# NCERT Solutions for Class 10 Maths Chapter 1 Real Numbers

Here, we have providing**Chapter 1 Real Numbers NCERT Solutions for Class 10 Maths**which will be beneficial for students. These solutions are updated according to 2020-21 syllabus. As NCERT Solutions are

**prepared by Studyrankers**experts, we have taken of every steps so you can understand the concepts without any difficulty. These

**Real Numbers Class 10 solutions**can be really helpful in the preparation of Board exams and will provide you with in depth detail of the chapter.

We have solved every question stepwise so you don't have to face difficulty in understanding the solutions. It will also give you concepts that are important for overall development of students. Class 10 Maths

**Real Number NCERT Solutions**will be useful in higher classes as well because variety of questions related to these concepts can be asked so you must study and understand them properly.

Page No: 7

**Exercise 1.1**

1. Use Euclid's division algorithm to find the HCF of:

(i) 135 and 225

(ii) 196 and 38220

(iii) 867 and 255

**Answer**

(i) 225 > 135 we always divide greater number with smaller one.

Divide 225 by 135 we get 1 quotient and 90 as remainder so that

225= 135 Ã— 1 + 90

Divide 135 by 90 we get 1 quotient and 45 as remainder so that

135= 90 Ã—Â 1 + 45

Divide 90 by 45 we get 2 quotient and no remainder so we can write it as

90 = 2 Ã—Â 45+ 0

As there are no remainder so divisor 45 is our HCF.

(ii)Â 38220 > 196 we always divide greater number with smaller one.

Divide 38220 by 196 then we get quotient 195 and no remainder so we can write it as

38220 = 196Â Ã—Â 195 + 0

As there is no remainder so divisor 196 is our HCF.

(iii)Â 867 > 255 we always divide greater number with smaller one.

Divide 867 by 255 then we get quotient 3 and remainder is 102Â so we can write it as

867 = 255Â Ã—Â 3 + 102

Divide 255 by 102 then we get quotient 2 and remainder is 51 so we can write it as

255 = 102Â Ã—Â 2 + 51

Divide 102 by 51 we get quotient 2 and no remainder so we can write it as

102 = 51 Ã—Â 2 + 0

As there is no remainder so divisor 51 is our HCF.

2. Show that any positive odd integer is of the form 6

*q*+ 1, or 6

*q*+ 3, or 6

*q*+ 5, where

*q*is some integer.

**Answer**

Let take

*a*as any positive integer and

*b*= 6.

Then using Euclidâ€™s algorithm we get a = 6

*q*+

*r*here

*r*is remainder and value of

*q*is more than or equal to 0 and

*r*= 0, 1, 2, 3, 4, 5 because 0Â â‰¤Â

*r*< b and the value of

*b*is 6Â

So total possible forms will 6

*qÂ*+ 0 , 6*qÂ*+ 1 , 6*qÂ*+ 2,6*qÂ*+ 3, 6*q*+ 4, 6*q*+ 56

*q*+ 0

6 is divisible by 2 so it is a even numberÂ

6

*q*+ 1Â

6 is divisible by 2 but 1 is not divisible by 2 so it is a odd numberÂ

6

*q*+ 2Â

6 is divisible by 2 and 2 is also divisible by 2 so it is a even numberÂ

6

*q*Â +3Â
6 is divisible by 2 but 3 is not divisible by 2 so it is a odd numberÂ

6

*q*+ 4Â

6 is divisible by 2 and 4 is also divisible by 2 it is a even number

6

*q*+ 5Â
6 is divisible by 2 but 5 is not divisible by 2 so it is a odd number

So odd numbers will in form of 6q + 1, or 6q + 3, or 6q + 5.

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?

**Answer**

HCF (616, 32) will give the maximum number of columns in which they can march.

We can use Euclid's algorithm to find the HCF.

616 = 32Â Ã—Â 19 + 8

32 = 8Â Ã—Â 4 + 0

The HCF (616, 32) is 8.

Therefore, they can march in 8 columns each.

4. Use Euclid's division lemma to show that the square of any positive integer is either of form 3

*m*or 3

*m*+ 1 for some integer m.

[Hint: Let

*x*be any positive integer then it is of the form 3

*q*, 3

*q*+ 1 or 3

*q*+ 2. Now square each of these and show that they can be rewritten in the form 3

*m*or 3

*m*+ 1.]

**Answer**

Let a be any positive integer and

*b*= 3.

