factors: **

**140****156****3825****5005****7429**

**Solutions: **

**(i) 140**

By Taking theÂ LCM of 140, we will get the product of its prime factor.

Therefore, 140 = 2 Ã— 2 Ã— 5 Ã— 7 Ã— 1 = 2^{2} Ã— 5 Ã— 7

**(ii) 156**

By Taking theÂ LCM of 156, we will get the product of its prime factor.

Hence, 156 = 2 Ã— 2 Ã— 13 Ã— 3 Ã— 1 = 2^{2} Ã— 13 Ã— 3

**(iii) 3825**

By Taking theÂ LCM of 3825, we will get the product of its prime factor.

Hence, 3825 = 3 Ã— 3 Ã— 5 Ã— 5 Ã— 17 Ã— 1 = 3^{2} Ã— 5^{2} Ã— 17

**(iv) 5005**

By Taking theÂ LCM of 5005, we will get the product of its prime factor.

Hence, 5005 = 5 Ã— 7 Ã— 11 Ã— 13 Ã— 1 = 5 Ã— 7 Ã— 11 Ã— 13

**(v) 7429 **

By Taking theÂ LCM of 7429, we will get the product of its prime factor.

Hence, 7429 = 17 Ã— 19 Ã— 23 Ã— 1 = 17 Ã— 19 Ã— 23

**2. Find the LCM and HCF of the following pairs of integers and verify that LCM Ã— HCF = product of the two numbers. **

**(i) 26 and 91 **

**(ii) 510 and 92 **

**(iii) 336 and 54**

**Solutions:**

**(i) 26 and 91 **

Expressing 26 and 91 as product of its prime factors, we get,

26 = 2 Ã— 13 Ã— 1

91 = 7 Ã— 13 Ã— 1

Therefore, LCM (26, 91) = 2 Ã— 7 Ã— 13 Ã— 1 = 182

And HCF (26, 91) = 13

Verification

Now, product of 26 and 91 = 26 Ã— 91 = 2366

And Product of LCM and HCF = 182 Ã— 13 = 2366

Hence, LCM Ã— HCF = product of the 26 and 91.

**(ii) 510 and 92 **

Expressing 510 and 92 as product of its prime factors, we get,

510 = 2 Ã— 3 Ã— 17 Ã— 5 Ã— 1

92 = 2 Ã— 2 Ã— 23 Ã— 1

Therefore, LCM(510, 92) = 2 Ã— 2 Ã— 3 Ã— 5 Ã— 17 Ã— 23 = 23460

And HCF (510, 92) = 2

Verification

Now, product of 510 and 92 = 510 Ã— 92 = 46920

And Product of LCM and HCF = 23460 Ã— 2 = 46920

Hence, LCM Ã— HCF = product of the 510 and 92.

**(iii) 336 and 54**

Expressing 336 and 54 as product of its prime factors, we get,

336 = 2 Ã— 2 Ã— 2 Ã— 2 Ã— 7 Ã— 3 Ã— 1

54 = 2 Ã— 3 Ã— 3 Ã— 3 Ã— 1

Therefore, LCM(336, 54) = = 3024

And HCF(336, 54) = 2Ã—3 = 6

Verification

Now, product of 336 and 54 = 336 Ã— 54 = 18,144

And Product of LCM and HCF = 3024 Ã— 6 = 18,144

Hence, LCM Ã— HCF = product of the 336 and 54.

**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**

**Solutions:**

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

**Solutions:**

**5. Check whether 6 ^{n} can end with the digit 0 for any natural number n.**

**Solutions: **If the numberÂ 6^{n}Â ends with the digit zero (0), then it should be divisible by 5, as we know any number with unit place as 0 or 5 is divisible by 5.

Prime factorization of 6^{n} = (2Ã—3)^{n}

Therefore, the prime factorization ofÂ doesnâ€™t contain prime number 5.

Hence, it is clear that for any natural number n, 6^{n} is not divisible by 5 and thus it proves that cannot end with the digit 0 for any natural number n.

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

**Solutions: **By the definition of composite number, we know, if a number is composite, then it means it has factors other than 1 and itself. Therefore, for the given expression;

**7 Ã— 11 Ã— 13 + 13 **

Taking 13 as common factor, we get,

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

Hence, 7 Ã— 11 Ã— 13 + 13 is a composite number.

Now letâ€™s take the other number,

**7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1 + 5 **

Taking 5 as a common factor, we get,

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

Hence, 7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1 + 5 is a composite number.

**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?**

**Solutions: **Since, Both Sonia and Ravi move in the same direction and at the same time, the method to find the time when they will be meeting again at the starting point is LCM of 18 and 12.

Therefore, LCM(18,12) = 2Ã—3Ã—3Ã—2Ã—1 = 36

Hence, Sonia and Ravi will meet again at the starting point after 36 minutes.

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### Class 10 Maths Chapter 1 Exercise 1.3 Page: 14

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**1. Prove that âˆš5 is irrational.**

**2. Prove that 3 + 2âˆš5 is irrational.**

**3. Prove that the following are irrationals: **

**(i) 1/âˆš2 **

**(ii) 7âˆš5**

**(iii) 6 + âˆš2 **

### Class 10 Maths Chapter 1 Exercise 1.4 Page: 14

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**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:**

**(i)13/3125 (ii)17/8 (iii) 64/455 (iv)15/1600 (v) 29/343 (vi)23/2 ^{3}5^{2} (vii)129/2^{2} 5 ^{7}7^{5} (viii)6/15 (ix)35/50 (x)77/210 **

**Solutions:**

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

**Solutions: **

**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. . . **

## NCERT Solutions for Class 10 Maths Chapter 1- Real Number

As Real Number is one of the important topics in maths, It has a weightage of 6 marks in class 10 (Number system â€“ Real Numbers) maths board exams. The average number of questions asked from this chapter is usually 3.

