1 : Real numbers

Real Numbers

Chapter 1. REAL NUMBERS

Class 10 Maths Chapter 1 Real numbers Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question and Solutions :

Class 10 Real Numbers SECTION = A

Question : If the LCM of and 18 is 36 and the HCF of and 18 is 2 , then

(a) 2

(b) 3

(d) 1

Solution : (c) 4

[Here, ,

We have,

]

Question : If the HCF of 65 and 117 is expressible in the form , then the value of is :

(a) 4

(b) 2

(d) 3

Solution: (b) 2

[Since, 65 = 5 × 13

And 117 = 3 × 3 × 13

HCF (70 , 125) = 13

A/Q ,

]

Question : The largest number which divides 70 and 125 , leaving remainders 5 and 8 respectively , is :

(a) 13

(b) 65

(d) 1750

Solution: (a) 13 .

[Since, 70 – 5 = 65 = 5 × 13

And 125 – 8 = 117 = 3 × 3 × 13

HCF (70 , 125) = 13 ]

Question : If two positive integers and are written as and are prime numbers, then HCF is :

(a)

(b)

(c)

(d)

Solution: (b) .

[Product of the smallest power of each common prime factor in the numbers .

HCF ]

Question : If two positive integers and can be expressed as and are prime numbers, then HCF is :

(a)

(b)

(c)

(d)

Solution : (a)

[ Product of the smallest power of each common prime factor in the numbers .

Here, and

HCF]

Question: According to Euclid’s Division lemma, given two positive integers a and b , there exist unique integers q and r such that - [ SEBA 2016]

(a)

(b)

(c)

(d)

Solution: (c)

Question: 120 can be expressed as a product of its prime factors as : [CBSE 2020 basic]

(a)

(b)

(c)

(d)

Solution : (d)

[ We have , ]

Question: The H.C.F of 8 , 9 , 25 is :

(a) 8

(b) 9

(d) 1

Solution: (d) 1

[ We have , ; ;

HCF (8 , 9 , 25) = 1 ]

Question: Which of the following is an irrational number ? [SEBA 2020]

(a) 0.142857142857142857……………

(b)

(d)

Solution: (c) π

Question: Which of the following is not irrational ?

(a)

(b)

(c)

(d)

Solution: (c)

[ We have, (c) is not irrational ]

Question: Which one of the following is a non-terminating repeating decimal ? [SEBA 2019]

(a)

(b)

(c)

(d)

Solution: (c)

[ We have, ]

Question: Given three statement such as :

(i) The sum or difference of a rational and an irrational number is irrational .

(ii) The product and quotient of a non-zero rational and irrational number is irrational .

(iii) The product of the two numbers is not equal to the product of their HCF and LCM .

(iv) The LCM is equal to the product of the greatest power of each common prime factor in the numbers.

(a) (i) , (ii) and (iii) are correct .

(b) (i) , (ii) and (iv) are correct .

(d) (ii) , (iii) and (iv) are correct .

Solution: (d) (ii) , (iii) and (iv) are correct .

Question: In the following real numbers, which one is non-terminating repeating decimal expansion ? [ SEBA 2015]

(a)

(b)

(c)

(d)

Solution: (c)

Question: Which number is not divisible by 11 ?

(a) 253

(b) 1771

(d) 91

Solution: (d) 196

[ We have, 253 = 11×23 ; 1771 = 7 × 11 × 23 , 286 = 2 × 11 × 13 ; 91 = 7 × 13 ]

Question: The largest number which divides 60 and 75 , leaving remainders 8 and 10 respectively,is

(a) 260

(b) 75

(d) 13

Solution: (a) 260

[ We have, 60 – 8= 52 = 2 × 2 × 13 ; 75 - 10 = 65 = 5 × 13

LCM (52 , 65) = 2 × 2 × 5 × 13 = 260 ]

Question: If LCM (91 , 26) = 182 , then HCF (91 , 26) is :[SEBA 2016]

(a) 13

(b) 26

(d) 9

Solution: (a) 13

[ We have, ]

Question: The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is

(a) 840

(b) 2520

(d) 420

Solution: (b) 2520

[ We have , LCM (1, 2 , 3 , 4 , ………., 10)

]

Question: When a number is divided by 7 , its remainder is always :

(a) greater than 7

(b) at least 7

(d) at most 7

Solution: (a) greater than 7 .

