1
Numbers and
Arithmetic Operations
You have already learnt:
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MATHS
FLASH
3
In this unit students will learn to:
• identify place values of digits up to one hundred million
• read and write numbers in words and numerals up to one hundred million
• compare and order numbers up to 8 digits
• add and subtract numbers up to 6 digits
• multiply numbers up to 5 digits by numbers up to 3 digits
• divide numbers up to 4 digits by numbers up to 2 digits
• solve simple sums involving four operations
• solve real-life problems involving four operations
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• t o read and write numbers in words and figures
up to 6 digits
• t o identify place value of numbers up to 6 digits
• t o compare and order 6-digit numbers
• t o represent numbers on a number line
• t o add and subtract numbers up to 4 digits
• t o multiply and divide 2-digit numbers by a 1-digit
number
1
1
KEY
vocabulary
million, place value,
compare, order,
operation, multiplier,
multiplicand,
product, divisor,
dividend, quotient,
remainder,
commutative law
Numbers and Arithmetic Operations
1
Numbers
Now we will work with numbers up to 9 digits that is hundred million.
Let us consider the following place value chart.
Millions
HM
M
HTh T Th
Ones
Th
H
T
U
1
0
0
0
0
0
6-digit number
1
0
0
0
0
0
0
7-digit number
1
0
0
0
0
0
0
0
8-digit number
0
0
0
0
0
0
0
0
9-digit number
3
1
TM
Thousands
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From the above chart we conclude that as the number of digits
increases the value of the number also increases. A 7-digit number is
a million, 8-digit number is ten million, and 9-digit number is hundred
million.
2
4
Th
H
9
6
8
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HTh TTh
T
U
5
2
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M
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Now as an example consider a 7-digit number 2 489 652 and place it in
the place value chart.
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This shows that a 7-digit number goes up to one million.
M
2
3
HTh TTh
8
9
Th
H
T
U
6
1
0
8
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TM
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When we place 23,896,108 in a place value chart we notice that a
8-digit number goes up to ten million.
While a 9-digit number 630 057 789 goes up to hundred million.
1
HM
TM
M
6
3
0
Numbers and Arithmetic Operations
HTh TTh
0
5
2
Th
H
T
U
7
7
8
9
1
Examples:
1. Write 682 952 602 in words.
Solution
First place the number in the place value chart.
HM
TM
M
6
8
2
HTh TTh
9
5
Th
H
T
U
2
6
0
2
Now we can easily write the number in word as:
2.
3
Six hundred and eighty-two million, nine hundred and fifty-two
thousand, six hundred and two.
Write the given number in figures:
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Twenty five million, nine hundred and nine thousand, four hundred
and thirty-seven.
5
9
Th
H
9
4
0
3.
T
U
3
7
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Hence the required number is 25 909 437.
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2
HTh TTh
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M
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TM
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Solution:
Make a place value chart and write the number in it.
Write the value of the ringed digit in the number 489 8 3 2 416
TM
M
4
8
9
HTh TTh
8
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HM
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Solution:
Place the number in the place value chart.
3
Th
H
T
U
2
4
1
6
The value of 3 is thirty thousand or 30 000.
4.
Write the number 469 825 136 in expanded form.
Solution:
400 000 000 + 60 000 000 + 9 000 000 + 800 000 + 20 000 + 5000 + 100 + 30 + 6
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3
Numbers and Arithmetic Operations
1
5.
The given number is written in expanded form, find the value.
500 000 000 + 90 000 000 + 6 000 000 + 000 000 + 20 000 + 1000 + 500 + 40 + 1
Solution:
The required value can be obtained by adding up all the numbers
vertically. The required number is 596 021 541
Comparing and ordering 8-digit numbers
Comparing numbers is the same as knowing which number is smaller
and which number is the bigger. Let us take an example of two 6-digit
numbers, 698 721 and 698 831
3
To compare these numbers, follow the given steps.
1. Make a place value chart and write both the numbers in it
TTh
Th
H
T
U
6
9
8
7
2
1
6
9
8
8
3
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HTh
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2.Now check the place value of each digit starting from the largest
place – hundred thousand.
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Symbolically, it is denoted as 698, 831 > 698 721.
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4.
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3.We see that the numbers in HTH, TTH and Th are equal, but in the
hundred place, 8 hundred is greater than 7 hundred which makes
698 831 greater than 698 721.
The open side of the symbol indicates the greater number.
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Now, let us compare two 8-digit numbers, 46 251 881 and 46 270 430.
Following the steps we write the numbers in the place values
TM
M
HTh TTh
4
6,
2
4
6,
2
Th
H
T
U
5
1,
8
8
1
7
0,
4
3
0
Starting from the left we find that 70 thousand in greater than 50
thousand.
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Numbers and Arithmetic Operations
4
1
Therefore,
or
46 270 430 > 46 261 881
46 251 881 < 46 270 430
The closed end of the symbol indicates the smaller number.
Using the same strategy, we can order a sequence of numbers in their
ascending or descending order.
Example:
Arrange the following numbers in ascending order.
62 345 801 ; 62 346 801 ; 62 341 801
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6 2 3 4 5 8 0 1
6 2 3 4 6 8 0 1
6 2 3 4 1 8 0 1
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Solution:
Arrange the given numbers as follows:
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Starting from the left we see that digits in the thousand place differ
from each other.
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We know that 1000 < 5000 < 6000.
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Therefore, the required ascending order is: 62 341 801, 62 345 801,
62 346 801
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or 62 341 801 < 62 345 801 < 62 346 801
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Example:
Arrange the following numbers in descending order.
73 921 016 ; 73 921 816 and 73 921 516
Solution:
Proceeding according to the steps given above we find that in the
hundreds place we have digits 0, 8, and 5.
We know that 800 > 500 > 000.
Therefore, the required descending order is:
73 921 816 , 73 92 1516 , 73 921 016
or 73 921 816 > 73 921 516 > 73 921 016
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5
Numbers and Arithmetic Operations
1
Exercise 1a
1. Fill in the blanks.
a.In 49 8 52 610, the place value of the ringed digit is
.
b.5 000 000 + 200 000 + 00 000 + 0000 + 400 + 50 + 9 =
c.Insert space to express 725895213 in International number
system.
.
d. The predecessor of 8 001 001 is
.
e. The successor of 2 316 999 is
.
.
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2. State whether the following are true or false.
)
a. The predecessor of 60 000 000 is 60 000 999. (
b. The successor to 189 561 100 is 189 561 200. (
)
c.A number at one million place has six zeroes on its right. (
d.Place value is the value of the position of a digit in a number.