Then a = 3

*q*+

*r*for some integer

*q*â‰¥ 0

And

*r*= 0, 1, 2 because 0 â‰¤

*r*< 3

Therefore,

*a*= 3

*q*or 3

*q*+ 1 or 3

*q*+ 2

Or,

*a*

^{2}= (3

*q*)

^{2}or (3

*q*+ 1)

^{2}or (3

*q*Â + 2)

^{2}

*a*

^{2}= (9

*q*)

^{2}or 9

*q*

^{2}Â + 6

*q*Â + 1 or 9

*q*

^{2}Â + 12

*q*Â + 4

= 3 Ã— (3

*q*

^{2}) or 3(3

*q*

^{2}Â + 2

*q*) + 1 or 3(3

*q*

^{2}Â + 4

*q*Â + 1)Â + 1

= 3

*k*

_{1}or 3

*k*

_{2}Â + 1 or 3

*k*

_{3}Â + 1

WhereÂ

*k*

_{1},Â

*k*

_{2}, andÂ

*k*

_{3}Â are some positive integers

Hence, it can be said that the square of any positive integer is either of the form 3

*m*or 3

*m*+ 1.

5. Use Euclid's division lemma to show that the cube of any positive integer is of the form 9

*m*, 9

*mÂ*+ 1 or 9

*m*+ 8.

**Answer**

Let a be any positive integer and b = 3

*a*= 3

*q*+

*r*, where

*q*â‰¥ 0 and 0 â‰¤

*r*< 3

âˆ´

*a*= 3q or 3

*q*Â + 1 or 3

*q*Â + 2

Therefore, every number can be represented as these three forms. There are three cases.

Case 1: When

*a*= 3*q*,*a*

^{3}= (3

*q*)

^{3}= 27

*q*

^{3}= 9(3

*q*)

^{3}= 9

*m*,

Where

*m*is an integer such that

*m*= 3

*q*

^{3}

Case 2: When

*a*= 3q + 1,

*a*

^{3}= (3

*q*+1)

^{3}

*a*

^{3}= 27

*q*

^{3}+ 27

*q*

^{2}+ 9

*q*+ 1

*a*

^{3}= 9(3

*q*

^{3}+ 3

*q*

^{2}+

*q*) + 1

*a*

^{3}= 9

*m*+ 1

Where

*m*is an integer such that

*m*= (3

*q*

^{3}+ 3

*q*

^{2}+

*q*)

Case 3: When

*a*= 3

*q*+ 2,

*a*

^{3}= (3

*q*+2)

^{3}

*a*

^{3}= 27

*q*

^{3}+ 54

*q*

^{2}+ 36

*q*+ 8

*a*

^{3}= 9(3

*q*

^{3}+ 6

*q*

^{2}+ 4q) + 8

*a*

^{3}= 9

*m*+ 8

Where

*m*is an integer such that

*m*= (3

*q*

^{3}+ 6

*q*

^{2}+ 4

*q*)