Question: If HCF (16,y) = 8 and LCM(16, y) = 48 , then the value of y is :

(a) 24

(b) 16

(d) 48

Solution: (a) 24

[ A/Q , HCF (16 ,y) × LCM (16 , y) = 16 × y

⇒ 16 × y = 8 × 48

⇒ y = 8 × 3 = 24 ]

Question: Find the least number of 3 digits , that will gives us remainder of 9 when divided by 2 and 5 respectively .

(a) 121

(b) 141

(d) 109

Solution: (d) 109

[ We have , 109 = 10 × 10 + 9 ]

Question: The ratio between the LCM and HCF of 5 , 15 , 20 is :

(a) 9 : 1

(b) 4 : 3

(d) 12 : 1

Solution: (d) 12 : 1

[ We have , 5 = 1 × 5 ; 15 = 3 × 5 ; 20 =

HCF (5 , 15 , 20) = 5

LCM (5 , 15 , 20) =

So, ]

Question: HCF of 52× 42 and 35 × 65 is :

(a) 52 × 35

(b) 5 × 33

(d) 7 × 13

Solution: (d) 7 × 13

[ We have, 52 × 42 = 2 × 2 × 13 × 2 × 3 × 7 =

and 35 × 65 = 5 × 7 × 5 × 13 = 52 × 7 × 13

HCF ( 52 × 42 , 35 × 65) = 7 × 13 ]

Question: Which one of the following is a rational number ? [SEBA 2018]

(a)

(b)

(c)

(d)

Solution: (d)

[ We have, ]

Question: The smallest number by which should be multiplied so as to get rational number is : [ SEBA 2017]

(a)

(b)

(c)

(d) 3

Solution: (c)

[ We have, ]

Filled in the blanks :

Question: The HCF of two number is 27 and their LCM is 162 , if one of the number is 54 , then the other number is ......................................... .

Solution: 81

[ The other number ]

Question: If product of two numbers is 2366 and their LCM is 26 , then their HCF is ...............................

Solution: 91

[ We have, ]

Question: The HCF and LCM of two numbers are 33 and 264 respectively , When the first number is completely divided by 2 and the quotient is 33 , then other number is...................................... .

Solution: 132

[ We have ,

Other number ]

Question: If the prime factorisation of a natural number is , then number is.........................

Solution: 8232

[ We have , ]

Question: is ( irrational / a rational number ) .

Solution: a rational number .

[ We have , is a rational ]

Question: If is expressed in the form , then values of is .................................. .

Solution: 4

[ We have , ]

Answers following the question :

Question: Find the LCM and HCF of 6 and 20 by prime factorization method .

Solution: We have, and

and .

Question: Express the number 0.104 in the form of rational number .

Solution: We have ,

is the form of rational number .

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

Solution: We have ,

Question: The LCM of two number is 182 and their HCF is 13 . If one of the numbers is 26 , find the other . [CBSE 2020 standard]

Solution: We have , LCM × HCF one number × other number

The other number

Question: Find HCF of 1001 and 385 .

Solution: We have,

and

Question: Find the LCM of the two digit smallest prime and smallest odd composite natural number .

Solution: The two digit smallest prime number is 11 and the smallest odd composite number is 15 .

So,

Question: Decompose 32760 into prime factors .

Solution: We have ,

Question: What is the HCF of smallest prime number and the smallest composite number ? [CBSE 2018]

Solution: The smallest prime number is 2 and the smallest composite number is 4 .