)
(
e. The place value of 6 in 10 682 817 is 600 000. (
)
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3. Select the correct answer from the given options.
a. The predecessor of 61 15 999 is
6 015 999
6 115 989
6 115 998
6 116 000
b.Eighty million, four hundred and ninety thousand in figures is
80 490 000
80 400 9000
804 900 000
8 049 000
c. The successor of 5 49 62 111 is
55 062 111
64 962 111
54 962 101
54 962 112
d. 8 463 110 written in expanded form is
8 + 463 + 110
8 000 000 + 400 000 + 60 000 + 3000 + 100 + 10
800 000 + 400 + 60 + 3 + 100 + 10
800 000 + 40 000 + 6000 + 300 + 10
e. The predecessor of 7 843 002 is
7 843 001
7 834 000
7 835 002
8 834 003
4. Write the given numbers in words.
a. 3 000 000 b. 140 000 000 c. 217 000 000
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Numbers and Arithmetic Operations
6
d. 50 000 000
1
5. Write in figures with correct spacings.
a. Nine million b. Twenty four million
c. Seven hundred and five million d. Ten million 9 hundred
6. Write the given numbers in words.
a. 1 289 056 b. 27 389 100 c. 613 421 005
d. 497 560
7. Write the place value of the circled digit.
a. 7 8 9 452 100 b. 14 0 09 838
c. 25 6 943 100 d. 86 24 7 120
b. 497 215 010
d. 630 713 455
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59 899 989
106 343 201
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9. Write the successors of:
a. 146 780 000 b.
c. 871 600 009 d.
10. Write the numbers in expanded form:
a. 7 200 017b.
c. 917 245 985 d.
83 153 206
164 205 789
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11. Fill the blank with >, < or =.
a. 6 984 529 _____________ 6 894 529
b. 45 800 100 _____________ 45 810 100
c. 359 412 025 _____________ 359 412 125
d. 6 800 152 _____________ 6 800 152
e. 100 985 263 _____________ 99 985 265
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8. Write the predecessor of:
a. 60 129 808
c. 256 943 100
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12. Write numbers that match the following expanded form.
a. 6 000 000 + 900 000 + 20 000 + 5000 + 000 + 20 + 7
b. 78 000 000 + 50 000 + 7000 + 600 + 50 + 3
c. 400 000 000 + 30 000 000 + 7 000 000 + 90 + 6
13. Arrange the given numbers in ascending order.
a. 45 298 653 ; 45 288 653 ; 45 308 653
b. 7 011 262 ; 70 111 162 ; 7 002 162
14. Arrange the given numbers in descending order.
a. 69 245 823 ; 69 645 823 ; 69 045 823
b. 10 153 215 ; 10 133 215 ; 10 143 215
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7
Numbers and Arithmetic Operations
1
Addition
Adding big numbers is simple, provided we remember to write our
columns carefully, and to work from right to left.
1 1 1 1 1
362 246
+ 324 321
686 567
2 36 134
+ 4 95 987
7 32 121
3
Do not forget to start with the ones, then add the tens, then the
hundreds, then the thousands, then the ten thousands, and lastly the
hundred thousands.
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Subtraction
If we write our columns carefully, and
remember to work from the ones column
first, we find that subtraction with big
numbers is easy.
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• 200 less than 1100
• 500 less than 3560
8 12 14 15 17 1
• 1000 less than 9686
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935 68 6
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– 447 79 7
487 88 9
Exercise 1b
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– 21 246
72 221
Write the number which is
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93 467
REMEMBER
1. Fill in the blanks.
a. 10 000 + 6000 + 900 + 5
.
b. 12 500 –
= 5600.
c. 100 more than 99 990 equals to
.
d.
must be added to 100 050 to make 102 550.
e. 49 396 –
= 10 086.
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Numbers and Arithmetic Operations
8
1
2. State whether the following are true or false.
a. 375 + 405 + 163 equals 943. (
)
b. 400 more than 69 843 is 69 443. (
)
c. 7000 less than 897 536 is 890 536. (
)
d.Asif has Rs 895 123 and Fahad has Rs 896 012. Fahad has more
money than Asif. (
)
e.Anas spent Rs 728 000 to purchase a car. Salman bought a car
for Rs 725 555. Salman paid more money for his purchase.
(
)
c.
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4. Add the following.
a.
43 846
+ 35 017
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3. Select the correct answer from the given options.
a.The sum of the largest 5-digit number and the smallest 4-digit
number is
19 999
100 999
10 999
1 000 999
b. 250 000 plus 25 000 is equal to
275 000
252 500
50 000
2525
c. 2400 + 180 + 9 is equal to
4290
2409
2580
2589
d. The difference between 99 999 and 1000 is equal to
89 999
80 001
98 999
8000
e. Take away 874 528 from 900 059
25 531
36 531
174 531
1 773 587
465 357
+ 32 986
9
b.
753 865
+ 169 530
d.
329 436
+ 281 274
Numbers and Arithmetic Operations
1
5. Write vertically and add.
a. 24 639 + 10 251
c. 216 154 + 158 309
e. 316 010 + 187 684 + 3472
b. 50 466 + 18 974
d.14 838 + 249 501 + 108 633
f. 204 469 + 20 496 + 2460
6. Write the number which is:
a. 200 more than 1 25 000
c. 900 more than 4 60 100
b. 1000 more than 200 000
d. 400 more than 509 600
7. Subtract the following.
c.
5 02 395
– 1 49 146
b.
1 05 003
–  87 675
d.
7 62 043
– 5 09 567
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7 46 389
–  94 247
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8. Write these vertically and subtract.
a. 36 738 – 12 849 b. 256 704 – 103 028
c. 2 00 031 – 1 89 764 d. 500 000 – 360 824
e. 750 000 – 125 255
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9. Subtract 100 from each of the following numbers.
a. 1 46 952 b. 500 000
c. 305 103 d. 810 000
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Word Problems
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10. Subtract 500 from each of the following numbers.
a. 385 620 b. 284 320
c. 505 400 d. 69 010
Read the problems carefully. Decide whether you should add or
subtract to solve them. Write complete statements.
1.A factory made 64 750 jute bags on Monday and 51 060 more on
Tuesday. How many bags were made altogether?
2.In an election, Mr Kamal got 156 720 votes and Mrs Abid got
158 986 votes. Who got more votes and how many more?
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Numbers and Arithmetic Operations
10
1
3.A library has 46 918 books of English and 108 625 books of Urdu.
What is the total number of books?
4.A shopkeeper earned Rs 170 920 in one year. If his expenses for
the same year were Rs 129 486, how much money did he save?
5.Find the difference between the largest 4-digit number and the
smallest 6-digit number.
6.A man bought a house for Rs 956 780. He sold it for Rs 120 000
less than the cost price. How much did he sell the house for?
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7.A school in the village needs Rs 400 000 for a new building.
Mr Abdul donated Rs 150 000. How much more money must be
collected?
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Multiplication
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We have learnt the method of multiplication of numbers in previous
classes. Now we will learn to multiply 5-digit numbers with 3-digit
numbers.
1458
9720
11178
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46
multiplier
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*
multiplicand
(243 * 6)
(243 * 40)
product
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243
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Each part of a multiplication sum has a special name in maths.
The multiplicand is the number or quantity to be multiplied.
The multiplier is the number or quantity by which the multiplicand is
to be multiplied.
The product is simply the end result (answer) of the multiplication.
1
11
Numbers and Arithmetic Operations
1
Example:
Find the product of 24 358 � 465.
24358
* 465
1
1
121790
(24358 * 5)
1461480
(24358 * 60)
+ 9743200
(24358 * 400)
11326470
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Division
3
We multiply by the ones of the multiplier first, then the tens of the
multiplier, and then the hundreds of the multiplier. Then add to get
the product.
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We have already done division of 2-digit numbers by 1-digit divisors.
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We have also learnt the special words used for the different parts of a
division.
