Vedanta mathematics _Book - 4

2. Divide 2800 by 400. Fundamental Operations
2800 ÷ 400 = 28 ÷ 4 = 7
Equal number of zeros from
2800 and 400 are cancelled.
Then 28 ÷ 4 = 7!

3. Divide 1200 by 30. Equal number of zeros from 1200 and
1200 ÷ 30 = 120 ÷ 3 = 40 30 are cancelled. Then divide 12 by 3
and add one zero to the quotient 4.

Classwork - Exercise

1. Let's tell and write the quotient as quickly as possible.

a) 80 ÷ 20 = b) 90 ÷ 30 = c) 80 ÷ 40 =

d) 100 ÷ 50 = e) 120 ÷ 40 = f) 140 ÷ 20 =

g) 1500 ÷ 300 = h) 1800 ÷ 600 = i) 2400 ÷ 400 =

j) 2700 ÷ 300 = k) 3200 ÷ 800 = l) 4500 ÷ 500 =

m) 800 ÷ 20 = n) 1600 ÷ 80 = o) 2800 ÷ 70 =

3.15 Division facts

Now, let's learn a few important facts about division.

(i) When a number is divided by 1, the quotient is the number itself.

4 ÷ 1 = 4, 7 ÷ 1 = 7, 12 ÷ 1 = 12, and so on.

(ii) When a number is divided by itself, the quotient is always 1.

5 ÷ 5 = 1, 8 ÷ 8 = 1, 15 ÷ 15 = 1, and so on.

(iii) When 0 is divided by any non-zero number, the quotient is always 0.

0 ÷ 6 = 0, 0 ÷ 9 = 0, 0 ÷ 18 = 0, and so on.

(iv) Dividend = Divisor × Quotient + Remainder

In 7 ÷ 2 = 3 is the quotient and 1 is remainder.

So, 7 = 2 × 3 + 1 = 6 + 1 = 7 = Divisor × Quotient + Remainder
55 vedanta Excel in Mathematics - Book 4

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3.16 Division of bigger numbers

1. Let's divide 54 by 4.

4 5 4 13 ← Quotient
54 ÷ 4 = = – 4 ↓

14
40 ÷ 4 12÷4 R 1 ten 3 ones Remainder – 1 2
2 ← Remainder

2. Let's divide 118 by 5.

118 ÷ 5 = = 5 1 1 8 23 ← Quotient
= – 10 ↓
100 ÷ 5 15 ÷ 5 R
18
–15

3 ← Remainder

3. Let's divide 7246 by 7. 4. Let's divide 9018 by 9.

7) 7246 )1035 Quotient 9) 9018 )1002 Quotient
–7 –9

02 00
–0 –0

24 01
– 21 –0

36 18
– 35 – 18

1 Remainder 0 Remainder



5. Let's divide 278 by 25.

25) 278 )11 Here, divisor 25 has two digits.
–25 So, at first try to divide two digits
27 of the dividend 278.
28 27 ÷ 25 = 1 time and 2 remainder.
– 25 Then continue the process.

3
Q = 11 and R = 3

6. Let's divide 5976 by 84. Here, we cannot divide 59 by 84.
So, let's divide 597 by 84.
84) 5976)71 Trick: Let's think 59 ÷ 8. It goes 7 times.
–588 Let's try 84 × 7 = 588.
So, 597 ÷ 84 = 7 times and remainder 9.
96 Then, continue the process.
– 84

12
Q = 71 and R = 12

vedanta Excel in Mathematics - Book 4 56

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Exercise - 3.3
Section A - Classwork
1. Let's tell and write the dividend and divisor. Then find the quotient.

a) b) c)

÷ = ÷ = ÷ =

2. Let's draw as many dots as to match the sums. Ten tell and write the
quotient as quickly as possible.

a) 15 ÷ 5 = =

b) 15 ÷ 3 = =

c) 24 ÷ 6 = =

d) 24 ÷ 4 = =

3. Let's write the total number of dots and divide by number of rows or
by number of columns. Then find the number of dots in each row and
column.

a) b)

÷ = dots in each row ÷ = dots in each column

4. Let's tell and write the missing dividend, divisor or quotient.

a) 36 ÷ = 9 b) 42 ÷ = 6 c) 56 ÷ = 7

d) 63 ÷ 7 = e) ÷ 9 = 8 f) ÷ 8 = 10

5. Let's tell and write the answer as quickly as possible. = Rs
a) The cost of 4 pencils is Rs 32. The cost of 1 pencil = ÷ = Rs
b) The cost of 5 sweets is Rs 35. The cost of 1 sweet = ÷

57 vedanta Excel in Mathematics - Book 4

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c) 40 cherries are shared between 8 children equally.

