Vedanta Excel in Mathematics Book 5 Final (2077)

Properties of Whole Numbers

c) Possible factors of 12
Possible factors of 18

In this way, the possible factors of 12 are 1, 2 , 3, 4 , 6 and 12. The possible
factors of 18 are 1, 2, 3, 6 and 18. Here, 2, 3 and 6 are the common factors of
12 and 18.
2. Let's tell and write the ϐirst 10 multiples of each pair of numbers. Then
circle the common multiples.

a) 2 o

3o

b) 4 o

6o

c) 8 o

10 o

Thus, the irst ten multiples of 8 are 8, 16, 24, …, 80. The irst ten
multiples of 10 are 10, 20, 30,…, 100. Here, 40 and 80 are the common
multiples of 8 and 10.

3.7 Highest Common Factor (H.C.F)

Classwork - Exercise

1. Let's tell and write the answer of these questions.

a) What are the all possible factors of 12 ?

b) What are the all possible factors of 18 ?

c) What are the common factors of 12 and 18 ?
d) Which one is the Highest Common Factor of 12 and 18 ?
Thus, the Highest Common Factor (H.C.F) of 12 and 18 is 6.
2. a) What is the H.C.F. of 2 and 4 ?

b) What is the H.C.F of 3 and 6 ?

c) What is the H.C.F. of 4 and 8 ?

49 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers 6

3.8 Lowest Common Multiple (L.C.M.)

Classwork - Exercise

1. a) Let's tell and write the irst 10 multiples of 2 and 3.
2
3

b) What are the common multiples of 2 and 3 ?
c) Which one is the Lowest Common Multiple of 2 and 3 ?
d) Thus, the Lowest Common Multiple (L.C.M.) of 2 and 3 is

2. a) What is the L.C.M. of 2 and 4 ?
b) What is the L.C.M. of 3 and 4 ?
c) What is the L.C.M. of 5 and 10 ?

3.9 Process of finding H.C.F.

Let's study the example and learn the process of inding H.C.F. of the given
numbers.

Example 1: Find the H.C.F. of 18 and 24.

Solution All Possible factors of 18 and 24 are:
18 o 1, 2, 3, 6, 9, 18
2 18 2 24 24 o 1, 2, 3, 4, 6, 8, 12, 24

39 2 12 H.C.F. of 18 and 24 = 6
3 2
6 And, 6 is the product of common prime factors
18 = 2 × 3 × 3 3 2 and 3.

24 = 2 × 2 × 2 × 3 So, H.C.F. = product of common prime
? H.C.F. = 2 × 3 = 6 factors

3.10 Process of finding L.C.M.

Now, let's learn the process of inding L.C.M. of the given numbers from the
following example.

Example 2 : Find the L.C.M. of 8 and 10.

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Properties of Whole Numbers

Solution 2 10 A few multiples of 8 and 10 are :
5 8 o 8, 16, 24, 32, 40, …, 80, …
28 10 o 10, 20, 30, 40, …, 80, …
24 L.C.M. of 8 and 10 = 40

2

8 =2×2×2 And, 40 is the product of common prime factor
10 = 2 × 5 2 and the remaining prime factors 2, 2 and 5.

? L.C.M. = 2 × 2 × 2 × 5 = 40
Alternative process (Division method)

2 8, 10 8 and 10 are divided by their common factor 2.

4, 5
? L.C.M. = 2 × 4 × 5 = 40

Example 3: Find the greatest number that divides 12 and 20 without
leaving remainder.

Solution

Here, the required greatest number is the H.C.F. of 12 and 20.

2 12 2 20 Only the common factors 2 and 4 of 12 and
20 can divide them exactly. Between 2 and
26 2 10 4, 4 is the Greatest one. So, 4 is the H.C.F. of
3 5 12 and 20.

12 = 2 × 2 × 3

20 = 2 × 2 × 5

? H.C.F. = 2 × 2 = 4

Hence, the required greatest, number is 4.

Example 4 : Find the smallest number which is exactly divisible by 12 and
16.

Solution

Here, the required smallest number is the L.C.M. of 12 and 16.

2 12, 16 Only the common multiples of 12 and 16
2 6, 8 are exactly divisible by 12 and 16. And, the
smallest multiple is the L.C.M. of 12 and 16.
3, 4

L.C.M. = 2 × 2 × 3 × 4 = 48
Hence, the required smallest number is 48.