Therefore, the cube of any positive integer is of the form 9

or 9

Page No: 11

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

(i) 140*m*, 9*m*+ 1,or 9

*m*+ 8.Page No: 11

**Exercise 1.2**1. Express each number as product of its prime factors:

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

**Answer**

(i) 140 = 2 Ã— 2 Ã— 5 Ã— 7 = 2

^{2}Ã— 5 Ã— 7

(ii)Â 156 = 2 Ã— 2 Ã— 3 Ã— 13 = 2

^{2}Ã— 3 Ã— 13

(iii)Â 3825 = 3 Ã— 3 Ã— 5 Ã— 5 Ã— 17 = 3

^{2}Ã— 5

^{2}Ã— 17

(iv)Â 5005 = 5 Ã— 7 Ã— 11 Ã— 13

(v)Â 7429 = 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 92Â

(iii) 336 and 54

**Answer**
(i) 26 = 2Â Ã—Â 13

91 =7Â Ã—Â 13

HCF = 13

LCM =2Â Ã—Â 7Â Ã—Â 13 =182

Product of two numbers 26Â Ã—Â 91 = 2366

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

Hence, product of two numbers = product of HCF Ã— LCM

91 =7Â Ã—Â 13

HCF = 13

LCM =2Â Ã—Â 7Â Ã—Â 13 =182

Product of two numbers 26Â Ã—Â 91 = 2366

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

Hence, product of two numbers = product of HCF Ã— LCM

(ii) 510 = 2Â Ã— 3 Ã— 5Â Ã—Â 17

92 =2Â Ã—Â 2Â Ã—Â 23

HCF = 2

LCM =2Â Ã—Â 2Â Ã—Â 3Â Ã—Â 5Â Ã—Â 17Â Ã—Â 23 = 23460

Product of twoÂ numbers 510Â Ã—Â 92 = 46920

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

Hence, product of two numbers = product of HCF Ã— LCM

(iii) 336 = 2Â Ã—Â 2Â Ã—Â 2Â Ã—Â 2Â Ã—Â 3Â Ã—Â 7

54 = 2Â Ã—Â 3Â Ã—Â 3Â Ã—Â 3

HCF = 2Â Ã—Â 3 = 6

LCM = 2Â Ã—Â 2Â Ã—Â 2Â Ã—Â 2Â Ã—Â 3Â Ã—Â 3Â Ã—Â 3Â Ã—Â 7 =3024

Product of two numbers 336Â Ã—Â 54 =18144

Product of HCF and LCM 6Â Ã—Â 3024 = 18144

Hence, product of two numbers = product of HCF Ã— LCM.

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

(i) 12, 15 and 21Â

(ii) 17, 23 and 29Â

(iii) 8, 9 and 25

**Answer**

(i) 12 = 2 Ã— 2 Ã— 3

15 =3 Ã— 5

21 =3 Ã— 7

HCF = 3

LCM = 2 Ã— 2 Ã— 3 Ã— 5 Ã— 7 = 420

(ii) 17 = 1 Ã— 17

23 = 1 Ã— 23

29 = 1 Ã— 29

HCF = 1

LCM = 1 Ã— 17 Ã— 19 Ã— 23 = 11339

(iii) 8 =1 Ã— 2 Ã— 2 Ã— 2

9 =1 Ã— 3 Ã— 3

25 =1 Ã— 5 Ã— 5

HCF =1

LCM = 1 Ã— 2 Ã— 2 Ã— 2 Ã— 3 Ã— 3 Ã— 5 Ã— 5 = 1800

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

**Answer**

We have the formula that

Product of LCM and HCF = product of number

LCMÂ Ã—Â 9 = 306Â Ã—Â 657

Divide both side by 9 we get

LCM = (306Â Ã—Â 657)Â / 9 = 22338

5. Check whether 6

*n*can end with the digit 0 for any natural number

*n*.

**Answer**

If any digit has last digit 10 that means it is divisible by 10 and the factors ofÂ 10 = 2Â Ã—Â 5.

So value 6

*n*should be divisible by 2 and 5 both 6*n*is divisible by 2 but not divisible by 5 SoÂ it can not end with 0.6. 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.

**Answer**

7 Ã— 11 Ã— 13 + 13

Taking 13 common, we get

13 (7 x 11 +1 )

13(77 + 1 )

13 (78)

It is product of two numbers and both numbers are more than 1 so it is a composite number.

7 Ã— 6 Ã— 5 Ã— 4 Ã— 3 Ã— 2 Ã— 1 + 5

Taking 5 common, we get

5(7Â Ã—Â 6Â Ã—Â 4Â Ã—Â 3Â Ã—Â 2Â Ã—Â 1 +1)

5(1008 + 1)

5(1009)

It is product of two numbers and both numbers are more than 1 so it 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?

**Answer**

They will be meet again after LCM of both values at the starting point.

18 = 2 Ã— 3 Ã— 3

12 = 2 Ã— 2 Ã— 3

LCM = 2 Ã— 2 Ã— 3 Ã— 3 = 36

Therefore, they will meet together at the starting point after 36 minutes.

Page No: 14

**Exercise 1.3**

1. Prove that âˆš5Â is irrational.

**Answer**

Let takeÂ âˆš5Â as rational number

If

*a*and

*b*are two co prime number and

*b*is not equal to 0.

We can writeÂ âˆš5Â =

*a*/

*b*

Multiply by b both side we get

*b*âˆš5Â =

*a*

To remove root, Squaring on both sides, we get

5

*b*

^{2}=

*a*

^{2}â€¦ Â

**(i)**

Therefore, 5 divides

That means 5 will divide

Putting value of

5

5

Divide by 25 we get

*a*^{2}and according to theorem of rational number, for any prime number*p*which is divides*a*^{2}then it will divide*a*also.That means 5 will divide

*a*. So we can write*a*= 5*c*Putting value of

*a*in equation**(i)**we get5

*b*^{2}= (5*c*)^{2}5

*b*^{2}= 25*c*^{2}Divide by 25 we get

*b*2/5 =

*c*

^{2}

Similarly, we get that

and we have already get that

but

HenceÂ âˆš5Â is not a rational number, it is irrational.

2. Prove that 3 + 2âˆš5Â is irrational.

*b*will divide by 5and we have already get that

*a*is divide by 5but

*a*and*b*are co prime number. so it contradicts.HenceÂ âˆš5Â is not a rational number, it is irrational.

2. Prove that 3 + 2âˆš5Â is irrational.

**Answer**

Let take that 3 + 2âˆš5Â is a rational number.