Each of them gets ÷= cherries.

d) 45 students are kept in 5 rows equally. The number of students in each row

=÷= students.

e) How many times are twos be subtracted from 10 to get 0?

÷ = times
f) How many times are fives be subtracted from 15 to get 0?

÷ = times

6. Let's tell and write the quotients as quickly as possible.

a) 100 cm = 1m So, 200 cm = 200 ÷ 100 = m

b) 1000 m = 1 km So, 4000 m = 4000 ÷ 1000 = km

c) 1000 g = 1 kg So, 5000 g = 5000 ÷ 1000 = kg

d) 1000 ml = 1 l So, 8000 ml = 8000 ÷ 1000 = l

e) 100 kg = 1 quintal So, 7000 kg = 7000 ÷ 100 = quintal

7. Let's tell and write the number of rupees notes.

a) How many Rs 5 notes are there in Rs 100? ÷=

b) How many Rs 10 notes are there in Rs 100? ÷=

c) How many Rs 10 notes are there in Rs 1000? ÷=

d) How many Rs 20 notes are there in Rs 500? ÷=

e) How many Rs 50 notes are there in Rs 1000? ÷=

f) How many Rs 100 notes are there in Rs 1000? ÷ =

8. Let's complete division by using number lines as shown.
a)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

12 ÷ 4 =

vedanta Excel in Mathematics - Book 4 58

Fundamental Operations

b)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

÷=

c)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

÷=

Section B
9. Let's rewrite the sums and find the quotients in your exercise book.

a) 80 ÷ 20 b) 800 ÷ 20 c) 8000 ÷ 20 d) 8000 ÷ 200

e) 60 ÷ 30 f) 600 ÷ 30 g) 6000 ÷ 30 h) 6000 ÷ 300

i) 120 ÷ 40 j) 250 ÷ 50 k) 360 ÷ 90 l) 4200 ÷ 60

m) 5600 ÷ 70 n) 32000 ÷ 400 o) 45000 ÷ 500 p) 72000 ÷ 800

Let's divide and find the quotient and remainder.

10. a) 58 ÷ 70 b) 74 ÷ 6 c) 65 ÷ 4 d) 84 ÷ 7 e) 96 ÷ 8
f) 213 ÷ 2 g) 326 ÷ 3 h) 416 ÷ 4 i) 530 ÷ 5 j) 927 ÷ 9
k) 429 ÷ 2 l) 638 ÷ 3 m) 853 ÷ 4 n) 767 ÷ 5 o) 852 ÷ 6

11. a) 4602 ÷ 4 b) 5963 ÷ 5 c) 6792 ÷ 6 d) 7856 ÷ 7 e) 3276 ÷ 3

f) 7156 ÷ 7 g) 8024 ÷ 8 h) 1348 ÷ 3 i) 2352 ÷ 4 j) 5257 ÷ 7

12. a) 168 ÷ 14 b) 185 ÷ 15 c) 216 ÷ 21 d) 360 ÷ 24 e) 550 ÷ 32

f) 152 ÷ 16 g) 210 ÷ 26 h) 1470 ÷ 35 i) 3358 ÷ 54 j) 5699÷ 78

Let's read these problems carefully and solve them.

13. a) The cost of 5 kg of rice is Rs 425. Find the cost of 1 kg of rice.

b) If the cost of 1 kg of rice is Rs 85, how many kilograms of rice can be
bought for Rs 425?

c) The cost of 8 l of milk is Rs 768. Find the cost of 1 l of milk.

d) If the rate of cost of milk is Rs 96 per litre, how many litres of milk can
be purchased for Rs 768?

59 vedanta Excel in Mathematics - Book 4

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14. a) In 1 dozen of pencils there are 12 pencils. How many dozens of pencils
are there in 180 pencils?

b) 1 crate eggs contains 30 eggs. How many crates of eggs are there in 750
eggs?

c) 5 dozens of cricket balls can be kept in a box. How many boxes are
needed to keep 1440 balls?