51 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

EXERCISE 3.3
Section A - Class work

1. Let's tell and write the possible factors of each pair of numbers. Circle the
Highest common factor (H.C.F.).

a) 2 o and 4 o

b) 3 o and 6 o

c) 4 o and 8 o

d) 6 o and 8 o

2. Let's tell and write the ϐirst 10 multiples of each pair of numbers. Then
circle the Lowest Common Multiple (L.C.M.).

a) 2 o

4o

b) 2 o

3o

c) 3 o

6o

d) 4 o

8o

3. Let's investigate the fact from the given illustrations. Then tell and write
the H.C.F. of each pair of numbers.

4 is a factor of 8. So, H.C.F. of 4 and 8 is 4.

5 is a factor of 15. So, H.C.F. of 5 and 15 is 5.

a) H.C.F. of 2 and 4 is b) H.C.F. of 2 and 4 is

c) H.C.F. of 4 and 12 is d) H.C.F. of 6 and 24 is

4. Let's investigate the facts from the given illustrations. Then tell and write
the L.C.M. of each pair of numbers.

(i) 4 is a multiple of 2. So L.C.M. of 2 and 4 is 4.

21 is a multiple of 7. So, L.C.M. of 7 and 21 is 21.

vedanta Excel in Mathematics - Book 5 52

Properties of Whole Numbers

a) L.C.M. of 3 and 6 is b) L.C.M. of 4 and 8 is

c) L.C.M. of 5 and 20 is d) L.C.M. of 6 and 18 is

(ii) 2 and 3 are prime numbers. So, L.C.M. of 2 and 3 = 2 × 3 = 6

a) L.C.M. of 2 and 5 is b) L.C.M. of 3 and 5 is

c) L.C.M. of 2 and 7 is d) L.C.M. of 5 and 7 is
Group B

5. Let's write the possible factors of each pair of numbers. Circle the
common factors and select the H.C.F.

a) 4, 6 b) 6, 9 c) 8, 12 d) 12, 18 e) 15, 20 f) 14, 21

6. Let's write the ϐirst ten multiples of each number of the pairs. Circle the
common multiples and select the L.C.M.

a) 2, 6 b) 4, 6 c) 3, 9 d) 6, 8 e) 5, 10 f) 4, 5

7. Let's ϐind the prime factors of each number of the pairs. Then ϐind their
H.C.F.

a) 8, 12 b) 6, 12 c) 10, 15 d) 12, 16 e) 10, 20

f) 12, 18 g) 16, 24 h) 18, 27 i) 15, 30 j) 24, 32

8. Let's ϐind the prime factors of each number of the pairs. Then ϐind their
L.C.M.

a) 4, 6 b) 6, 8 c) 4, 8 d) 6, 9 e) 8, 10

f) 9, 12 g) 10, 15 h) 10, 20 i) 12, 16 j) 18, 24

9. Let's ϐind the L.C.M. of these pairs of numbers by division method.

a) 6, 12 b) 8, 12 c) 4, 10 d) 9, 15 e) 12, 18

f) 14, 21 g) 15, 20 h) 16, 24 i) 18, 27 j) 20, 30

10. a) Find the greatest number that divides 12 and 18 without leaving remainder.

b) Find the greatest number that exactly divide 16 and 24.

c) Find the greatest number that divide 20 and 30 without leaving remainder.
11. a) Find the smallest number which is exactly divisible by 6 and 8.

b) Find the least number that can be divided by 9 and 12 without leaving
remainder.

c) Find the smallest number which is exactly divisible by 10 and 15.
53 vedanta Excel in Mathematics - Book 5

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It's your time - Project Work !

12. Let's play a game of ϐinding H.C.F. of any two numbers.

H. C. F. of 4 and 8 Equal number of
circles in both sides.
4 is less than 8. So,
remove 4 circles from 8.

48 48

So, H. C. F. is 4.

H. C. F. of 6 and 9 3 is less than 6. So, remove Equal number of
3 circles from 6. circles in both sides.
6 is less than 9. So,
remove 6 circles from 9. 69 69

69 So, H. C. F. is 3.

Now, let's ind the H.C.F. of these numbers by playing the games.
a) 2 and 4 b) 3 and 6 c) 6 and 8 d) 6 and 7 e) 10 and 5

13. Let's play a game of inding L.C.M. of any two numbers from 2 to 10. Make
number cards of the irst ten multiples of the numbers 2 to 10. Arrange the
multiple cards of each number separately in order.

Now, Let's play to ind the L.C.M. of 4 and 6.

At irst, pull the multiple card of 4 4 then 6 6

Again, pull the multiple card of 4 8 then 6
4 8 6 12
Again, pull the multiple card of 4
12

12
now stop !