Three questions were asked from this chapter in the previous year board examination (2018).

- One out of three questions in part A (1 marks).
- One out of three questions in part B (2 marks).
- One out of three questions in part C (3 marks).

This chapter talks about

- Euclidâ€™s Division Algorithm
- The Fundamental Theorem of Arithmetic
- Revisiting Rational & Irrational Numbers
- Decimal Expansions

**List of Exercises in class 10 Maths Chapter 1:**

Exercise 1.1 Solutions 5 Question ( 4 long, 1 short)

Exercise 1.2 Solutions 7 Question ( 4 long, 3 short)

Exercise 1.3 Solutions 3 Question ( 3 short)

Exercise 1.4 Solutions 3 Question ( 3 short)

NCERT solutions for class 10 maths chapter 1 â€“ Real Numbers is the first chapter of class 10 Maths. Real Numbers is introduced in class 9 and this is discussed more in details in class 10. The chapter discusses the real numbers and their applications. The divisibility of integers using Euclidâ€™s division algorithm says that any positive integer a can be divided by another positive integer b such that the remainder will be smaller than b. On the other hand, The Fundamental Theorem of Arithmetic works on multiplication of positive integers.

The chapter starts with the introduction of real numbers in section 1.1 followed by two very important topics in section 1.2 and 1.3

- Euclidâ€™s Division Algorithm â€“ It includes 5 questions based on Theorem 1.1 â€“ Euclidâ€™s Division Lemma.
- The Fundamental Theorem of Arithmetic â€“ Explore the applications of this topic which talks about the multiplication of positive integers, through solutions of the 7 problems in Exercise 1.2.

Next, it discusses the following topics which were introduced in class 9.

- Revisiting Rational & Irrational Numbers â€“ In this the solutions for 3 problems in Exercise 1.3 is given which also use the topic in last Exercise 1.2.
- Decimal Expansions â€“ It explores when the decimal expansion of a rational number is terminating and when it is recurring. It includes a total of 3 problems with subparts in Exercise 1.4

### Key Features of NCERT Solutions for Class 10 Maths Chapter 1- Real Number

- These NCERT Solutions let you solve and revise the whole syllabus of class 10.
- After going through the stepwise solutions given by our subject expert teachers, you will be able to score more marks.
- It follows NCERT guidelines which help in preparing the students accordingly.
- It contains all the important questions from the examination point of view.
- It helps in scoring well in maths in exams.

## Frequently Asked Questions on Chapter 1- Real Number

### Euclidâ€™s division algorithm to find the HCF of 135 and 225?

**135 and 225**

As you can see, from question 225 is greater than 135. Therefore, by Euclidâ€™s division algorithm, we have,

225 = 135 Ã— 1 + 90

Now, remainder 90 â‰ 0, thus again using division lemma for 90, we get,

135 = 90 Ã— 1 + 45

Again, 45 â‰ 0, repeating the above step for 45, we get,

90 = 45 Ã— 2 + 0

The remainder is now zero, so our method stops here. Since, in the last step, the divisor is 45, therefore, HCF (225,135) = HCF (135, 90) = HCF (90, 45) = 45.

Hence, the HCF of 225 and 135 is 45.

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

Let a be any positive integer and b = 6. Then, by Euclidâ€™s algorithm, a = 6q + r, for some integer q â‰¥ 0, and r = 0, 1, 2, 3, 4, 5, because 0â‰¤r<6.

Now substituting the value of r, we get,

If r = 0, then a = 6q

Similarly, for r= 1, 2, 3, 4 and 5, the value of a is 6q+1, 6q+2, 6q+3, 6q+4 and 6q+5, respectively.

If a = 6q, 6q+2, 6q+4, then a is an even number and divisible by 2. A positive integer can be either even or odd Therefore, any positive odd integer is of the form of 6q+1, 6q+3 and 6q+5, where q is some integer.

### Express each number as a product of its prime factors 5005?

5005

By Taking the LCM of 5005, we will get the product of its prime factors.

Hence, 5005 = 5 Ã— 7 Ã— 11 Ã— 13 Ã— 1 = 5 Ã— 7 Ã— 11 Ã— 13

### Find the LCM and HCF of the following pairs of integers and verify that LCM Ã— HCF = product of the two numbers 26 and 91 ?

6 and 91

Expressing 26 and 91 as product of its prime factors, we get,

26 = 2 Ã— 13 Ã— 1

91 = 7 Ã— 13 Ã— 1

Therefore, LCM (26, 91) = 2 Ã— 7 Ã— 13 Ã— 1 = 182

And HCF (26, 91) = 13

**Verification**

Now, product of 26 and 91 = 26 Ã— 91 = 2366

And Product of LCM and HCF = 182 Ã— 13 = 2366

Hence, LCM Ã— HCF = product of the 26 and 91.

### 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 43.123456789?

43.123456789

Since it has a terminating decimal expansion, it is a rational number in the form of p/q and q has factors of 2 and 5 only.

Thanks

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