15. a) If 12 quintals of weight is equal to 1200 kg, how many kilograms are
there in 1 quintal?

b) If 9 metric tons weight is equal to 9000 kg, how many kilograms are
there in 1 metric ton?

c) 15 bottles of equal capacity can hold 7500 ml of liquid. Find the capacity
of each bottle in millilitres.

16. a) 154 children of 14 football teams of different schools are playing
football in an inter-school football match. How many players are there
in each team?

b) There are 11 players in a football team. 154 children of different school
teams are playing football in an inter-school football match. How many
teams are playing the match?

17. a) 540 students are arranged in 18 rows with the equal number of students
in each row. How many students are there in 1 row?

b) 480 students are arranged in some columns with 32 students in each
column. How many columns of students are there?

18. a) There are 7 days in one week. How many weeks are there in 364 days?

b) There are 12 months in one year. How many years are there in 300
months?

c) There are 365 days in one year. How many years are there in 4380
days?

It's your time - Project work!

19. a) Let's write any three 2-digit numbers. Divide them separately by any
three 1-digit divisor. Then show that

Dividend = Divisor × Quotient + Remainder

b) Let's write any three 3-digit numbers. Divide them separately by any
three 1-digit divisor. Then show that

Dividend = Divisor × Quotient + Remainder
Let's stick your findings on the school's wall-magazine!

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c) Let's draw 12 circles in 1 row, 2 rows, 3 rows, 4 rows, 6 rows and 12
rows with equal number of circles in each row in a chart paper. Find
the number of circles in each row by using division process. Stick your
findings on the school's wall-magazine!

3.17 Simplification - A single solution of a mixed operation

Classwork - Exercise

1. Let's listen to your teacher! Tell and write the answer as quickly as
possible.

a) Add 6 and 8, then subtract 5. 6 + 8 – 5 = 14 – 5 = 9

b) Add 4 and 7, then subtract 3. +–=–=

c) Subtract 7 from 15, then add 9. –+=+=

d) Subtract 6 from 16, then again subtract 4.

– – = – =

e) Multiply 5 and 4, then add 7. ×+=+=

f) Divide 27 by 3 and subtract 6. ÷–=–=

The problems given above are the mixed operations. Because these problems
have more than one operations. To workout these mixed operations, we
should perform division, multiplication, addition and subtraction in order
and get a single answer. The order of performing such mixed operation to
get a single and simple answer is called simplification.

2. First divide, then multiply, add and subtract.

a) 8 ÷ 2 × 3 + 4 – 6 b) 10 + 7 – 15 ÷ 5 × 4

= × + – =+–×

= + – = + –

= – = –

= =
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3.18 Order of operations

Order of addition and subtraction

Example 1: Simplify a) 12 + 6 – 8 b) 12 – 8 + 6 c) 15 – 6 – 3

Solution

a) 12 + 6 – 8 = 18 – 8 b) 12 – 8 + 6 = 4 + 6 c) 15 – 6 – 3 = 9 – 3

= 10 = 10 =6

Order of multiplication, addition and subtraction

Example 2: Simplify a) 4 × 7 + 2 b) 2 + 4 × 7 c) 20 – 5 × 3

Solution 4 × 7 + 2 = 4 × 9 = 36
Which is the wrong order!
a) 4 × 7 + 2 = 28 + 2
2 + 4 × 7 = 6 × 7 = 42
= 30 Which is the wrong order!

b) 2 + 4 × 7 = 2 + 28 20 – 5 × 3 = 15 × 3 = 45
Which is the wrong order!
= 30

c) 20 – 5 × 3 = 20 – 15

=5

Order of division and multiplication

Example 3: Simplify a) 24 ÷ 4 × 2 b) 2 × 24 ÷ 4

Solution 24 ÷ 4 × 2 = 24 ÷ 8 = 3
Which is the wrong order!
a) 24 ÷ 4 × 2 = 6 × 2
Another process
= 12 2 × 24 ÷ 4 = 48 ÷ 4 = 12

b) 2 × 24 ÷ 4 = 2 × 6

= 12

Example 4: Simplify 3 × 10 ÷ 5 + 12 – 4

Solution

3 × 10 ÷ 5 + 12 – 4 Another process Short process

= 3 × 2 + 12 – 4 3 × 10 ÷ 5 + 12 – 4 3 × 10 ÷ 5 + 12 – 4

= 6 + 12 – 4 = 30 ÷ 5 + 12 – 4 = 3 × 2 + 8

= 18 – 4 = 6 + 12 – 4 =6+8

= 14 = 18 – 4 = 14 = 14

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3.19 Use of brackets in simplification

Let's read these illustrations carefully and learn to use brackets in
simplifications.

