R S AGGARWAL AND V AGGARWAL Solutions for Class 10 Maths Chapter 1 - Real Numbers

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Chapter 1 - Real Numbers Exercise Ex. 1A

Question 1

What do you mean by Euclid's division lemma?

Solution 1

For any two given positive integers a and b there exist unique whole numbers q and r such that  

 

Here, we call 'a' as dividend, b as divisor, q as quotient and r as remainder.

Dividend = (divisor quotient) + remainder

Question 2

A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.

Solution 2

By Euclid's Division algorithm we have:

Dividend = (divisor × quotient) + remainder

= (61 27) + 32 = 1647 + 32 = 1679

Question 3

By what number should 1365 be divided to get 31 as quotient and 32 as remainder?

Solution 3

By Euclid's Division Algorithm, we have:

Dividend = (divisor quotient) + remainder

Question 4

Using Euclid's algorithm, find the HCF of:

(i) 405 and 2520

(ii) 504 and 1188

(iii) 960 and 1575

Solution 4

(i)

On dividing 2520 by 405, we get

Quotient = 6, remainder = 90

2520 = (405 6) + 90 

Dividing 405 by 90, we get

Quotient = 4,

Remainder = 45

405 = 90 4 + 45

Dividing 90 by 45

Quotient = 2, remainder = 0

90 = 45 2

H.C.F. of 405 and 2520 is 45


(ii) Dividing 1188 by 504, we get

Quotient = 2, remainder = 180

1188 = 504 2+ 180

Dividing 504 by 180

Quotient = 2, remainder = 144

504 = 180 × 2 + 144

Dividing 180 by 144, we get

Quotient = 1, remainder = 36

Dividing 144 by 36

Quotient = 4, remainder = 0

H.C.F. of 1188 and 504 is 36


(iii) Dividing 1575 by 960, we get

Quotient = 1, remainder = 615

1575 = 960 × 1 + 615

Dividing 960 by 615, we get

Quotient = 1, remainder = 345

960 = 615 × 1 + 345

Dividing 615 by 345

Quotient = 1, remainder = 270

615 = 345 × 1 + 270

Dividing 345 by 270, we get

Quotient = 1, remainder = 75

345 = 270 × 1 + 75

Dividing 270 by 75, we get

Quotient = 3, remainder =45

270 = 75 × 3 + 45

Dividing 75 by 45, we get

Quotient = 1, remainder = 30

75 = 45 × 1 + 30

Dividing 45 by 30, we get

Remainder = 15, quotient = 1

45 = 30 × 1 + 15

Dividing 30 by 15, we get

Quotient = 2, remainder = 0

H.C.F. of 1575 and 960 is 15


Question 5

Show that every positive integer is either even or odd.

Solution 5

Question 6

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

Solution 6

Question 7

Show that any positive odd integer is of the form (4m + 1) or (4m + 3), where in is some integer.

Solution 7

Chapter 1 - Real Numbers Exercise Ex. 1B

Question 1(i)

Using prime factorization, find the HCF and LCM of:

36, 84

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(i)

Question 1(ii)

Using prime factorization, find the HCF and LCM of:

23, 31

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(ii)

Question 1(iii)

Using prime factorization, find the HCF and LCM of:

96, 404

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(iii)

Question 1(iv)

Using prime factorization, find the HCF and LCM of:

144,198

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(iv)

Question 1(v)

Using prime factorization, find the HCF and LCM of:

396, 1080

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(v)

Question 1(vi)

Using prime factorization, find the HCF and LCM of:

1152, 1664

In each case, verify that:

HCF x LCM = product of given numbers.