Here, 12 is the L.C.M. of 4 and 6.

Now, Let's play the game with a friend and ind the L.C.M. of the following
pairs of numbers. Remember, you should pull the multiple cards of the smaller
number at irst.

a) 2 and 3 b) 3 and 4 c) 5 and 6 d) 6 and 8 e) 8 and 10

vedanta Excel in Mathematics - Book 5 54

Properties of Whole Numbers

14. Rolling number cubes (or dice)

You can play this game with a friend. Take turns rolling 22 1122 33 3300
two numbered cubes (or two dice). Find the LCM of 2300 55 1155 66
the two numbers rolled and circle in the square with 3300 1100 44 2200
the answer. The irst person to get 4 circles in a row is 66 11 99 1122
the winner !

3. 11 Square and square root

1 It is a square of 1 by 1 = 1 × 1 = 12 = 1 Square box
1 It is a square of 2 by 2 = 2 × 2 = 22 = 4 Square boxes

2

2

3 It is a square of 3 by 3 = 3 × 3 = 32 = 9 Square boxes

3 I got it !
When a number is multiplied
Similarly, by itself, we get the square
The square of 4 = 4 × 4 = 42 = 16 number of the given number !
The square of 5 = 5 × 5 = 52 = 25
The square of 6 = 6 × 6 = 62 = 36
and so on.

Again, let's study the following illustrations and investigate the idea about
square root of a square number.

1 Square of 1 = 12 = 1 and 1 = 1 is the square root of 1.

1

2 Square of 2 = 22 = 4 and 4 = 2 is the square root of 4.

2

3 Square of 3 = 32 = 9 and 9 = 3 is the square root of 9.

3

55 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

Similarly,
Square of 4 = 42 = 16 and 16 = 4 is the square root of 16

Square of 5 = 52 = 25 and 25 = 5 is the square root of 25, and so on .

In this way, when a number is multiplied by itself, the product is called the
square of the number. And, the number itself is the square root of the square

number. I got it !

We write square root of 4 as 4 = 2. 9 = 3, 36 = 6

The radical sing ( ) is the sign of square root. 49 = 7, 100 = 10

3.12 Process of finding square and square root

Let's learn about the process of inding square and square root of a given
number from the following examples.

Example 1: Find the square of a) 24 b) 50

Solution 52 = 25 Then write two zeros at the
Square of 24 = 242 = 24 × 24 = 576 end of 25.

Square of 50 = 502 = 50 × 50 = 2500

Example 2: a) Find the square root of 324
b) Find the prime factors of 8100.

Solution

a) 2 324 The same prime factors of 324
2 162 are arranged in pairs.

324 = 2 × 2 × 3 × 3 × 3 ×3

3 81 From each pair, one factor is
3 27 taken as square root and they
are multiplied.

39 ? 324 = 2 × 3 × 3 = 18

3

324 = 2 × 2 × 3 ×3 × 3 ×3

324 = 2 × 3 × 3 = 18

b) 3 81 56
3 27
39
3

vedanta Excel in Mathematics - Book 5

81 = 3 × 3 × 3 × 3 Properties of Whole Numbers
81 = 3 × 3 = 9
8100 = 90 We understood!
At ϔirst, we should ϔind the square root of none zero
number. So, root 81 = 9. Then we should write half
number of zero at the end of 9. 8100 = 90 !!

Example 3: If 7 students are kept in each row of 7 rows in a school
assembly, how many students are there in the assembly?

Solution

Here, the required number of students is the
square of 7.

72 = 7 × 7

= 49

Hence, there are 49 students in the assembly.

Example 4 : 36 marbles are arranged in the same number of rows and
columns. Find the number of marbles in each row or in
column.

Solution

Here, the required number of marbles is the
square root by 36.

Now, inding the prime factors of 36.

2 36
2 18
39

3
36 = 2 × 2 × 3 × 3

∴ 36 = 2 × 3 =6

Hence, there are 6 marbles in each row or in column.