Example 5: Multiply the sum of 5 and 4 by 7.
Solution

Here, the mathematical expression is (5 + 4) × 7 but not 5 + 4 × 7

(5 + 4) × 7 = 9 × 7 But 5 + 4 × 7 = 5 + 28 = 33 and it is the
wrong answer for the given problem
= 63

In this problem, at first we need to find the sum of 5 and 4. Then the sum is
multiplied by 7. Therefore, to find the sum at first we enclose 5 + 4 in the
brackets ( ).

Example 6: Find 5 times the difference of 16 and 12.
Solution

Here, the difference of 16 and 12 = 16 – 12

So, 5 × (16 – 12) = 5 × 4

= 20

Example 7: Simplify a) 40 ÷ 5 × 2 + 12 – 7 b) 40 ÷ (5 × 2) + 12 – 7
Solution

a) 40 ÷ 5 × 2 + 12 – 7 b) 40 ÷ (5 × 2) + 12 – 7

= 8 × 2 + 12 – 7 = 40 ÷ 10 + 12 – 7

= 16 + 5 = 4 + 5

= 21 = 9

Exercise - 3.4
Section A - Classwork

1. Let's read the instructions and make mathematical expressions. Then
simplify.

a) Add 5 and 7, then subtract 8. +–=–=

b) Subtract 6 from 18, then add 4. – + = + =

c) Subtract 9 from 20, then subtract 3. – – = – =

d) Multiply 6 and 7, then add 10. × + = + =

e) Divide 72 by 9, then multiply by 5. ÷ ×=×=
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2. Let's simplify and tell and write the answer as quickly as possible.

a) 4 + 7 – 2 = – = b) 10 – 3 + 8 = + =

c) 16 – 5 – 6 = – = d) 4 × 6 + 3 = + =

e) 5 × 7 – 10 = – = f) 9 + 7 × 2 = + =

g) 30 – 4 × 3 = – = h) 27 ÷ 9 × 6 = × =

i) 4 × 10 ÷ 5 = × = j) 4 × 10 ÷ 5 = ÷ =

3. Let's insert the appropriate sign (+, – , × or ÷) in the blank spaces to get
the given answer.

a) 4 5 2 = 7 b) 4 5 2 = 18

c) 4 5 2 = 22 d) 4 5 2 = 14

e) 3 4 2 = 6 f) 3 4 2 = 5

g) 3 4 2 = 10 h) 3 4 2 = 24

i) 12 6 3 = 3 j) 12 6 3 = 5

k) 12 6 3 = 10 l) 12 6 3 = 9

Section B b) 5 × 7 – 20 + 10
Let's simplify these mixed operations.
d) 60 ÷ 6 + 3 × 4
4. a) 6 × 4 + 8 – 9 f) 10 × 4 ÷ 2 – 5 + 6
h) 54 ÷ 6 – 5 × 3 + 7
c) 30 ÷ 5 × 2 + 7 j) 4 × 18 ÷ 9 – 3 × 5 + 12
e) 9 × 8 ÷ 4 – 3
g) 56 ÷ 7 + 4 × 5 – 8
i) 30 + 2 × 10 ÷ 5 × 3 – 15

5. a) 9 + (8 – 3) b) 15 – (7 + 5)

c) 4 × (20 – 12) d) 36 ÷ (8 + 4)
e) (50 – 20) ÷ 6 f) (6 + 4) × 8 ÷ 2
g) (5 + 3) × (7 + 2) h) (12 + 16) ÷ (18 – 11)
i) 72 ÷ (15 – 6) × (12 – 8) j) 80 ÷ (14 – 9) × 2 + 7
k) (8 + 4 × 8) ÷ (6 × 3 – 10) l) (60 – 3 × 4) ÷ (7 × 5 – 29)

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6. Let's enclose the operation to perform it first by using brackets. Then

simplification get the given answer.

a) (3 + 2) × 5 = 25 b) 4 + 3 × 2 = 14 c) 4 × 10 – 8 = 8

d) 24 ÷ 4 × 2 = 3 e) 20 ÷ 7 + 3 = 2 f) 49 – 7 ÷ 7 = 6

g) 45 + 18 ÷ 9 = 7 h) 3 × 6 + 4 ÷ 2 = 15 i) 24 ÷ 6 – 2 × 5 = 30

7. Let's make mathematical expressions and simplify.

a) You have Rs 10 and mother gives you Rs 5 more. If you spend Rs 8, how
much money do you have now?