Solution 1(vi)

Question 5(i)

Using prime factorization, find the HCF and LCM of:

8, 9, 25

Solution 5(i)

Question 5(ii)

Using prime factorization, find the HCF and LCM of:

12, 15, 21

Solution 5(ii)

Question 5(iii)

Using prime factorization, find the HCF and LCM of:

17, 23, 29

Solution 5(iii)

Question 5(v)

Using prime factorization, find the HCF and LCM of:

30, 72, 432

Solution 5(v)

Question 5(vi)

Using prime factorization, find the HCF and LCM of:

21, 28, 36, 45

Solution 5(vi)

Question 6

Is it possible to have two numbers whose HCF is 18 and LCM is 760? Give reason.

Solution 6

Question 7

Find the simplest form of:

(iv)begin mathsize 12px style 368 over 496 end style

Solution 7

(iv)

begin mathsize 12px style 368 over 496 equals fraction numerator 2 cross times 2 cross times 2 cross times 2 cross times 23 over denominator 2 cross times 2 cross times 2 cross times 2 cross times 31 end fraction
space space space space space space space space space equals 23 over 31 end style

Question 8

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.

Solution 8

Question 9

The HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is 725, find the other.

Solution 9

Question 12

The HCF of two numbers is 18 and their product is 12960. Find their LCM.

Solution 12

Question 15

Find the largest number which divides 438 and 606, leaving remainder 6 in each case.

Solution 15

Question 16

Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively.

Solution 16

Subtracting 5 and 7 from 320 and 457 respectively:

 320 - 5 = 315,

457 - 7 = 450

Let us now find the HCF of 315 and 405 through prime factorization:

 

The required number is 45.

Question 17

Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case.

Solution 17

Question 18

Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.

Solution 18

Question 19

Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520.

Solution 19

Question 20

Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.

Solution 20

Question 24

Find the missing numbers in the following factorisation:

Solution 24

By going upward

5 11= 55

55 3= 165

1652 = 330

330 2 = 660

Question 25

In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively. Find the minimum number of rooms required, if in each room, the same number of participants are to be seated and all of them being in the same subject.

Solution 25

Question 26

Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subjectwise and the height of each stack is the same. How many stacks will be there?

Solution 26

Let us find the HCF of 336, 240 and 96 through prime factorization:

 

 

 

Each stack of book will contain 48 books

Number of stacks of the same height

Question 27

Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank?

Solution 27

The prime factorization of 42, 49 and 63 are:

42 = 2 3 7, 49 = 7 7, 63 = 3 3 7

H.C.F. of 42, 49, 63 is 7

Hence, greatest possible length of each plank = 7 m

Question 28

Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm.

Solution 28

7 m = 700cm, 3m 85cm = 385 cm

12 m 95 cm = 1295 cm

Let us find the prime factorization of 700, 385 and 1295:

Greatest possible length = 35cm

Question 29

Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.

Solution 29

Let us find the prime factorization of 1001 and 910:

1001 = 11 7 13

910 = 2 5 7 13

 

H.C.F. of 1001 and 910 is 7 13 = 91

Maximum number of students = 91

Question 30

Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad.

Solution 30

 

 

Question 31

Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.

Solution 31

Let us find the LCM of 64, 80 and 96 through prime factorization:

 

 

 

L.C.M of 64, 80 and 96

=

 Therefore, the least length of the cloth that can be measured an exact number of times by the rods of 64cm, 80cm and 96cm = 9.6m

Question 32

An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?

Solution 32

Interval of beeping together = LCM (60 seconds, 62 seconds)

The prime factorization of 60 and 62:

60 = 30 2, 62 = 31 2

L.C.M of 60 and 62 is 30 31 2 = 1860 sec = 31min

electronic device will beep after every 31minutes

After 10 a.m., it will beep at 10 hrs 31 minutes

Question 33

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 hours, then at what time will they again change simultaneously?

Solution 33

 

 

Question 34

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together?

Solution 34

Chapter 1 - Real Numbers Exercise Ex. 1C

Question 1(i)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

  

Solution 1(i)

Question 1(ii)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

Solution 1(ii)

Question 1(iii)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

  

Solution 1(iii)

Question 1(iv)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

Solution 1(iv)

Question 1(v)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

Solution 1(v)

Question 1(vi)

Without actual division, show that each of the following rational number is a terminating decimal. Express each in decimal form.