So,

Question: Find a rational number between and

Solution: We have , and

Thus , the rational number between and is .

Question: Find one irrational number between and .

Solution: We have , and

Thus , the one irrational number between and is 1.21021002100021………….. .

Question: If HCF(336 , 54) = 6 , find LCM (336 , 54) . [CBSE 2019]

Solution: We have ,

Question : Given 29 and 53 are two number, then HCF (29 , 53) will be :

(a) 29 × 53

(b) 53

(d) 1

Solution : (d) 1 [since 29 and 53 both are prime numbers .] .

Question : What is the LCM of 36 , 72 and 120 ?

(a) 620

(b) 720

(d) 260

Solution: (c) 360 .

Question : The ratio of the two number is 3 : 7 . If the HCF is 13 , then the number are :

(a) 29 , 91

(b) 39 , 72

(d) 29 , 82

Solution: (c) 39 , 92

[ let the two number are and .

and

HCF

and ]

Class 10 Real Numbers SECTION = B

Question : Prove that is an irrational number .

Solution: Let us assume , to the contrary , that is rational . We can find co-prime and ( ) such that

Since 7 , and are integers , is rational and so, is rational .But this contradicts the fact that is irrational .So , we conclude that is irrational .

Q1. Find the LCM and HCF of 120 and 144 by using Fundamental theorem of Arithmetic . [CBSE2012]

Solution: We have ,

and

Q2. Check whether can end with the digit 0 for any natural number .

Solution: We have,

Prime factors of are only 2 and 3 .

So, the prime factors of does not contain , where and are positive integers .

Therefore, does not end with the digit 0 .

Q3. Prove that is irrational . [SEBA 2016]

Solution: let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that

Since, 2 , and are integers ,is rational and so, is rational .

But this contradicts the fact that is irrational . So , is irrational .

Q4. Find the largest number that will divides 398 , 436 and 542 leaving remainders 7 , 11 and 15 respectively .

Solution: We have ,

398 - 7 = 391 = 17 × 23

436 - 11 = 425 = 5 × 5 × 17

542 - 15 = 527 = 17 × 31

HCF(391 , 425 , 527) = 17

Therefore, 17 is the largest number that will divide given numbers.

Q5. Express 5050 as product of its prime factors . Is it unique ?

Solution: We have,

It is not unique .

Q6. Check whether can end with the digit 0 for any natural number .

Solution: Since,

Prime factors of are only 2 and 3 .

So, the prime factors of does not contain , where and are positive integers .

Therefore, does not end with the digit 0 .

Q7. Find the HCF and LCM of 12 , 15 and 21 ,using the prime factorization method .

Solution: We have ,

and

Question : Prove that is irrational .

Solution: let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that,

Since and are integers , is rational and so, is rational .

But this contradicts the fact that is irrational . So, is irrational .

Question : Write the smallest number which is divisible by both 306 and 657 . [CBSE 2019]

Solution: We have , 306 = 2 × 3 × 3 × 17

and 657 = 3 × 3 × 73

HCF of 306 and 657 is 9 .

Question : A bell rings at every 18 seconds , another bell rings at every 60 seconds . If these two bells ring simultenously an instant , then find after how many seconds will the bells ring simultenously again .

Solution : We need to determine the LCM of the time intervals between the bell rings.

The first bell rings every 18 seconds, and the second bell rings every 60 seconds.

Find the LCM of 18 and 60.

We have,

and

LCM

So, the bells will ring simultaneously again after 180 seconds

Question : A radio station plays ‘Assam Sangeet ’ once every two days . Another radio station plays the same song once every three days . How many times in 30 days will both the radio stations play the same song on the same day .

Solution : To find out how many times both radio stations will play the same song on the same day in 30 days, we need to determine the LCM of their respective schedules.

One radio station plays the song once every two days, and the other plays it once every three days.