58
Quotient
8 469
Divisor
Dividend
–40
69
–64
5
Remainder
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Division of 4-digit numbers by 2-digit numbers
It is easy to work with 4-digit dividends if we carry out our division
steps carefully.
Example:
6914 ÷ 34
First, we look at the thousands.
34 6914
6 < 34, so we put the 6 thousands together with the 9 hundreds.
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Numbers and Arithmetic Operations
12
1
69 > 34, so now we can divide.
2
34 6914
–68
1
(34 � 2)
Next, we take the remainder of 1 hundred with the 1 in the tens
column.
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20
34 6914
–68
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11 < 34, so we write 0 in the tens column of the quotient.
Then, we work as shown below:
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203
Divisor 34 6914
–68
114
–102
12
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114 > 34, so now we can easily divide.
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20
34 6914
–68
114
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Next, we take the remainder of 11 tens with the 4 in the ones column:
Quotient
dividend
(34 � 3)
remainder
6914 ÷ 34 = 203 rem 12
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13
Numbers and Arithmetic Operations
1
Exercise 1c
1. Fill in the blanks.
a. In 3256 * 928 ; 3256 is called the
.
b.The quantity by which the multiplicand is to be multiplied is
known as
.
c. The number which divides a dividend is called a
.
d. 18 400 ÷ 100 =
.
e.If the dividend is 348 and the divisor is 12, then the remainder is
.
3
2. State whether the following are true or false.
a. When 1485 is divided by 6 the remainder is zero. (
b. In 429 ÷ 8; the number 429 is the dividend. (
)
)
c. If we divide 3696 by 12, the quotient is 308. (
)
e. 1008 * 35 is 350 280. (
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d.The product of two numbers is 274 800. If one of the numbers is
100, the other number will be 27 480. (
)
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)
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3. Select the correct answer from the given options.
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b. 1500 * 200 equals
300
3000
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a. The result of multiplication of two numbers is called the
multiplier sum
product quotient
300 000
30 000
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c. If we divide 4592 by 26 the remainder is
6 26
16 0
d. The number left in the end of a long division is called the
divisor dividend
quotient
remainder
e.If the cost of one book is Rs 35, then to find the cost of 13 books
we will
multiply
add
divide subtract
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Numbers and Arithmetic Operations
14
1
4. Multiply the following.
a. 61 964
* 279
b.
38 547
* 473
8501
c.
* 906
5. Write these vertically and multiply.
a. 72 562 * 169 b. 80 093 * 200
c. 40 981 * 624 d. 34 571 * 843
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6. Find the product.
a. multiplicand 5829, multiplier 431
b. multiplicand 601 835, multiplier 297
7.Divide the following.
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e. 43 5629 f. 39 2084
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c. 31 6485 d. 18 1002
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a. 28 4042 b. 26 1064
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8. Write in long division form and divide.
a. 4618 ÷ 29 b. 9320 ÷ 86
c. 3047 ÷ 53 d. 6593 ÷ 72
e. 3271 ÷ 49 f. 4818 ÷ 35
check.
Dividend 4082, divisor 62
Dividend 5641, divisor 13
Dividend 82 721, divisor 94
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9. Divide the following, then multiply to
a. Dividend 399, divisor 17 b.
c. Dividend 6351, divisor 49 d.
e. Dividend 3001, divisor 81 f.
10. Write the quotients in the blanks.
a. 4900 ÷ 49 =
b. 6500 ÷ 10 =
c. 8000 ÷ 20 =
d. 5100 ÷ 17 =
e. 4800 ÷ 40 =
f. 8400 ÷ 12 =
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15
Numbers and Arithmetic Operations
1
Word Problems
Solve the problems, writing complete statements.
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1.A leaking tap wastes 3750 ml of water in 1 hour. How much water
will be wasted in a day?
2.Rehan found out that he can save 4726 ml of water every day
by not leaving the tap open continuously, during his bath. How
much water can be saved in 25 days in this manner?
3.There are 1440 minutes in a day. How many minutes are there in
8 weeks?
4.Thirty children made 5280 paper bags. How many did each child
make?
5.How many pieces of cloth can be cut from a 5 m length of cloth,
if each piece is 50 cm long?
6.From several orchards, delicious apples are packed in boxes of 72
each. How many boxes will be required for 936 apples?
7.Five doctors travel from Karachi to Lahore to attend a conference.
The total bill for their train tickets is Rs 9675. How much does
each doctor pay?
8.How many hours are there in 2700 minutes?
9.Four friends share a flat. Their total expenses for January are
Rs 9256. How much money must each friend contribute?
Using four operations
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Addition, subtraction, multiplication, and division are known as the
four operations.
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We should be able to perform them quickly and accurately as we
tackle more difficult and exciting maths problems.
We follow the given rules when mixed operations are used.
Rule
1
When there are two operations (‘+’ and ‘–’ only) or (‘*’ and ‘÷’
only) in the given expression, we start working from the left and
operations are used in sequence.
1
Numbers and Arithmetic Operations
16
1
Examples:
1.
12 – 5 + 8
2.
3+4–2
Solution
12 – 5 + 8 (first subtract)
= 7 + 8
(then add)
= 15
Solution:
3 + 4 – 2 (add first)
= 7 – 2
(then subtract)
=5
3.
4.
18 ÷ 2 * 3
Rule
2
3
Solution:
18 ÷ 2 * 3 (first divide)
=9*3
(then multiply)
= 27
5×4÷2
Solution:
5*4÷2
= 20 ÷ 2
= 10
(first multiply)
(then divide)
Solution:
10 + 9 * 5
= 10 + 45
= 55
4.
55 � 15 ÷ 5 – 25
Solution:
55 � 15 ÷ 5 – 25
= 55 � 3 – 25
= 165 – 25
= 140
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6 – 55 ÷ 11
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2.
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10 + 9 * 5
Solution:
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1.
6 – 55 ÷ 11
=6–5
=1
5.
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Examples:
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When there are two (‘*’ with ‘+’ or ‘–’) or (‘÷’ with ‘+’ or ‘–’) or
more than two operations then we use DMAS rule, where D is
for division, M is for multiplication, A is for addition, and S is for
subtraction. The operations are used in the same sequence as given
in DMAS.
85 + 24 ÷ 3 – 11
Solution
85 + 24 ÷ 3 – 11
= 85 + 8 – 11
= 93 – 11
= 82
17
3.
8–4*1+3
Solution:
8–4*1+3
=8–4+3
=4+3
= 7
6.
12 + 8 � 2 – 6 ÷ 2
Solution:
12 +
= 12 +
= 12 +
= 28 –
= 25
8�2–6÷2
8�2–3
16 – 3
3
Numbers and Arithmetic Operations
1
Exercise 1d
1. Fill in the blanks.
a. 4 � 5 –
= 12
b.9 + 5 – 4 � 2 =
c.