Solution 1(vi)

Question 2(i)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(i)

Question 2(ii)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(ii)

Question 2(iii)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(iii)

Question 2(iv)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(iv)

Question 2(v)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(v)

Question 2(vi)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(vi)

Question 2(vii)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(vii)

Question 2(viii)

Without actual division, show that each of the following rational number is a nonterminating repeating decimal.

Solution 2(viii)

Question 3

Express each of the following as a fraction in simplest form:

Solution 3


Chapter 1 - Real Numbers Exercise Ex. 1D

Question 1

Define (i) rational numbers, (ii) irrational numbers, (iii) real numbers.

Solution 1

Question 2

Classify the following numbers as rational or irrational:

Solution 2

Question 3

Prove that each of the following numbers is irrational:

Solution 3







Question 7

Solution 7

Question 8

Solution 8

Question 10

Solution 10

Question 11

Solution 11

Question 12

Prove that is irrational.

Solution 12

 

 

Question 13

Solution 13

Question 16

(i) Give an example of two irrationals whose sum is rational.

(ii) Give an example of two irrationals whose product is rational.

Solution 16

 

Question 17

State whether the given statement is true or false:

(i) The sum of two rationals is always rational.

(ii) The product of two rationals is always rational.

(iii) The sum of two irrationals is an irrational.

(iv) The product of two irrationals is an irrational.

(v) The sum of a rational and an irrational is irrational.

(vi) The product of a rational and an irrational is irrational.

Solution 17

(i) The sum of two rationals is always rational - True

(ii) The product of two rationals is always rational - True

(iii) The sum of two irrationals is an irrational - False

(iv) The product of two irrationals is an irrational - False

(v) The sum of a rational and an irrational is irrational - True

(vi) The product of a rational and an irrational is irrational - True

Chapter 1 - Real Numbers Exercise Ex. 1E

Question 1

State Euclid's division lemma.

Solution 1

Question 2

State fundamental theorem of arithmetic.

Solution 2

Question 3

Express 360 as product of its prime factors.

Solution 3

Question 4

If a and b are two prime numbers then find HCF(a, b).

Solution 4

Question 5

If a and b are two prime numbers then find LCM(a, b).

Solution 5

Question 6

If the product of two numbers is 1050 and their HCF is 25, find their LCM.

Solution 6

Question 7

What is a composite number?

Solution 7

A whole number that can be divided evenly by numbers other than 1 or itself.

Question 8

If a and b are relatively prime then what is their HCF?

Solution 8

Question 9

If the rational number  has a terminating decimal expansion, what is the condition to be satisfied by b?

Solution 9

Question 10

  

Solution 10

Question 11

Solution 11

Question 12

Show that there is no value of n for which (2n x 5n) ends in 5.

Solution 12

Question 13

Is it possible to have two numbers whose HCF is 25 and LCM is 520?

Solution 13

Question 14

Give an example of two irrationals whose sum is rational.

Solution 14

Question 15

Give an example of two irrationals whose product is rational.

Solution 15

Question 16

If a and b are relatively prime, what is their LCM?

Solution 16

Question 17

The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why?

Solution 17

Question 18

Express  as a rational number in simplest form.

Solution 18

Question 19

Express  as a rational number in simplest form

Solution 19

Question 20

Explain why 0.15015001500015 ... is an irrational number.

Solution 20

Question 21

  

Solution 21

Question 22

Write a rational number between and 2.

Solution 22

Question 23

Explain why  is a rational number.

Solution 23

Chapter 1 - Real Numbers Exercise MCQ

Question 1

Which of the following is a pair of co-primes?

(a) (14, 35)

(b) (18, 25)

(c) (31,93)

(d)(32, 62)

Solution 1

Question 2

If a = (22×33×54) and b = (23×32×5) then HCF (a, b) = ?