We have,

and

LCM

So, the LCM of the two radio stations' schedules is 6 days.

This means they will play the same song on the same day every 6 days.

In 30 days, they will play the same song on the same day times.

Question: 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 ?

Solution : The required number of minutes is LCM (18 , 12) .

We find the LCM by prime factorization method , we have

and

Therefore, the LCM (18 , 12)

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

Class 10 Real Numbers SECTION = C

Q1. Find LCM and HCF of 6 , 72 and 120 by prime factorization method . Is HCF × LCM of three numbers equal to the product of the three numbers ?

Solution: We have,

LCM (6 , 72 , 120)

HCF (6 , 72 , 120)

LCM (6 , 72 , 120 ) × HCF (6 , 72 , 120 )

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

Solution: We have,

is a composite number.

and

is a composite number .

So, the product of three numbers is not equal to the product of their HCF and LCM .

Q2. Find the LCM and HCF of 144 , 112 and 418 by prime factorization .

Solution : We have,

LCM (144 , 112 , 418)

HCF (144 , 112 , 418)

Q4. Prove that is an irrational .

Solution: let us assume , to the contrary , that is rational .

We can find co-prime and ( ) such that

–

Since , 2 , and are integers , is rational and so, is rational . But this contradicts the fact that is irrational . So , is irrational .

Q5. Given that is an irrational , prove that is an irrational number . [CBSE 2018]

Solution: let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that,

Since 3 , and are integers , is rational and so, is rational .

Given is irrational . So, is irrational .

Q6. Prove that is irrational and hence show that is also irrational .

Solution: let us assume , to the contrary that is rational .

So, we can find integers and () such that [ and are co-prime]

Therefore , is divisible by 5 and so is also divisible by 5 .

let , , for some integer .

From and , we get

Therefore , is divisible by 5 and so is also divisible by 5 .

So, and have at least 5 as a common factor .

But this contradicts the fact that and are co-prime . So, is irrational .

Since is an irrational number . Therefore , is also irrational .

Q7. Find HCF and LCM of 404 and 96 and verify that HCF × LCM Product of the two given numbers . [SEBA 2019 , CBSE 2018]

Solution: We have, and

HCF (96 , 404)

LCM (96 , 404)

HCF (96 , 404) × LCM (96 , 404)

Verified.

Q8. Rajesh has two vessels containing 720 ml and 405 ml of milk respectively . Milk from these containers is poured into glasses of equal capacity to their brim . Find the minimum number of glasses that can be filled .

Solution: We have ,

Therefore, the number of glasses is 45 .

Q11. Prove that is irrational .

Solution: let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that ,

Since, 2 , and are integers , is rational and so, is rational .

But this contradicts the fact that is irrational . So, is irrational .

CBSE Previous Year Questions and Solution , 2007

1. Real numbers

Q2. Calculate the HCF of 33×5 and 3²×5² . [2007]

Answer : We have, 33×5 and 3²×5²

Therefore, the HCF of 33×5 and 3²×5² is 32×5=9×5=45 .

CBSE Previous Year Questions and Solution , 2008

Q1. Complete the missing in the following factor tree : [2008] 1M

Solution: We have,

Therefore, the missing numbers are 21 and 42 .

11 Prove that is an irrational number . [2008 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and ( ) such that

Since and are integers, is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

CBSE Previous Year Questions and Solution , 2009

Q2. Find the (HCF × LCH) for the numbers 100 and 190 . [2009] [1M]

Answer : We have, and

Therefore,

Q26. Prove that is an irrational number . [2009 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and () such that

Since and are integers, is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

Q20. Prove that is an irrational number . [2009 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and () such that

Since and are integers, is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

CBSE Previous Year Questions and Solution , 2010

Q2. Find the least number that is divisible by all numbers between 1 and 10 (both inclusive) . [2010]

Solution : We have, ; ;; ; ; ; ; and

Q27. Prove that is an irrational number . [2010 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and () such that

Since and are integers, is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

Q27. Prove that is an irrational number . [2010 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and ( ) such that

Since and are integers , is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

Q27. Prove that is an irrational number . [2010 3M]

Solution : Let us assume ,to the contrary ,that is rational . We can find co-prime and ( ) such that

Since and are integers, is rational and so, is rational .But this contradicts the fact that is irrational .Therefore, is irrational .