� 5 ÷ 5 = 10
d. 8 + 9 – 6 – 5 =
e. 6 +
–8=0
3
2. State whether the following are true or false.
a. 9 + 1 – 6 = 4 (
)
b. 4 � 5 – 3 = 8 (
)
c. 49 ÷ 7 + 10 = 17 (
)
)
d. 54 – 4 � 10 ÷ 5 = 46 (
e. 3 � 2 + 4 – 3 = 9 (
)
1
15
22
13
56
b. 831 * 500 ÷ 100
d. 144 ÷ 9 * 4
5. Solve
a. 140 * 65 + 130
c. 70 – 7 * 10
b. 100 – 75 ÷ 5
d. 28 ÷ 4 + 24
6. Solve
a. 9 * 5 – 15 + 8
d. 7 * 6 – 8 + 15 ÷ 3
g. 120 ÷ 12 * 3 – 15
b. 4 + 9 ÷ 3 * 5
e. 42 + 8 – 9 * 5
h. 36 – 63 ÷ 7 + 8
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4. Solve
a. 524 + 48 – 62
c. 91 – 45 – 30
Numbers and Arithmetic Operations
23
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3. Select the correct answer from the given options.
a. 3 * 5 + 8
16 39
120
b. 2 * 4 + 8 – 9
6 9
7
c. 10 ÷ 5 * 6 + 5
35
55
17
d. 18 – 4 + 9
23
31
5
e. 4 + 8 * 3 – 20
16 8
140
18
c. 72 + 8 – 4 * 2
f. 36 * 9 ÷ 3 + 8
i. 21 – 10 * 2 ÷ 2
1
2
Factors and Multiples
• times tables
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You have already learnt:
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MATHS
FLASH
3
In this unit students will learn to:
• use divisibility tests for 2, 3, 5, and 10 on numbers up to 5 digits
• differentiate between prime and composite numbers
• list factors of a number up to 50
• list the first twelve multiples of a 1-digit number
• differentiate between factors and multiples
• factorise a number by using prime factors
• find common factors and multiples of two or more 2-digit numbers
• find the HCF of two or more 2-digit numbers using a Venn diagram and prime
factorisation
• find the LCM by common multiples and prime factorisation
• solve real-life problems involving HCF and LCM
• to identify even and odd numbers
• t hat even numbers can be arranged in pairs
without leaving a remainder.
For example, 8 can be arranged as:
• that odd numbers cannot be arranged in
pairs as they always leave a remainder.
For example, 9 can be arranged as:
1
19
KEY
vocabulary
divisibility, prime,
composite, factors,
prime factors, Highest
common factor,
HCF, multiples, Least
common multiple,
LCM,
Factors and Multiples
2
Tests of divisibility
Divisibility rules of whole numbers are very useful to find out if a
number can be divided by 2, 3, 4, 5, 6, 7, 8, 9, 10. With the help of
these rules we do not need to do long divisions. These rules help us
to test if one number is divisible by another number without doing
a lengthy calculation. We use these tests to find out the factors that
make up a number.
Rules of divisibility
on
ls
Rule
Solution:
Yes, because 138 is an even
number and it has 8 at its
units place. According to rule
1, it is divisible by 2.
2
Factors and Multiples
rF
az
Fo
1. Is 138 divisible by 2?
Any number with 0 at the
unit place is divisible by 10.
aia
Any number with 0 or 5 at
the unit place is divisible
by 5.
Examples
4
Sc
3
If the digits of any number
add up to a number which
is divisible by 3, then the
original number is also
divisible by 3.
ly
Any number with 0, 2, 4, 6, 8
at the unit place is divisible
by 2. All even numbers are
divisible by 2.
Rule
2
Rule
ho
o
1
3
Rule
2. Is 105 divisible by 3?
Solution:
Add the digits.
1 + 0 + 5 = 6; 6 is divisible by 3.
herefore, 105 is also divisible
T
by 3.
20
1
3.
Is 593 divisible by 3?
do you
know?
Solution:
Add the digits: 5 + 9 + 3 = 17
17 is not divisible by 3.
• ‘Divisible by’ and ‘can
be exactly divisible
by’ means the same.
• Every number is
divisible by 1.
Hence, the original number 593 is not
divisible by 3.
b.
Is 195 divisible by 5?
Solution:
Yes, 195 is divisible by 5, because it has 5 at the unit place.
Is 230 is divisible by 5?
Solution:
Yes, 230 is divisible by 5, because it has 0 at the unit place.
ly
a.
3
4.
ls
on
c.Which of the following numbers are divisible by 5? Circle
them and give reason.
Sc
Is 240 divisible by 10?
aia
5.
ho
o
Solution:
780 , 225 , 192, 263
780 and 225 are divisible by 5, because they have 0 and 5 at
their unit places.
a.
Is 4055 divisible by 5 and 10?
Fo
6.
rF
az
Solution:
Yes, 240 is divisible by 10 because it has 0 at the unit place.
Solution:
4055 is divisibly by 5 because it has 5 at the unit place, but
4055 is not divisible by 10 because its unit place is not 0.
b.
Is 1850 divisible by 5 and 10?
Solution:
1850 is divisible by 5 because it has 0 at the unit place.
1850 is divisible by 10, because it has 0 at the unit place.
1
21
Factors and Multiples
2
Exercise 2a
1. Fill in the blanks.
a. A number is divisible by
b. 156 is divisible by
if it has 5 at the unit place.
,
, and
.
c.If a number has 0 at the unit place, it is divisible by
and
.
,
d.If the sum of digits is divisible by 3, then the number itself is
divisible by
.
e. All even numbers are divisible by
.
3
2. State whether the following are true or false.
a. 20 589 is divisible by 3. (
)
b. 160 is divisible by 2. (
)
c. 225 is divisible by 5 and 10. (
ly
)
e. 5563 is divisible by 5. (
)
on
d. Any number which is divisible by 10 is divisible by 5. (
)
Fo
rF
az
aia
Sc
ho
o
ls
3. Select the correct answer from the given options.
a. 6234 is divisible by
5
2 and 3
2 and 10
2 and 5
b. 4600 is divisible by
5 only
2 and 10
2 and 5
2, 5 and 10
c. 525 is divisible by
5 and 3
2 and 5
5 only
10
d. A number can be divided by 5 if the digit at the unit place is
even
5 or 0
5 only
odd but not 5
e. 5 * 3 is divisible by
5 and 3
5 only
3 only
2 only
4. Which of the following numbers are divisible by 2?
a. 200b. 126c. 427d. 187
e. 134 f. 2032 g. 139 h. 345
2
5. Which of the following numbers are divisible by 3?
a. 624b. 2358c. 130
d. 3612
6. Which of the following numbers are divisible by 5?
a. 2900 b. 6954 c. 4085
d. 840,050
Factors and Multiples
22
1
7. Which of the following numbers are divisible by 10?
a. 7281 b. 81 080 c. 10 000 d. 90 005
e. 52 010 f. 40 185 g. 4002h. 8760
8. Use the rules of divisibility to check whether each given number is
divisible by 2, 3, 5, or 10. Write Yes or No.
Divisible by
Numbers
1872
2
3
5
10
Yes
Yes
No
No
53 250
3
673 655
2971
on
ly
4720
ls
Prime and composite numbers
Sc
ho
o
Prime numbers
A prime number has only two factors that is 1 and the number itself.
We can say if a number cannot be divided exactly by any other
number except 1 or itself, then it is a prime number.
2
3
1
1
1, 2
1, 3
Number
rF
az
1
Factors
Fo
Number
aia
Look at the following table.