So we can write this number as

3 + 2âˆš5Â =

*a*/

*b*

Here a and b are two co prime number and b is not equal to 0

Subtract 3 both sides we get

2âˆš5Â =

*a*/

*b*â€“ 3

2âˆš5Â = (

*a*-3

*b*)/

*b*

Now divide by 2, we get

âˆš5Â = (

*a*-3

*b*)/2

*b*

Here

*a*and

*b*are integer so (

*a*-3

*b*)/2

*b*is a rational number soÂ âˆš5Â should be a rational number ButÂ âˆš5Â is aÂ irrational number so it contradicts.

Hence, 3 + 2âˆš5Â is a irrational number.

3. Prove that the following are irrationals:

(i) 1/âˆš2 (ii) 7âˆš5Â (iii) 6 + âˆš2

**Answer**

(i) Let take that 1/âˆš2Â is a rational number.

So we can write this number as

1/âˆš2Â =

*a*/

*b*

Here

*a*and

*b*are two co prime number and

*b*is not equal to 0

Multiply byÂ âˆš2Â both sides we get

1 = (

*a*âˆš2)/

*b*

Now multiply by

*b*

*b*=

*a*âˆš2

divide by a we get

*b*/

*a*=Â âˆš2

Here

*a*and

*b*are integer so b/a is a rational number soÂ âˆš2Â should be a rational number ButÂ âˆš2Â isÂ aÂ irrational number so it contradicts.

Hence, 1/âˆš2Â is a irrational number

(ii) Let take that 7âˆš5Â is a rational number.

So we can write this number as

7âˆš5Â =

*a*/

*b*

Here

*a*and

*b*are two co prime number and

*b*is not equal to 0

Divide by 7 we get

âˆš5Â =Â

*a*/(7

*b*)

Here

*a*and

*b*are integer so

*a*/7

*b*is a rational number soÂ âˆš5Â should be a rational number butÂ âˆš5Â isÂ aÂ irrational number so it contradicts.

Hence, 7âˆš5Â is a irrational number.

(iii) Let take that 6 +Â âˆš2Â is a rational number.

So we can write this number as

6 +Â âˆš2Â =

*a*/

*b*

Here a and b are two co prime number and b is not equal to 0

Subtract 6 both side we get

âˆš2Â =

*a*/

*b*â€“ 6

âˆš2Â = (

*a*-6

*b*)/

*b*

Here

*a*and

*b*are integer so (a-6

*b*)/

*b*is a rational number soÂ âˆš2Â should be a rational number.

ButÂ âˆš2Â isÂ aÂ irrational number so it contradicts.

Hence, 6 +Â âˆš2Â is a irrational number.

Hence, 6 +Â âˆš2Â is a irrational number.

Page No: 17

**Exercise 1.4**

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) 17/8Â

(iii) 64/455Â

(iv) 15/1600Â

(v) 29/343Â

(vi) 23/2

^{3Â }Ã—Â 5^{2}Â
(vii) 129/2

^{2Â }Ã—Â 5^{7Â }Ã—Â 7^{5}Â
(viii) 6/15Â

(ix) 35/50Â

(x) 77/210

(i) 13/3125

Factorize the denominator we get

3125 =5Â Ã—Â 5Â Ã—Â 5Â Ã—Â 5Â Ã—Â 5 = 5

**Answer**(i) 13/3125

Factorize the denominator we get

3125 =5Â Ã—Â 5Â Ã—Â 5Â Ã—Â 5Â Ã—Â 5 = 5

^{5}
So denominator is in form of 5

^{m}so itÂ is terminating .(ii) 17/8

Factorize the denominator we get

8 =2Â Ã—Â 2Â Ã—Â 2 = 2

^{3}

So denominator is in form of 2

^{m}Â so itÂ is terminating .

(iii) 64/455

Factorize the denominator we get

455 =5Â Ã—Â 7Â Ã—Â 13

There are 7 and 13 also in denominator so denominator is not in form of 2

^{m}Â Ã—Â 5

^{n}Â . so itÂ is not terminating.

(iv) 15/1600

Factorize the denominator we get

1600 =2Â Ã—Â 2Â Ã—Â 2Â Ã—2Â Ã—Â 2Â Ã—Â 2Â Ã—Â 5Â Ã—Â 5 = 2

^{6}Â Ã—Â 5

^{2}

so denominator is in form of 2

^{m}Â Ã—Â 5

^{n}

Hence itÂ is terminating.