Solution

Rs10+Rs5–Rs8=

So, I have Rs now.

b) Bishwant had 9 sweets. He ate 4 sweets and again he bought 7 sweets.
How many sweets did he have now?

c) There are 25 students in a school bus. 6 students get down at one place
and 8 students get down at another place. How many students are left
in the bus?

d) There are 7 rows of 5 chairs in each row and 4 more chairs in a room.
How many chairs are there in the room?

Solution

7 × 5 + 4 =

So, there are chairs in the room.

e) Bina Magar had 7 colour pencils. She bought a few more colour pencils
for Rs 40 at the rate of Rs 8 each. How many colour pencils does she
have now?

f) On Friday, there were 27 students in class four. 16 of them were girls
and the rest were boys. If only 3 boys were absent on that day, find the
number of boys in class four.

g) Ram, Hari and Laxmi were in a race. When Ram finished the race, Hari
was 5 metres behind Ram and Laxmi was 9 metres behind Hari. How
far away from the finish line was Laxmi?

h) After buying 5 chocolates at Rs 10 each, Kalpana Rai had Rs 25 left.
How much money did she have at first?

Let's make mathematical expressions using brackets. Then simplify.

8. a) The sum of 6 and 10 is subtracted from 25.
Solution
25 – (6 + 10) =

65 vedanta Excel in Mathematics - Book 4

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b) The sum of 4 and 5 is multiplied by 6.
c) The difference of 20 and 8 is divided by 3.
d) The product of 6 and 4 is divided by 8.
e) 56 is divided by the sum of 3 and 4.

9. a) A sick person takes 20 ml of medicine twice a day. How much medicine
does she/he take in a week?

Solution

7 × (20 ml + 20 ml) =

So, she/he takes ml of medicine in a week.

b) The distance between your house and your school is 4 km. How many
kilometres do you travel in 6 days?

c) The highway distance between Birendra Bazar and Lahan is 55 km.
Mr. Mahato travelled 25 km by a taxi, 27 km by a bus and the remaining
distance he walked. How many kilometres did he walk?

d) You buy a pen for Rs 30 and a box for Rs 85. If you give Rs 150 to the
shopkeeper, what changes does the shopkeeper return you?

e) There are 4 girls and 3 boys in each group. How many students are
there in 5 groups?

f) Mrs. Shrestha earns Rs 3500 in a week. She spends Rs 200 everyday to
run her house. How much money does she save in a week?

g) Teacher divided 50 sweets equally between 12 boys and 13 girls of
class four. How many sweets will each student get?

It's your time - Project work!

10. a) Let's rewrite these simplifications and find the mistakes. Then complete
the simplification in the correct way. You can display your work in your
school's wall-magazine.

20 – 10 – 5 5 + 4 × 3 14 – 5 × 2 30 ÷ 3 × 2

= 20 – 5 = 9 × 3 = 9 × 2 = 30 ÷ 6

= 15 = 27 = 18 =5

b) Let's make any four your own mixed expressions by using all 4 signs
(+, –, ×, ÷) in each expressions. Simplify them and get the correct answer.

?

vedanta Excel in Mathematics - Book 4 66

Unit Factors and Multiples

4

4.1 Divisibility Test
Let's discuss about the answer of these questions.
a) Are 2, 4, 6, 8 and 10 exactly divisible by 2?
b) Are 3, 5, 7 and 9 exactly divisible by 2?
c) Are 3, 6, 9, 12, 15 and 18 exactly divisible by 3?
d) Are 4, 5, 7, 8, 10 and 11 exactly divisible by 3?

When a dividend is divisible by a divisor with no remainder, the dividend
is called exactly divisible by the divisor.

10 ÷ 2 = 5 quotient with no remainder. 10 is exactly divisible by 2.

13 ÷ 2 = 6 quotient with 1 remainder. 13 is not exactly divisible by 2.