57 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

EXERCISE 3.4
Section A - Class work
1. Let's tell and write the square of the given numbers.

a) 12 = b) 22 = c) 32 =

d) 42 = e) 52 = f) 62 =

g) 72 = h) 82 = i) 92 = j) 102 =

2. Let's tell and write the square of the given numbers.

a) 102 = b) 202 = c) 302 =

d) 402 = e) 502 = f) 602 =

g) 702 = h) 802 = i) 902 =

3. Let's tell and write the square roots of the given numbers.

a) 1 = b) 4 = c) 9 =

d) 16 = e) 25 = f) 36 =

g) 49 = h) 64 = i) 81 = j) 100 =

4. Let's investigate from the given example, how the numbers are written in
the box and in circles. Then complete the remaining sums.

a) b) c) d)
9 16 25 36

3×3 4× ×5 ×
e)
f) g) h)
64 100

7× × ×9 ×

5. Let's write the correct number under the sign of square root ( ).

a) = 3 b) = 8 c) = 5 d) =6

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Properties of Whole Numbers

6. Insert the sign of square root ( ) to the appropriate number.

a) 16 = 4 b) 7 = 49 c) 10 = 100 d) 400 = 20

Section B
7. Let's answer the following questions ?

a) What do you mean by the square number of 5 is 25 ?

b) What do you mean by the square root of 81 is 9 ?

c) How do we ind the square number of a given natural number ?

d) What is the least natural square number ?

e) What is the greatest natural square number ?

f) What is a number which is a square and square root itself ?
8. Let's ϐind the square of the following numbers.

a) 11 b) 12 c) 13 d) 14 e) 15 f)16

g) 17 h) 18 i) 19 j) 24 k) 25 l) 36

m) 120 n) 130 o) 140 p) 150 q) 100 r) 200

9. Let's ϐind the prime factors of these square numbers. Then ϐind their
square roots.

a) 16 b) 25 c) 36 d) 64 e) 81 f) 100

g) 144 h) 196 i) 225 j) 256 k) 324 l) 441

10. a) If 8 students are kept in each row of 8 rows in a school assembly, how many
students are there in the assembly?

b) 9 chairs are arranged in each column of 9 columns in a room. How many
chairs are there in the room ?

c) There are 10 potted lowers in each row of a garden. If there are the same
number of rows and columns, ind the total number of potted lowers in the
garden.

11. a) 64 children are arranged in the same number of rows and columns in the
ground. Find the number of children in each row or in column.

b) Class ive students collected a sum of Rs. 900 to support the 'Poor students
Helping Fund.' If every student donated the equal amount of money as their
number, ind the amount donated by each student.

59 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

c) There are 144 unit square rooms in square graph. Find the number of unit
square rooms along the length or breath of the graph.

It's your time- Project work !

12. a) Look at the pattern of square numbers.
1, 1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25,

What types of numbers are 3, 5, 7, 9, ?

Let's follow the pattern and ind the square numbers upto 100.

b) Let's write the square of the numbers from 1 to 10. Investigate the fact

whether the digit at ones place of any of these square number can be 2, 3, 7

and 8.

c) Let's investigate the interesting facts about square numbers. Then complete

the pyramid in a chart paper.

12 1 = 1
22 1 + 2 + 1 = 4
32 1 + 2 + 3 + 2 + 1 = 9
42 =
52 =
62 =
72 =

d) Let's observe the patterns of some square numbers. Then draw circles
(or dots) in a chart paper to show the similar patterns of the following
numbers.

22 = = 4 52 = = 25

(i) 32 (ii) 42 (iii) 62 (iv) 72 (v) 82 (vi) 92 (vii) 102

3.13 Cube and cube root

Let's study the following illustrations and learn about the cube of a given
number.

1 The cube of 1 =13 = 1 × 1 × 1 = 1
11

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Properties of Whole Numbers

The cube of 2 = 23 = 2 × 2 × 2 = 8 A cube number
always represents
2 a cube shape!
22

The cube of 3 = 33 I got it!
When I multiply a number
3 =3×3×3 three times by itself, I get
= 27 a cube number.

3
3

Similarly,
The cube of 4 = 43 = 4 × 4 × 4 = 64
The cube of 5 = 53 = 5 × 5 × 5 = 125 and so on.

Thus, the cube of a number is the product obtained by multiplying the number
three times by itself.

Now, let's learn about cube root of a cube number.

Cube of 1 = 13 = 1 So, 1 is the cube root of 1.
Cube of 2 = 23 = 8 So, 2 is the cube root of 8.
Cube of 3 = 33 = 27 So, 3 is the cube root of 27 and so on.

In this way, we obtain a cube number by multiplying the same three numbers,
and each identical number is the cube root of the cube number.

3.14 Process of finding cube and cube root

Let's learn about the process of inding cube and cube root of a given number
from the following examples.

Example 1: Find the cube of a) 12 b) 30 It's easy !
Solution In 303, cube of 3 is 27. Then
a) Cube of 12 = 123 = 12 × 12 × 12 = 1728 I should write three zeros
b) Cube of 30 = 303 = 30 × 30 × 30 at the end of 27 !!