Q1. Write whether on simplification gives a rational or an irrational number . [2010]

Solution : We have,

is a rational number.

Q3. Prove that is irrational . [CBSE 2009 , 2010]

Solution: let us assume , to the contrary that is rational .

So, we can find integers and () such that , [ and are co-prime]

Therefore , is divisible by 3 and so is also divisible by 3 .

let , , for some integer .

From and , we get

Therefore , is divisible by 3 and so is also divisible by 3 .

So, and have at least 3 as a common factor .

But this contradicts the fact that and are co-prime . So, is irrational .

Q3. Prove that is an irrational number . [2009,2010]

Solution : let, us assume to the contrary that is rational . There exists co-prime integers and () such that

Therefore , is divisible by 5 . So, is also divisible by 5 .

Let , for some integer .

From and , we get

Therefore , is divisible by 5 . So, is also divisible by 5 .

Therefore, and have at least as a common factor . But this contradicts the fact that and are co-prime . This contradiction has arisen because of our incorrect assumption that is rational . So, we conclude that is irrational .

Q1. Prove that is an irrational number . [2010 ,19 3M]

Solution : let, us assume to the contrary that 2 is rational . There exists co-prime integers and () such that

Therefore, is divisible by 2 . So, is also divisible by 2 .

Let , for some integer c .

From and , We get

Therefore , is divisible by 2 . So, is also divisible by 2 .

CBSE Previous Year Questions and Solution , 2012

Q23. Find the HCF and LCM of 90 and 144 by the method of prime factorization . [2012]

Solution : We have,

and

CBSE Previous Year Questions and Solution , 2013

Q22. Given that HCF(306,1314) = 18 . Find LCM (306,1314) . [2013]

Solution: Here, and

We have,

CBSE Previous Year Questions and Solution , 2015

Q24. Show that is an irrational number . [2015]

Solution : Let us assume , to the contrary , that is rational . We can find co-prime and ( ) such that

Since 5 , and are integers, is rational and so, is rational .But this contradicts the fact that is irrational . Therefore, is irrational .

CBSE Previous Year Questions and Solution , 2018

Q4. What is the HCF of smallest prime number and smallest composite number ? [2018] 1M

Solution : Since, the smallest prime number is 2 and smallest composite number is 4 .

We have, and

Q25. Given that, is irrational , prove that is an irrational number . [2018 2M]

Solution : Let us assume , to the contrary, that is rational . We can find co-prime and ( ) such that

Since and are integers, is rational and so, is rational .But, given is an irrational . Therefore, is irrational .

Q28. Find HCF and LCM of 404 and 96 and verify that Product of the two given numbers .[2018 3M]

Solution : We have, and

CBSE Previous Year Questions and Solution , 2019

Q5. Find a rational number between and . [2019] 1M

Solution : We have, and

Therefore, the rational number between and is 1.5 .

Q29. Prove that is an irrational number . [2019 3M]

Solution: Given : 2010

Q23. If , find . [2019 old]

Solution : Here, and

We have,

Q45. Write the smallest number which is divisible by both 306 and 657 . [2019]

Solution : We have,

and

Therefore, the smallest number which is divisible by both 306 and 657 is 9 .

CBSE Previous Year Questions and Solution , 2020

Q6. HCF of 144 and 198 is [2020] 1M

(a) 9 (b) 18 (c) 6 (d) 12

Solution : (a) 9

[ We have, and

HCF of 144 and 198 is 9 ]

Q7. 225 can be expressed as [2020] [1M]

(a) (b) (c) (d)

Solution : (c) .