Now, let's learn a few rules of divisibility test.

Exactly Rules of divisibility test

divisible by

The digit at ones place of any number is 0 or even number

2 (2, 4, 6, 8). So, 90, 132, 754, 3616, 5978, ... are exactly

divisible by 2.

The sum of the digits of any number is exactly divisible by

3 3. In 225, 2 + 2 + 5 = 9 and 9 is exactly divisible by 3. So,

225 is exactly divisible by 3.

The number formed by last two digits of any number is

4 exactly divisible by 4. So, 92, 308, 924, 2564, ... are exactly

divisible by 4.

5 The digit at ones place is 0 or 5. So, 80, 170, 245, 4195, ...
are exactly divisible by 5.

6 Any even number exactly divisible by 3 are also exactly
divisible by 6. So, 84, 198, 2580, ... are exactly divisible by 6.

The sum of the digits of any number is exactly divisible by

9 9. In 486, 4 + 8 + 6 = 18 and 18 is exactly divisible by 9.

So, 486 is exactly divisible by 9.

10 The digit at ones place is 0. So, 70, 350, 830, 4120, ... are
exactly divisible by 10.

67 vedanta Excel in Mathematics - Book 4

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Classwork - Exercise

1. Let's circle the numbers which are exactly divisible by the given
numbers.

a) by 2 → 50 92 117 354 700 945 1681 4218

b) by 3 → 62 87 102 460 540 893 2154 5307

c) by 4 → 48 94 108 282 628 714 3236 8560

d) by 5 → 56 70 145 378 500 903 4785 7129

e) by 6 → 72 93 234 416 738 825 5160 9204

f) by 9 → 93 108 319 558 6300 7210 2457 6980

g) by 10 → 60 95 130 305 550 1002 3700 5670

2. Let's use the rule of divisibility test to find whether these numbers are
exactly divisible by 3 or 9.

a) Is 174 exactly divisible by 3? 1 + 7 + 4 = 12 Yes.
b) Is 289 exactly divisible by 9? 2 + 8 + 9 = 19 No.

c) Is 253 exactly divisible by 3? ++=

d) Is 414 exactly divisible by 3? ++=

e) Is 153 exactly divisible by 9? ++=

4.2 Factors and multiples

Let's investigate the ideas of factors and multiples from the following
examples.

In how many ways can you In how many ways can you
make 12 by multiplication? make 18 by multiplication?

1 × 12 2×6 1 × 18 2×9

12 18

3×4 3×6

So, 1, 2, 3, 4, 6 and 12 are the So, 1, 2, 3, 6, 9 and 18 are the
factors of 12. factors of 12.
18 is the multiple of 1, 2, 3, 6,
12 is the multiple of 1, 2, 3, 4, 9 and 18.
6 and 12.

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Factors and Multiples

The factors of a number can exactly divide the number.
12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1

18 ÷ 1 = 18, 18 ÷ 2 = 9, 18 ÷ 3 = 6, 18 ÷ 6 = 3, 18 ÷ 9 = 2 and 18 ÷ 18 = 1

4.3 Prime Factors
All possible factors of 15 are 1, 3, 5 and 15. Among these factors 3 and 5 are

the prime factors because 3 and 5 are prime numbers.
Similarly, all possible factors of 20 are 1, 2, 4, 5, 10 and 20. Among these

factors, 2 and 5 are the prime factors.

Classwork - Exercise

1. Let's find the multiples. Tell and write the all possible factors of the
multiples. Then list the prime factors.

a) 1 × 4 = , 2 × 2 =

All possible factors of 4 are , and

The prime factors of 4 is

b) 1 × 6 = , 2 × 3 = ,, and
All possible factors of 6 are and
The prime factors of 6 are

c) 1 × 10 = , 2 × 5 =
All possible factors of 10 are , , and
The prime factors of 10 are and
2. Let's tell and write the all possible factors of these numbers. Then circle

the prime factors.
a) All possible factors of 8 are , , and

b) All possible factors of 9 are , and

c) All possible factors of 15 are , , and

d) All possible factors of 20 are , , , , and

4.4 Process of finding prime factors
Let's study these examples and investigate the idea of finding prime factors

of the given numbers.

69 vedanta Excel in Mathematics - Book 4

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Let's find the prime factors of 12.