Factors
10
1, 2, 5, 10
11
1, 11
12
1, 2, 3, 4, 6, 12
4
1, 2, 4
13
1, 13
5
1, 5
14
1, 2, 7, 14
6
1, 2, 3, 6
15
1, 3, 5, 15
7
1, 7
16
1, 2, 4, 8, 16
8
1, 2, 4, 8
17
1, 17
9
1, 3, 9
18
1, 2, 3, 6, 9, 18
23
Factors and Multiples
2
Number of factors
This result can be shown on a column graph.
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
Number
3
We see that 2, 3, 5, 7, 11, 13 and 17 have only two factors, that is 1
and the number itself.
ly
Numbers which have only two different factors—the number itself and
1 have a special name. They are called prime numbers.
Co-prime numbers
aia
What are the factors of this pair?
factors of 2: 1, 2
factors of 3: 1, 3
Sc
Now look at this pair of numbers: 2 and 3
ho
o
ls
on
Number 1 is not a prime number because it has only one factor.
Fo
rF
az
There is only one common factor here, and that common factor
is 1. Two numbers which have only 1 as their common factor are
called co-prime numbers.
Example:
Are 9 and 16 co-prime numbers?
Solution:
factors of 9: 1, 3, 9
factors of 16: 1, 2, 4, 8, 16
Yes, 9 and 16 are co-prime numbers.
2
Factors and Multiples
24
1
A magic way of showing all prime numbers from 1–100.
1
2
3
4
5
6
7
8
9
10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
3
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
ly
81 82 83 84 85 86 87 88 89 90
ls
on
91 92 93 94 95 96 97 98 99 100
ho
o
Take a colour pencil or crayon and follow the given steps.
Step
1
Step
2
Step
3
Step
4
5 is a prime number. Except for 5 itself, colour all squares
containing multiples of 5 (10, 15, and so on). You may
have already coloured some of these numbers.
Step
5
7 is a prime number. Except for 7 itself, colour all squares
containing multiples of 7 (14, 21, and so on). Many of
these squares will already be coloured.
1
Sc
Do not colour the cell containing the number 1.
rF
az
aia
2 is a prime number. Except for 2 itself, colour all cells
containing multiples of 2 (4, 6, 8, 10, and so on).
Fo
3 is a prime number. Except for 3 itself, colour all cells
containing multiples of 3 (6, 9, 12, and so on). You may
have already coloured some of the numbers.
25
Factors and Multiples
2
1.
Look at your chart.
Why are we not asked to colour multiples of 4 or 6?
2.Look at the chart again and write down all the numbers which
have been left uncoloured. How many such numbers are there
altogether?
All the numbers which are coloured are composite numbers.
3
All the numbers left uncoloured are prime numbers. Make
a list of these. Each has only 2 factors: itself and 1.
on
ly
Composite numbers
We already know that prime numbers are numbers which have only
two different factors.
ho
o
ls
All other numbers (except the number 1) have three or more different
factors. They, too, have a special name: composite numbers.
aia
Sc
Composite means made up of several or many different parts.
rF
az
We can say that if a number can be divided exactly by numbers other
than 1 and the number itself, it is a composite number.
Fo
By arranging pebbles, we discover that composite numbers can always
be arranged in exact rectangles.
9
12
15
Prime numbers cannot be arranged in this way.
5
2
Factors and Multiples
11
26
1
Exercise 2b
1. Fill in the blanks.
a. Composite numbers have more than
factors.
b. Each prime number has exactly
c.
factors.
is neither prime nor a composite number.
d. 50 is a
number.
e. Every prime number except
is odd.
2. State whether the following are true or false.
a. 10 is a prime number. (
)
3
b. 1, 3, and 13 are factors of 13. (
)
c. 8 is a composite number because it has four factors. (
d. 45 is a composite number. (
)
)
)
on
ly
e. 18 is a prime number because it has two factors only. (
3. Select the correct answer from the given options.
1
3
b. The composite numbers have
one factor
two factors
more than two factors
uncountable factors
aia
Sc
ho
o
ls
a. The smallest prime number is.
6
2
rF
az
c. The largest prime number less than 30 is
23
21
20 29
a factor of 50
a composite number
e. 38 is
a composite number
a prime number
a multiple of 9
an odd number
1
Fo
d. 47 is
a prime number
an even number
27
Factors and Multiples
2
4.Check whether the numbers given in the table below are prime
or composite. Write (P) for each prime number and (C) for each
composite number.
a. 15
b. 31
c. 24
d. 21
e. 32
f. 25
g. 29
h. 3
i. 37
j. 17
k. 45
l. 11
m. 18
n. 33
o. 25
p. 54
q. 35
r. 83
s. 57
t. 41
5. Which of the following are co-prime numbers?
c. 4 and 9
d. 14 and 25
e. 36 and 49
f. 72 and 55
g. 40 and 4
h. 5 and 10
on
Factors
ly
b. 31 and 62
3
a. 3 and 5
5.
aia
rF
az
4 * 3 = 12
4.
Fo
in threes
in twos
2.
12 * 1 = 12
3.
ho
o
in ones
Sc
1.
ls
We can arrange objects in groups. In the table given below 12 pebbles
have been arranged in different ways.
6.
in sixes
2 * 6 = 12
6 * 2 = 12
in fours
3 * 4 = 12
in twelves
1 * 12 = 12
The 12 pebbles have been arranged in ones, twos, threes, fours, sixes,
and twelves, with none left over.
2
Factors and Multiples
28
1
From the given example, we conclude that 12 can be divided
completely by 1, 2, 3, 4, 6, and 12.
12
12
12
12
12
12
÷ 1 = 12rem
÷ 2 = 6rem
÷ 3 = 4rem
÷ 4 = 3rem
÷ 6 = 2rem
÷ 12 = 1rem
0
0
0
0
0
0
NOTE
When a number is
divided by one of its
factors, there is no
remainder.
3
Number which can divide a given number leaving no remainder are
called factors of the given number.
15 ÷ 1 = 15 rem 0
or
15 ÷ 3 = 5 rem 0
on
ls
65
1 � 15 = 15
3 � 5 = 15
Stop when numbers begin to repeat.
15 ÷ 5 = 3 rem 0
15 ÷ 15 = 1 rem 0
Fo
∴ factors of 15 are 1, 3, 5, and 15.
b.
d.
ho
o
Solution
72
Sc
15
c.
rF
az
a.
49
aia
Examples:
Find all the factors of
b.
a. 15
ly
Here 1, 2, 3, 4, 6, and 12 are called factors of the number 12.
49
Solution
49 ÷ 1 = 49 rem 0
49 ÷ 7 = 7 rem 0
49 ÷ 49 = 1 rem 0
or
1 � 49 = 49
7 � 7 = 49
(factors will repeat from here onwards)
∴ factors of 49 are 1, 7, and 49.
1
29
Factors and Multiples
2
c.
72 (use rules of divisibility to find factors.)
Solution:
72 ÷ 1 = 72 or 72 ÷ 72 = 1 rem 0
or
1 � 72 = 72
72 ÷ 2 = 36 or 72 ÷ 36 = 2 rem 0
2 � 36 = 72
72 ÷ 3 = 24 or 72 ÷ 24 = 3 rem 0
3 � 24 = 72
72 ÷ 4 = 18 or 72 ÷ 18 = 4 rem 0
4 � 18 = 72
72 ÷ 6 = 12 or 72 ÷ 12 = 6 rem 0
6 � 12 = 72
72 ÷ 8 = 9 or 72 ÷ 9 = 8 rem 0
8 � 9 = 72
3
(factors will repeat from here
onwards)
∴ factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.
