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

## Chapter 1 - Real Numbers Exercise Ex. 1A

What do you mean by Euclid's division lemma?

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

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

By Euclid's Division algorithm we have:

Dividend = (divisor × quotient) + remainder

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

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

By Euclid's Division Algorithm, we have:

Dividend = (divisor quotient) + remainder

Using Euclid's algorithm, find the HCF of:

(i) 405 and 2520

(ii) 504 and 1188

(iii) 960 and 1575

(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

Show that every positive integer is either even or odd.

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

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

## Chapter 1 - Real Numbers Exercise Ex. 1B

Using prime factorization, find the HCF and LCM of:

36, 84

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

23, 31

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

96, 404

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

144,198

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

396, 1080

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

1152, 1664

In each case, verify that:

HCF x LCM = product of given numbers.

Using prime factorization, find the HCF and LCM of:

8, 9, 25

Using prime factorization, find the HCF and LCM of:

12, 15, 21

Using prime factorization, find the HCF and LCM of:

17, 23, 29

Using prime factorization, find the HCF and LCM of:

30, 72, 432

Using prime factorization, find the HCF and LCM of:

21, 28, 36, 45

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

Find the simplest form of:

(iv)

(iv)

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

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

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

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

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

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.

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

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

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

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

Find the missing numbers in the following factorisation:

By going upward

5 11= 55

55 3= 165

1652 = 330

330 2 = 660

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.

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?

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

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?

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

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.

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

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.

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

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.

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.

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

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?

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

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?

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?

## Chapter 1 - Real Numbers Exercise Ex. 1C

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

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

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

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

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

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

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

## Chapter 1 - Real Numbers Exercise Ex. 1D

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

Classify the following numbers as rational or irrational:

Prove that each of the following numbers is irrational:

Prove that is irrational.

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

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

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.

(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

State Euclid's division lemma.

State fundamental theorem of arithmetic.

Express 360 as product of its prime factors.

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

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

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

What is a composite number?

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

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

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

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

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

Give an example of two irrationals whose sum is rational.

Give an example of two irrationals whose product is rational.

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

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

Express as a rational number in simplest form.

Express as a rational number in simplest form

Explain why 0.15015001500015 ... is an irrational number.

Write a rational number betweenand 2.

Explain why is a rational number.

## Chapter 1 - Real Numbers Exercise MCQ

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

(a) (14, 35)

(b) (18, 25)

(c) (31,93)

(d)(32, 62)

If a = (2^{2}×3^{3}×5^{4}) and b = (2^{3}×3^{2}×5) then HCF (a, b) = ?

(a) 90

(b) 180

(c) 360

(d)540

HCF of (2^{3}×3^{2}×5), (2^{2}×3^{3}×5^{2}) and (2^{4}×3×5^{3}×7) is

(a) 30

(b) 48

(c) 60

(d)105

LCM of (2^{3}×3×5) and (2^{4}×5×7) is

(a) 40

(b) 560

(c) 1120

(d)1680

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

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

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

(a) 32

(b) 64

(c) 128

(d)256

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

(a) 13

(b) 9

(c) 3

(d)585

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

(a) 15

(b) 16

(c) 9

(d)5

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

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

Which of the following is an irrational number?

(a)

(b) 3.1416

(c)

(d) 3.141141114 …

𝜋 is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

(a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

2.13113111311113… is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

The number 3.24636363 … is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

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

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) four decimal places

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d)four decimal places

The number 1.732 is

(a) an irrational number

(b) a rational number

(c) an integer

(d)a whole number

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

(a) a rational number

(b) an irrational number

(c) a terminating decimal

(d)a nonterminating repeating decimal

(a) a fraction

(b) a rational number

(c) an irrational number

(d)none of these

(a) an integer

(b) a rational number

(c) an irrational number

(a) none of these

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

## Chapter 1 - Real Numbers Exercise FA

(a) a terminating decimal

(b) a nonterminating, repeating decimal

(c) a nonterminating and nonrepeating decimal

(d)none of these

Which of the following has a terminating decimal expansion?

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

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

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

Which of the following numbers are irrational?

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

Give an example of two irrationals whose sum is rational.

Give prime factorization of 4620.

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

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

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

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

### Other Chapters for CBSE Class 10 Mathematics

Chapter 2- Polynomials Chapter 3- Linear equations in two variables Chapter 4- Quadratic Equations Chapter 5- Arithmetic Progressions Chapter 6- Co-ordinate Geometry Chapter 7- Trigonometric Ratios of Complementary Angles Chapter 8- Trigonometric Identities Chapter 9- Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive Chapter 12- Circles Chapter 13- Constructions Chapter 14- Height and Distance Chapter 15- Probability Chapter 17- Perimeter and Areas of Plane Figures Chapter 18- Areas of Circle, Sector and Segment Chapter 19- Volume and Surface Areas of Solids### R S AGGARWAL AND V AGGARWAL Solutions for CBSE Class 10 Subjects

### Kindly Sign up for a personalised experience

- Ask Study Doubts
- Sample Papers
- Past Year Papers
- Textbook Solutions

#### Sign Up

#### Verify mobile number

Enter the OTP sent to your number

Change