Here, 12 ÷ 2 = 6 and 6 ÷ 2 = 3 Factor tree

2 12 → 12 ÷ 2 = 6 12
2 2 → 6 ÷ 2 = 3 2×6

3 2× 2×3
So, 12 = 2 × 2 × 3

So, 12 = 2 × 2 × 3

Thus, to find the prime factors of a given number, we should start to divide

the number by the lowest prime number. We should continue division till

the quotient becomes a prime number. Factor tree
Again, let's find the prime factors of 24. 24

2 24 → 24 ÷ 2 = 12 2 × 12
2 12 → 12 ÷ 2 = 6

2 6 → 6 ÷ 2 = 3 2 × 2 ×6

3 2 × 2× 2 × 3
So, 24 = 2 × 2 × 2 × 3 So, 24 = 2 × 2 × 2 × 3

Classwork - Exercise

1. Let's divide the given numbers by the prime numbers till the quotient

becomes a prime numbers.

a) 2 18 b) 3 27 c) 2 30

3 3

18 = × × 27 = × × 30 = × ×

2. Let's tell and write the correct numbers in the empty circles. Then write
the number as the product of its prime factors from the factor tree.

a) 18 b) 28 c) 36

×9 2× × 18

× ×3 × ×7 ××

2× × ×3

18 = × × 28= ×× 36 = × × ×
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4.5 Highest Common Factor (H. C. F.)

Classwork - Exercise
Let's tell and write the correct answer as quickly as possible.
1. a) What are the all possible factors of 8? , , ,
b) What are the all possible factors of 12? , , , , ,
c) What are the Common Factors of 8 and 12? , ,
d) Which one is the Highest Common Factor of 8 and 12?
So, the Highest Common Factor (H. C. F.) of 8 and 12 is 4.
2. a) What are the all possible factors of 10? , , ,
b) What are the all possible factors of 15? , , ,
c) What are the Common Factors of 10 and 15? , ,
d) Which one is the Highest Common Factor of 10 and 15?
So, the Highest Common Factor (H. C. F.) of 10 and 15 is 5.
4.6 Process of finding multiples of a given number
Let's study these examples and investigate the process of finding multiples

of a given number.

2 × 1 = 2 2 × 2 = 4 2 × 3 = 6 2 × 4 = 8 2 × 5 =10
So, 2, 4, 6, 8, 10, ... are the first five multiples of 2.

3 × 1 = 3 3 × 2 = 6 3 × 3 = 9 3 × 4 = 12 3 × 5 =15
So, 3, 6, 9, 12, 15, ... are the first five multiples of 3.

4.7 Lowest Common Multiple (L. C. M.)
Classwork - Exercise

Let's tell and write the first ten multiples of these pairs of numbers.

1. a) Multiples of 2 are , , , , , , , , ,

b) Multiples of 3 are , , , , , , , , ,
71 vedanta Excel in Mathematics - Book 4

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c) Common Multiples of 2 and 3 are , ,
d) The Lowest Common Multiple (L. C. M.) of 2 and 3 is
2. a) Multiples of 4 are , , , , , , , , ,
b) Multiples of 6 are , , , , , , , , ,
c) Common Multiples of 4 and 6 are? , ,
d) The Lowest Common Multiple (L. C. M.) of 4 and 6 is

Exercise - 4.1
Section A - Classwork
1. Let's tell and write the correct answer as quickly as possible.

a) Numbers between 59 and 71 which are exactly divisible by 2 are

, , , , ,

b) Numbers between 89 and 100 which are exactly divisible by 3.

, , ,

c) Any 5 three-digit numbers which are exactly divisible by 4.

, , , ,

d) Any 5 three-digit numbers between 100 and 150 which are exactly
divisible by 5.

, , , ,

e) Any 5 numbers between 10 and 50 which are exactly divisible by 6.

, , , ,

f) Any 5 three-digit numbers which are exactly divisible by 9.

, , , ,

g) Any 5 three-digit numbers less than 300 and exactly divisible by 10.