5 � 13
= 13
65 ÷13 = 5
65 ÷ 65 = 1
ly
65 ÷ 5
on
1 � 65
REMEMBER
• Factors of a number are
limited.
• Every number is a factor of
itself.
• 1 is a factor of every number.
ls
Solution:
65 ÷ 1 = 65
∴ factors of 65 are 1, 5, 13, and 65.
ho
o
65
Sc
d.
aia
In the above examples we notice that the number 1 and the number
itself appear in every list.
1. Fill in the blanks.
a.
Fo
Exercise 2c
rF
az
1 � 15 = 15; 1 � 49 = 49; 1 � 72 = 72, and 1 � 65 = 65.
is a factor of all the numbers.
b. Factors of 6 are
.
c. 1, 2 are factors of
.
d. The factors of 20 are 1, 2, 4, 5, 10, and
e. 13 is a factor of
2
Factors and Multiples
.
.
30
1
2. State whether the following are true or false.
a. 7 is a factor of 45. (
)
b.If a number divides another number completely, then the first
number is a factor of the other number. (
)
c. 6 and 8 have an equal number of factors. (
)
d. A number is always a factor of itself. (
)
e. 45 is a factor of 9. (
)
ly
on
ls
ho
o
Sc
aia
3
3. Select the correct answer from the given options.
a. 2 and 7 are factors of
3 7
14 2
b. 10 is not a factor of
32
240
20 60
c. All possible factors of 50 are
10, 5
1, 2, 5, 10, 25 and 50
1, 5, 25, 50
5, 10, 50
d. 8 is not a factor of
2 8
16 32
e. Which of the following is a factor of every number?
0 1
2 10
a.
b.
c.
d.
e.
Is
Is
Is
Is
Is
12
7
6
20
16
a
a
a
a
a
factor
factor
factor
factor
factor
of
of
of
of
of
36?
45?
66?
100?
64?
Fo
5. Write Yes or No.
rF
az
4. Find the factors of the following numbers.
a. 9b. 11c. 18
d. 21e. 24f. 35
6. Write down the factors of these numbers. How many factors does
each number have?
a. 25 b. 36 c. 54 d. 32 e. 45 f. 50
1
31
Factors and Multiples
2
Prime factorisation
Let us take the composite
number 12 and think of it as
the topmost part of a tree.
Let us now go down the tree
by thinking of the factors
that make 12:
3
Here, the factors are 2 and 6.
2 is a prime number, but 6 is
a composite number. We can
break 6 down to its prime
factors, 2 and 3.
Example:
12
Find Prime factors of 18
Solution:
2 * 6
2 * 3
18
18
2 * 9
3 * 6
3 * 3
12
3 * 2
3 * 4
2 * 2
Prime factors of 18 are
2�3�3
ly
Here, the factors are 3 and 4. 3 is a prime number, but 4 is a composite
number. We can break 4 down to its prime factors, 2 and 2.
on
2 * 2 * 3 are the prime factors of 12.
ho
o
ls
When we break down a number to its prime factors, we call that the
prime factorisation of that number.
Sc
Prime factors can never be composite numbers.
aia
An easy method to break up larger numbers into their prime factors is
the division method.
rF
az
Let us first find the prime factors of 36 and 45, using the division method.
2 ) 36
Fo
36 is an even number, so we divide by 2:
2 ) 18
we divide by 2 again
3 )   9
we divide by 3
3) 3
we divide by 3 again
1
The prime factors of 36 are 2, 2, 3, and 3.
2
Factors and Multiples
32
1
45 is an odd number, but it can be divided by 3.
3 ) 45
3 ) 15
we divide by 3 again
5) 5
we divide by 5 again
1
The prime factors of 45 are 3, 3, and 5.
.
ls
on
1. Fill in the blanks.
a. 2, 3, 5 are prime factors of
.
b. The prime factors of 41 are 1 and
.
c. There are
prime factors of 58.
d. The prime factors of 10 are
and
e. Prime factor can never be a
number.
ly
3
Exercise 2d
1
Fo
rF
az
aia
Sc
ho
o
2. State whether the following are true or false.
a.The prime factor of a number divides the number completely.
)
(
b. The prime factors of 100 are 2, 3, 5, 5, 10. (
)
c. A factor tree is used to find prime factors. (
)
d. Prime factors of numbers are prime numbers. (
)
e. 2 is the only prime factor of the number 6. (
)
3. Select the correct answer from the given options.
a. 3, 3, 3 and 5 are prime factors of
27 45
15
135
b. The prime factors of 110 are
5, 11
2, 5, 11
2, 2, 5, 5
2 and 55
c. The prime factors of 50 are 2, 5 and
2 1 5
10
d. The prime factor is always
a prime number
the smallest factor
a composite number
the biggest factor
33
Factors and Multiples
2
e. Prime factorisation means
dividing the number into two parts
finding the product of the number
multiplying the number with prime numbers
breaking a number into prime factors
4. Draw factor trees to
a. 15 b.
e. 21f.
i. 42j.
show the prime factors of the given numbers.
20c. 14d. 8
22g. 16h. 27
30
5.Draw as many different factor trees as you can to show the prime
factors of these numbers.
b.
48
3
a. 40
c.
60
d.
24
e.
50
f.
72
6.Find the prime factors of these numbers using the division method.
117 c.
333
d. 126 e.
f.   99
ls
Highest Common Factor (HCF)
ho
o
Common factors
Sc
Look at these 2 sets of factors:
factors of 12:
1, 2, 3, 4, 6, 12
factors of 18:
1, 2, 3, 6, 9, 18
aia
520
ly
b.
on
a.  84
rF
az
Did you notice that some of the factors, that is 1, 2, 3, and 6, appear in
both sets?
Fo
Because these factors are common to two different numbers, we call
them common factors. Common factors can also be found using a Venn
diagram.
Look at the prime factors of 36 and 45:
factors of 36 = 2 * 2 * 3 * 3
factors of 45 = 3 * 3 * 5
The common factor of 36 and 45 is 3 * 3 = 9.
This can be illustrated using a diagram.
2
Factors and Multiples
34
1
The rectangle below contains two loops with all prime factors of 36 and 45.
prime factors of 36
2
2
prime factors of 45
3
3
3
3
5
36 2
2
3
3
REMEMBER
5 45
HCF of two or more 2-digit
numbers can be found using
a Venn diagram.
ly
common factors
ls
∴ the common factor of 36 and 45 is 3 * 3 = 9.
on
3
Let us now combine the two loops, one showing the prime factors of 36
and the other showing the prime factors of 45.
ho
o
Such diagrams as shown above are called Venn diagrams.
Sc
Now look at these sets of factors:
factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
aia
factors of 18: 1, 2, 3, 6, 9, 18
rF
az
The common factors of 24 and 18 are: 1, 2, 3, and 6.
Fo
The greatest, or Highest Common Factor of numbers 24 and 18 is 6. This
is also called HCF.