(a) 90

(b) 180

(c) 360

(d)540

Solution 2

Question 3

HCF of (23×32×5), (22×33×52) and (24×3×53×7) is

(a) 30

(b) 48

(c) 60

(d)105

Solution 3

Question 4

LCM of (23×3×5) and (24×5×7) is

(a) 40

(b) 560

(c) 1120

(d)1680

Solution 4

Question 5

The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?

(a) 36

(b) 45

(c) 9

(d)81

Solution 5

Question 6

The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is

(a) 8000

(b) 1600

(c) 320

(d)1605

Solution 6

Question 7

What is the largest number that divides each one of 1152 and 1664 exactly?

(a) 32

(b) 64

(c) 128

(d)256

Solution 7

Question 8

What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?

(a) 13

(b) 9

(c) 3

(d)585

Solution 8

Question 9

What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?

(a) 15

(b) 16

(c) 9

(d)5

Solution 9

Question 10

Solution 10

Question 11

Euclid's division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy

(a) 1 < r < b

(b) 0 < r  b

(c) 0 r  < b

(d)0 < r < b

Solution 11

Question 12

A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?

(a) 0

(b) 1

(c) 3

(d)5

Solution 12

Question 13

Which of the following is an irrational number?

(a)

(b) 3.1416

(c) 

(d) 3.141141114 …

Solution 13

Question 14

𝜋 is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 14

Question 15

(a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

Solution 15

Question 16

2.13113111311113… is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 16

Question 17

The number 3.24636363 … is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 17

Question 18

Which of the following rational numbers is expressible as a terminating decimal?

Solution 18

Question 19

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) four decimal places

Solution 19

Question 20

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d)four decimal places

Solution 20

Question 21

The number 1.732 is

(a) an irrational number

(b) a rational number

(c) an integer

(d)a whole number

Solution 21

Question 22

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a+b) is

(a) 2

(b) 3

(c) 5

(d)8

Solution 22

Question 23

(a) a rational number

(b) an irrational number

(c) a terminating decimal

(d)a nonterminating repeating decimal

Solution 23

Question 24

(a) a fraction

(b) a rational number

(c) an irrational number

(d)none of these

Solution 24

Question 25

(a) an integer

(b) a rational number

(c) an irrational number

(a) none of these

Solution 25

Question 26

What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)

(a) 100

(b) 1260

(c) 2520

(d) 5040

Solution 26

Chapter 1 - Real Numbers Exercise FA

Question 1

(a) a terminating decimal

(b) a nonterminating, repeating decimal

(c) a nonterminating and nonrepeating decimal

(d)none of these

Solution 1

Question 2

Which of the following has a terminating decimal expansion?

Solution 2

Question 3

On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if (3n - 1) is divided by 9?

(a) 1

(b) 2

(c) 3

(d)4

Solution 3

Question 4

Solution 4

Question 5

Show that any number of the form 4n, n N can never end with the digit 0.

Solution 5

Question 6

The HCF of two numbers is 27 and their LCM is 162. If one of the number is 81, find the other.

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Which of the following numbers are irrational?

Solution 9

Question 10

Solution 10

Question 11

Find the HCF and LCM of 12, 15, 18, 27.

Solution 11

Question 12

Give an example of two irrationals whose sum is rational.

Solution 12

Question 13

Give prime factorization of 4620.

Solution 13

Question 14

Find the HCF of 1008 and 1080 by prime factorization method.

Solution 14

Question 15

Solution 15

Question 16

Find the largest number which divides 546 and 764, leaving remainders 6 and 8 respectively.

Solution 16

Question 17

Solution 17

Question 18

Show that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q.

Solution 18

Question 19

Show that one and only one out of n, (n+2) and (n+4) is divisible by 3, where n is any positive integer.

Solution 19

Question 20

Solution 20