, , , ,

vedanta Excel in Mathematics - Book 4 72

Factors and Multiples

2. Let's tell and write these numbers as the product of their prime factors.

a) 4 = × b) 6 = × c) 8 = × ×

d) 9 = × e) 10 = × f) 12 = × ×

g) 14 = × h) 15 = × i) 20 = × ×

3. The common factors of each pair of numbers are given. Let's circle and
write the Highest Common Factor (H. C. F.).

a) Common factors of 2 and 4 are 1, 2. So, H. C. F. of 2 and 4 =

b) Common factors of 3 and 6 are 1, 3. So, H. C. F. of 3 and 6 =

c) Common factors of 4 and 8 are 1, 2, 4. So, H. C. F. of 4 and 8 =

d) Common factors of 8 and 12 are 1, 2, 4. So, H. C. F. of 8 and 12 =

4. A few common multiples of each pair of numbers are given. Let's circle
and write the Lowest Common Multiple (L. C. M.).

a) Common multiples of 2 and 4 are 4, 8, 12, ... So, L. C. M. of 2 and 4 =

b) Common multiples of 3 and 6 are 6, 12, 18, ... So, L. C. M. of 3 and 6 =

c) Common multiples of 4 and 8 are 8, 16, 24, ... So, L. C. M. of 4 and 8 =

d) Common multiples of 6 and 9 are 18, 36, 54, ... So, L. C. M. of 6 and 9 =

Section B

5. Let's use the rules of divisibility test. Then, identify which of the
following numbers are exactly divisible by 3 or 3 and 9 both.

a) 153 b) 276 c) 387 d) 489 e) 5967

6. Rewrite and complete these Factor Trees. Then write the numbers as
the product of their prime factors.

a) 16 b) 24 c) 36
× 12 2×
2×

×× × 2× × ×
×
2× × ×3 2× ×

16 = 2 × 2 × 2 × 2 24 = 36 =
73
vedanta Excel in Mathematics - Book 4

Factors and Multiples

d) 30 e) 40 f) 54

2× 2× 2 × 27

×× ×× ××

×× ×5 ×× ×

30 = 40 = 54 =

7. Let's use the process of finding prime factors of these numbers. Then
write the numbers as the product of their prime factors.

a) 2 16 b) 12 c) 15 d) 18 e) 20 f) 24
h) 28 i) 30 j) 32 k) 35
2 8 g) 27 m) 40 n) 42 o) 45 p) 48

2 4 l) 36

2

So, 16 = 2 × 2 × 2 × 2

8. Let's write the all possible factors of each of the following pairs of
numbers. Then find their H. C. F.

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

g) 12, 16 h) 12, 18 i) 14, 21 j) 18, 24 k) 20, 30 l) 24, 30

9. Let's write the first ten multiples of each of the following pairs of
numbers. Then find their L. C. M.

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

g) 5, 10 h) 4, 8 i) 6, 9 j) 6, 8 k) 4, 10 l) 8, 10

It's your time - Project Work

10. Let's play a game of finding H. C. F. of any two numbers.

H. C. F. of 4 and 6 2 is less than 4. So, Equal number
4 is less than 6. So, remove 2 circles of circles in both
remove 4 circles from 4. sides.
from 6.
46 46
46

H. C. F. is 2.

74

vedanta Excel in Mathematics - Book 4

H. C. F. of 3 and 4 1 is less than 3. So, Factors and Multiples
3 is less than 4. So, remove 2 circles
remove 3 circles from 3. Equal number
from 4. of circles in both
34 sides.
34
34

H. C. F. is 1
Now, let's find the H. C. F. of these numbers by playing the games.

a) 2 and 4 b) 6 and 8 c) 6 and 9 d) 5 and 6 e) 8 and 12

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

Now, let's play to find the L. C. M. of 2 and 3. 4 8 7 14
3 6 6 12
2At first, pull the multiple card of 2 2 4 5 10

3Then, pull the multiple card of 3

4Again, pull the multiple card of 2

6Then, pull the multiple card of 3

6Again, pull the multiple card of 2 is the L. C. M. of 2 and 3.

Similarly, you can find the L. C. M. of 3 and 4, 4 and 6, 5 and 7 and so on...
You can play the game with a friend also. Remember, you should pull the
multiple cards of the smaller number at first.

12. Rolling number cubes (or dice)

You can play this game with a friend. Take turns 22 1122 33 3300
rolling two number cubes (or two dice). Find the
L. C. M. of the two numbers rolled and circle in the 2300 55 1155 66

square with the answer. The first person to get 4 3300 1100 44 2200

circles in a row is the winner! 66 11 99 1122

?

75 vedanta Excel in Mathematics - Book 4