So far, we have been looking at the common factors and HCF of pairs
of numbers. Now let us take 3 numbers: 16, 32, and 40.
factors of 16: 1, 2, 4,
8, 16
8, 16, 32
factors of 32: 1, 2, 4, factors of 40: 1,
2, 4, 5, 8, 10, 20, 40
The common factors of 16, 32, and 40 are: 1, 2, 4, and 8.
The HCF of 16, 32, and 40 is 8.
1
35
Factors and Multiples
2
To find HCF of two or more numbers using prime factorisation method
we must first find the prime factors of the given numbers.
Example:
Find the HCF of 12 and 21 using prime
factorisation method.
Solution:
hint
2 12
3 21
2
6
7
3
3
Start with the smallest
possible prime number
which completely
divides the number
leaving no remainder.
7
1
3
1
Keep on dividing till
you reach 1.
on
ly
Prime factors of 12 = 2 * 2 * 3
Prime factors of 21 = 3 * 7
Common factors of 12 and 21 = 3
ho
o
ls
∴ HCF of 12 and 21 is 3.
aia
2 78
2 130
2 26
3 39
5
13 13
13 13
13 13
1
1
rF
az
2 52
65
1
Fo
Solution:
Sc
Example:
Find HCF of 52, 78, and 130 by prime factorisation method.
Prime factors of 52 = 2 * 2 * 13
Prime factors of 78 = 2 * 3 * 13
Prime factors of 130 = 2 * 5 * 13
Common factors of 52, 78, and 130 = 2 * 13
∴ HCF of 52, 78, and 130 is 26.
2
Factors and Multiples
36
1
We can solve real-life problems using HCF. To solve word problems
involving HCF we should remember some keywords such as maximum,
greatest, and equal.
Example:
A 12 feet long and 8 feet wide room’s floor is to be covered with tiles.
What should be the maximum size of the tile to be used?
Solution:
Maximum is the keyword, so we will find the HCF of 12 and 8, by
finding their prime factors.
2
8
2
6
2
4
3
3
2
2
Prime factors of 8:
2 *2*2
Sc
2 *2 *3
aia
Prime factors of 12:
1
ho
o
1
ly
12
on
2
ls
3
Find the prime factors of 12, 8.
rF
az
HCF of 12 and 8 is 2 * 2 = 4.
Exercise 2e
1. Fill in the blanks.
Fo
∴ the size of the tile should be 4 square units.
a. The common factor of 2 and 3 is
.
b. The common factors of 32 and 4 are 2 and
c. The common factors of 8 and 16 are
d. The HCF of 8, 10, and 12 is
e.
1
.
.
.
is the HCF of 12 and 18.
37
Factors and Multiples
2
2. State whether the following are true or false.
a. 4 and 6 have no common factors. (
)
b.1, 2, 3, and 4 are the only common factors of 12 and 24. (
)
c. 20 is the common factor of 10 and 20. (________)
d.The HCF of two or more prime numbers is equal to one. (
e. The HCF of 20, 30, and 40 is 240. (
)
ls
on
ly
The greatest number which divides 42, 51, and 63 is
7 6 3
21
The HCF of 17, 19, and 31 is
31
1 19
31
The highest common factor of 81 and 45 is
3
9
5
15
The common factors of 26 and 78 are
13 only
1
1, 2 and 13 only
1, 2, 13 and 26
The HCF of 85 and 95 is
5
17 15
19
3
3. a.
b.
c.
d.
e.
)
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4.Write the factors of these pairs of numbers, and circle the common
factors.
a. 10, 18 b. 12, 1
c. 25, 15 d. 16, 20
e. 10, 32 f. 14, 21
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5. Find the HCF of the following pairs of numbers.
a. 32, 24 b. 50, 25
c. 48, 30 d. 42, 70
e. 64, 88 f. 76, 28
6. Find the common factors and HCF of each of the following sets.
a. 9, 12, and 15 b. 5, 25, and 35
c. 18, 20, and 24 d. 18, 27, and 39
e. 12, 16, and 20f. 14, 49, and 28
7.Use the division method to find the prime factors of these numbers.
(Hint: Use rules of divisibility)
a. 230 b. 3200
c. 459 d. 4545
2
Factors and Multiples
38
1
8. Using the division method, find the HCF of the following numbers.
a. 12 and 20b. 36 and 24
c. 14 and 35 d. 48, 30, and 40
e. 15, 21, and 33 f. 121, 132, and 165
9. Circle the pairs that are co-prime numbers.
a. 4 and 8 b. 2 and 10
c. 5 and 9 d. 2 and 17
e. 5 and 10 f. 14 and 4
Word Problems
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3
Solve the problems, writing complete statements.
1.Arif brought 40 chocolate bars and 60 marshmallows to distribute
among his friends on his birthday. Find the maximum number of
students to whom he can divide the sweets evenly.
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ls
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2.There are two pieces of ribbons having length 75 metres and 90
metres. If equal pieces are cut from the two pieces of ribbons,
what will be the maximum length of each piece?
3.Find the greatest number which divides 208 and 64.
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What is the greatest number that divides 30, 36, and 96 exactly?
Multiples
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5.
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4.Two drums contain 350 l and 450 l of water respectively. What
will be the maximum capacity of a container that exactly
measures the water in two drums?
100 ÷ 10 = 10
1000 ÷ 10 = 100
10,000 ÷ 10 = 1000
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When we divide each of the following numbers by 10 we get a
quotient with no remainder.
rem 0
rem 0
rem 0
Here 100, 1000, and 10,000 are known as multiples of 10.
A multiple is therefore a number which can be divided by another
number, without leaving any remainder.
1
39
Factors and Multiples
2
Examples:
6 is a multiple of 2.
6 ÷ 2 = 3 rem 0
challenge
a.If we add an even
number to another
even number, we get
an ____ number.
6 is also a multiple of 3.
6 ÷ 3 = 2 rem 0
56 is a multiple of 7.
56 ÷ 7 = 8 rem 0
56 is also a multiple of 8.
56 ÷ 8 = 7 rem 0
3
We learned earlier that even numbers can be
made into pairs, while odd numbers cannot
be made into pairs, leaving a remainder.
We can now describe even numbers.
b.If we add an odd
number to another
odd number, we get
an ____ number.
c.If we add an even
number to an odd
number, we get an
____ number.
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All even numbers are multiples of 2.
REMEMBER
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Example: List the first ten even and odd
numbers.
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• Multiples of a number
are unlimited.
• Every number is a
multiple of 1.
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Solution:
Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19
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Common multiple
Look at these two sets of the first ten multiples of 2 and 3.
What is special about the sets?
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2, 4, 6, 8, 10, 12, 14, 16, 18
3, 6, 9, 12, 15, 18, 21, 24, 27, 30
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multiples of 2:
multiples of 3:
Some multiples appear in both of them. We can show the pairs by
looping them like this:
multiples of 2:
2, 4, 6 , 8, 10, 12 , 14, 16, 18
multiples of 3:
3, 6 , 9, 12 , 15, 18 , 21, 24, 27, 30
Because 6, 12, and 18 are all multiples of both 2 and 3, we give them a
special name. We call them common multiples.
2
Factors and Multiples
40
1
Let us find common multiples using a Venn diagram.