(v) 29/343

Factorize the denominator we get

343 = 7Â Ã—Â 7Â Ã—Â 7 = 7

^{3}

There are 7 also in denominator so denominator is not in form of 2

^{m}Â Ã—Â 5

^{n}

Hence it is non-terminating.

(vi) 23/(2

^{3}Â Ã—Â 5

^{2})

Denominator is in form of 2

^{m}Â Ã—Â 5

^{n}

Hence it is terminating.

(vii) 129/(2

^{2}Â Ã—Â 5

^{7}Â Ã—Â 7

^{5})

Denominator has 7 in denominator so denominator is not in form of 2

^{m}Â Ã—Â 5

^{n}

Hence it is none terminating.

(viii) 6/15

divide nominator and denominator both by 3 we get 2/5

Denominator is in form of 5

^{m}so it is terminating.

(ix) 35/50 divide denominator and nominator both by 5 we get 7/10

Factorize the denominator we get

10=2Â Ã—Â 5

So denominator is in form of 2

^{m}Â Ã—Â 5

^{n}Â so it is terminating.

(x) 77/210

simplify it by dividing nominator and denominator both by 7 we get 11/30

Factorize the denominator we get

30=2Â Ã—Â 3Â Ã—Â 5

Denominator has 3 also in denominator so denominator is not in form of 2

^{m}Â Ã— 5

^{n}

Hence it is none terminating.

Page No: 18

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

Page No: 18

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

**Answer**

(i) 13/3125 = 13/5

^{5}= 13Ã—2

^{5}/5

^{5}Ã—2

^{5}= 416/10

^{5}= 0.00416 (ii) 17/8 = 17/2

^{3}= 17Ã—5

^{3}/2

^{3}Ã—5

^{3}= 17Ã—5

^{3}/10

^{3}= 2125/10

^{3}= 2.125

(iv) 15/1600 = 15/2

^{4}Ã—10

^{2}= 15Ã—5

^{4}/2

^{4}Ã—5

^{4}Ã—10

^{2}= 9375/10

^{6}= 0.009375

(vi) 23/2

^{3}5

^{2}= 23Ã—5

^{3}Ã—2

^{2}/2

^{3}5

^{2}Ã—5

^{3}Ã—2

^{2}= 11500/10

^{5}= 0.115

(viii) 6/15 = 2/5 = 2Ã—2/5Ã—2 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7.

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*you say about the prime factors of

*q*?

(i) 43.123456789

(ii) 0.120120012000120000...

(iii) 43.123456789**Answer**

(i) Since this number has a terminating decimal expansion, it is a rational number of the form

*p*/q, and

*q*is of the formÂ 2

^{m}Â Ã— 5

^{n}.

(iii) Since the decimal expansion is non-terminating recurring, the given number is a rational number of the formÂ

*p*/q, andÂ

*q*Â is not of the formÂ 2

^{m}Â Ã— 5

^{n}.

**Go Back To NCERT Solutions for Class 10 Maths**

**NCERT Solutions for Class 10 Maths Chapter 1 - Real Numbers**

**In chapter 1 Real Numbers, we will restart our explorationÂ of the world of real numbers. We will study Euclidâ€™s division algorithm and the Fundamental Theorem of Arithmetic. Also, we will see decimal representation of real numbers.**

Euclid's Division Algorithm is a technique to compute HCF or GCD (greatest common divisor) of two given positive integers saya and b (a > b) by successive use of Euclid's division lemma.

The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. It says very composite number can be factorised as a product of primes, and this factorisation is unique, apart from the order in which prime factors occur.

We have also provided Notes of Chapter 1 Real Numbers Class 10 Maths, MCQ Questions of Chapter 1 Real Numbers which will supplement your preparation in a better way.

There are total four exercises in this chapter. We have provided Class 10 Maths Solutions of every exercises step by step. You can check solutions of each exercises just by clicking on the links given below.

CBSE NCERT Solutions are best way to check your understanding of any chapter as it contains all the basic to advance questions. Studyrankers experts have solved all the questions so you can check them whenever you find it difficult.

### NCERT Solutions for Class 10 Maths Chapters:

**FAQ onÂ Chapter 1 Real Numbers**

#### How many exercises in Chapter 1 Real Numbers

There are only 4 exercises in the chapter. In order to solve problem in a better way, first you need to understand the concepts. We have provided NCERT Solutions for Chapter 1 Real Numbers Class 10 MathsÂ of every question exercisewise in step by step manner so you can score more in the examinations. The answers of every questions is provided in step by step manner so a student do not have to face difficulty in getting them.

#### What is Lemma?

A lemma is a proven statement used for proving another statement..

#### What do you mean by Fundamental Theorem of Arithmetic.

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

#### What is Algorithm?

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.