[We have, ]

Q8. is [2020] 1M

(a) an Integer (b) a rational number (c) an irrational number (d) a natural number is [2020] 1M

Solution : (b) a rational number .

[ We have , ]

Q9. The total number of factors of a prime number is [2020 1M]

(a) 1 (b) 0 (c) 2 (d) 3

Solution : (c) 2 [The factor of a any prime number are 1 and itself ( i.e., 2). ]

Q10 The HCF and the LCM of 12 ,21 and 15 respectively are [2020 1M]

(a) 3 , 140 (b) 12 , 420 (c) 3 , 420 (d) 420 , 3

Solution : (c) 3 , 420

[ We have, , and

, ]

Q30. Given that, is irrational , prove that ( ) is an irrational number . [2020 3M]

Solution : Let us assume , to the contrary, that is rational . We can find co-prime and ( ) such that

Since and are integers, is rational and so, is rational . But, given is an irrational . Therefore, is irrational .

Q31. An army contingent of 612 members is to march behind an army band of 48 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 ? [2020 3M]

Solution : .We find the maximum number of column of (616 , 32) .

We have

Thus , the maximum number of column is 8 .

CBSE Previous Year Questions and Solution , 2021

Q11. HCF of 92 and 152 is [2021 1M]

(a) 4 (b) 19 (c) 23 (d) 57

Solution : (a) 4

[We have

and

]

Q13. can also be written as [2021 1M]

(a) 5.213213213 ….

(b) 5.2131313…..

(d) 5.213

Solution : (a) 5.213213213…….

Q14. HCF of two consecutive even numbers is [2021 1 M]

(a) 0 (b) 1 (c) 2 (d) 4

Solution : (c) 2

[For Example : ; ]

Q16. The (HCF × LCM) for the numbers 50 and 20 is [2021 1M]

(a) 1000 (b) 50 (c) 100 (d) 500

Solution : (a) 1000

[ We have, and

and

]

Q17. For which natural n , ends with digit zero ? [2021 1 M]

(a) 6 (b) 5 (c) 0 (d) None

Solution : (a) 6

[ If , then ; If , then ]

17. The exponent of 5 in the prime factorization of 3750 is [2021 1M]

(a) 3 (b) 4 (c) 5 (d) 6

Solution : (b) 4

[We have,

So, The exponent of 5 in the prime factorization of 3750 is 4. ]

Q19. What is the greatest possible speed at which a girl can walk 95 m and 171 m in an exact number of minutes ? [2021 1M]

(a) 17m/min (b) 19m/min (c) 23m/min (d) 13m/min

Solution : (b) 19m/min

[ We have, and

]

Q20. Three alarm clocks ring their alarms at regular intervals of 20 min , 25 min and 30 min respectively . If they first beep together at 12 noon, at what time will they beep again for the first time ? [2021 1M]

(a) 4 :00 pm (b) 4:30 pm (c) 5:00 pm (d) 5:30 pm

Solution : (c) 5:00 pm

[ We have, and

]

Q21. The greatest number which when divides 1251, 9377 and 15628 leave remainder 1 , 2 and 3 respectively is [2021 1M]

(a) 575 (b) 450 (c) 750 (d) 625

Solution: (d) 625

[ We have, ,

And

]

Q22. If a and b are two co-prime numbers, then and are [2021 1M]

(a) Co-prime (b) Not co-prime (c) Even (d) odd

Solution : (a) co-prime

[ For example : Let and . So, and

]

Q23. In is a natural number, then always ends with [2021 1M]

(a) 1 (b) 4 (c) 3 (d) 2

Solution : (d) 2

[ We have,

If , then

If , then ]

Q24. The LCM of two number is 2400 . which of the following cannot be their HCF ? [2021 1 M]

(a) 300 (b) 400 (c) 500 (d) 600

Solution : (c) 500

[ We have, ]