= 900 × 30 = 27000 a) 216 b) 64000
Example 2: Find the cube root of

Solution

a) Finding the prime factors of 216,

61 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

2 216 The same three prime factors of 216
2 108 are grouped.
2 54
3 27 216 = 2 × 2 × 2 × 3 × 3 × 3
39
From each group , one factor is taken
3 as cube root and they are multiplied.

So, cube root of 216 = 2 × 3 = 6

216 = 2 × 2 × 2 × 3 × 3 × 3

? Cube root of 216 = 2 × 3 = 6

b) Finding the prime factors of 64,

2 64 I got it !
2 32 At ϔirst, I should ϔind the cube root
2 16 of none zero number. So, cube
28 root of 64 is 4. Then I should write
24 one-third number of zero at the end
of 4. So, cube root of 64000 = 40 !!
2

64 = 2 × 2 × 2 × 2 × 2 × 2

Cube root of 64 = 2 × 2 = 4

? Cube root of 64000 = 40

Example 3: The length, breath and height of a small wooden cubical block
is 3 cm each. Find its volume in cubic centimetres (cm3).

Solution

Length = breath = height of the cubical block = 3 cm 3 cm 3 cm
Volume of the block = (3 cm)3 3 cm

= 3 cm × 3 cm × 3 cm
= 27 cubic centimetres (cm3)

Hence, the volume of the wooden cubical block is 27 cm3.

Example 4 : If the volume of a cubical die is 125 cm3, ϔind the length of the
die.

Solution

The length (or breath or height) of the cubical die is the cube root of 125 cm3.

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Properties of Whole Numbers

Finding the prime factors of 125, V=125cm3 l
l l
5 125
5 25

5

125 = 5 × 5 × 5
? Cube root of 125 = 5
Hence, the length of the cubical die is 5 cm.

EXERCISE 3.5
Section A - Class work

1. Let's tell and write the square numbers and cube numbers separately.

1 36 25 27 8 Square numbers Cube numbers

16 64 49 9 125

2. Let's tell and write the cube of each of these numbers.

a) 13 = b) 23 = c) 33 = d) 43 =

e) 53 = f) 103 = g) 203 = h) 303 =

3. Let's tell and write the cube root of each of these cube numbers.

a) cube root of 1 = cube root of 1000 =

b) cube root of 8 = cube root of 8000 =

c) cube root of 27 = cube root of 27000 =

d) cube root of 64 = cube root of 64000 =

4. Each of the following blocks is made up of unit cubical block. Write the
cube number represented by each block.

a) b) c) d)

63 vedanta Excel in Mathematics - Book 5

Properties of Whole Numbers

Section B
5. Let's answer the following questions.

a) What do you mean by the cube of 3 is 27 ?
b) What do you mean by the cube root of 8 is 2 ?
c) How do we find the cube of a given natural number ?
d) What is the least natural cube number ?
e) What is the greatest natural cube number ?
f) What is a number which is a square, square root, cube and cube root itself ?
6. Let's find the cube of each of the following numbers.

a) 5 b) 6 c) 7 d) 8 e) 9 f) 11

g) 12 h) 15 i) 30 j) 40 k) 60 l) 100

7. Let's find the prime factors of these cube numbers. Then find their cube
roots.

a) 8 b) 27 c) 64 d) 125
e) 216 f) 343 g) 512 h) 729
i) 2744 j) 8000 k) 27000 l) 125000

8. Let's simplify

a) 2 + 22 + 23 b) 33 – 32 – 3 c) 32 + 22 – 23
d) 42 – 22 + 32 e) 43 – 23 – 33 f) 52 + 23 – 42

g) 33 – 52 + 22 h) 53 – 52 – 102 i) 92 – 43 – 23

9. a) The length, breath and height of a small wooden cubical block is 2 cm each.
Find it's volume in cubic centimetres (cm3).

b) A small metallic cubical block is 4 cm long. Find it's volume in cubic
centimetres (cm3).

c) A solid block is in the shape of a cube. If it is 10 cm high, find it's volume in
cubic centimetres (cm3).

d) The length, breath and height of the given cube is 4 cm
each. How many cubes of 1 cm length can be made from
this cube ?

10. a) If the volume of a cubical die is 64 cm3, find the length of the die.
b) The volume of a solid wooden block is 216 cm3. Find the height of the block.
c) If the product of three identical numbers is 125, find the number.

vedanta Excel in Mathematics - Book 5 64