The rectangle below contains all whole numbers from 1 to 20, and a
closed loop with no numbers at all.
12345
6 7
8 9
10 11
12 13
14 15 16 17 18 19 20
7
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11
1 3 5
2468
9
10 12 14 16
13
18 20
15 17 19
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3
Let us place all the multiples of 2 inside the loop.
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All the numbers left outside the loop are not multiples of 2.
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10
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8
13 14 16 17 19 20
11
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7
1245
3 6 9
12 15 18
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o
Let us place all the multiples of 3 inside another loop in another
diagram.
Fo
Let us now combine the two loops, one showing the multiples of 2 and
the other showing the multiples of 3.
multiples of 2 multiples of 3
24
8 10 14
16 20
6
12
18
39
15
1 5 7 11 13 17 19
These are the common multiples of 2 and 3.
1
41
Factors and Multiples
2
Lowest Common Multiple (LCM)
Now let us find the lowest common multiple of two or more than two
numbers.
Example 1: Find the LCM of 6 and 8 by using the first 10 multiples of
each number.
Solution:
multiples of 6: 6, 12, 18, 24 , 30, 36, 42, 48 , 54, 60
multiples of 8: 8, 16, 24 , 32, 40, 48 , 56, 64, 72, 80
common multiples: 24, 48
3
∴ LCM of 6 and 8 is 24.
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We can also find the LCM of more than two numbers by using the
same method.
on
Example 2: Find the LCM of 2, 3, and 4 by using the first ten (or more)
multiples of each number.
2, 4, 6, 8, 10, 12 , 14, 16, 18, 20, 22, 24
multiples of 3:
3, 6, 9, 12 , 15, 18, 21, 24 , 27, 30
multiples of 4:
4, 8, 12 , 16, 20, 24 , 28, 32, 36, 40
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Solution:
multiples of 2:
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common multiples: 12, 24
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∴ LCM of 2, 3, and 4 is 12.
Example 3: Find the LCM of 21, 14 and 7 by listing first six multiples.
21, 42 , 63, 84, 105, 126
multiples of 14:
14, 28, 42 , 56, 70, 84
multiples of 7:
7, 14, 21, 28, 35, 42
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Solution:
multiples of 21:
common multiple: 42
42 is the only common multiple in the given list and it is the lowest
common multiple.
∴ the LCM of 21, 14, and 7 is 42.
2
Factors and Multiples
42
1
Example 4:
Calculate the LCM of 12, 16 and 24 considering the first 8 multiples.
Solution:
multiples of 12:
12, 24, 36, 48 , 60, 72, 84, 96
multiples of 16:
16, 32, 48 , 64, 80, 96 , 112, 128
multiples of 24:
24, 48 , 72, 96 , 120, 144, 168, 192
common multiples: 48, 96
3
48 < 96, therefore, 48 is the lowest common multiple (LCM) of 12, 16
and 24.
on
multiples of 5
REMEMBER
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multiples of 3
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Look carefully at this Venn diagram. It shows the multiples of 3 and 5
and their common multiples (up to the number 35).
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5 10
3 6 9 12
15
20
18 21 24 30
25 35
27 33
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A whole number is always a multiple
of itself. 3 is a multiple of 3, and 5 is
a multiple of 5, because
3 * 1 = 3 and 5 * 1 = 5.
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1 2 4 7 8 11 13 14 16 17
19 22 23 26 28 29 31 32 34
Now look at the common multiples of 3 and 5.
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There are 2 common multiples shown: 15 and 30.
15 < 30, therefore, 15 is the Lowest Common Multiple (LCM) of 3 and 5.
We can find the lowest common multiple (LCM) by prime factorisation.
Finding the LCM of bigger numbers like 144 and 96 would be tedious if
we use the multiple method.
An easy method is to break up large numbers into their prime factors,
and then find the LCM by the prime factorisation method.
1
43
Factors and Multiples
2
Example: Find the LCM of 48 and 45.
Solution: Let us first find the prime factors of 48 and 45.
2
48
3
45
Prime factors of 48 = 2 * 2 * 2 * 2 * 3
2
24
3
15
Prime factors of 45 = 3 * 3 * 5
2
12
5
5
2
6
3
3
To find the LCM of 48 and 45, we
multiply together all their prime
factors. However, we include their
common prime factors only once.
1
3
1
Hence, the LCM of 48 and 45 = 2 * 2 * 2 * 2 * 3 * 3 * 5 = 720.
on
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Now let us find the LCM of three numbers by the prime factorisation
method.
5
5
1
2
72
3
27
2
36
3
9
2
18
3
3
3
9
1
3
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15
54
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3
2
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30
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2
3
1
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Solution:
ls
Example: Find the LCM of 30, 54, and 72.
REMEMBER
Take only one of the
common factors and
multiply by all the
other factors.
30 = 2 * 3 * 5
54 = 2 * 3 * 3 * 3
72 = 2 * 3 * 2 * 2 * 3
LCM of 30, 54, and 72 is: 2 * 3 * 2 * 2 * 3 * 3 * 3 * 5
LCM of 30, 54, and 72 = 3240.
2
Factors and Multiples
44
1
We can solve real-life problems using the LCM.
To solve word problems involving LCM, we should remember some
keywords, such as together, least, same, and all.
Example: The signal lights on two towers flashed after every 30 and
40 seconds. If they flashed together at 7:30 p.m., when will they next
flash together?
Solution
‘Together’ is the key word, so we will find the LCM of 30 and 40, by
finding their prime factors.
5
1
20
2
10
5
5
1
2*2*2*5
∴ L CM of 30 and 40 =
2 * 2 * 2 * 3 * 5 = 120 seconds.
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5
2
prime factors of 40:
Hence the two signals will flash
together after every 120 seconds or
2 minutes.
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15
40
5
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3
2
2 * 3 *
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o
30
3
2
prime factors of 30:
Sc
Hence, after 7:30 p.m., they will flash together at 7:32 p.m.
aia
Exercise 2f
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1. Fill in the blanks.
a. The fourth multiple of 15 is _______.
b. 4 x 9 = 36, so 36 is the multiple of _______ and _______.
c. The common multiple of 5 and 6 are _______.
d. The LCM of 3, 6, and 9 is _______.
e. The LCM of 5, 10, and 25 is _______.
2. State whether the following are true or false.
a.The LCM of two or more prime numbers is equal to the product
of the number. (________)
b. The LMC of 14 and 42 is 14. (________)
c. Multiples of 17 are 17, 34, 51, 68. (________)
d. The LCM of 5, 10, 15 is 30. (________)
e. Multiples of a number are finite. (________)
1
45
Factors and Multiples
2
3
3. Select the correct answer from the
a. The LCM of 7, 14, 28 is
1 14
b. 36, 48, 72 are multiples of
8
12
c. The LCM of 20, 25, 35 is
350
70
d. The first 4 multiplies of 15 are
15, 30, 40, and 50
15, 30, 45, and 60
e. The LCM of 8 and 9 is
72 1
given options.
28
7
9
48
140
700
30, 60, 90, and 105
3, 5, 15, and 30
8
9
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4. Find the LCM of these pairs of numbers, using the prime
factorisation method.
a. 18 and 24 b. 12 and 16 c. 30 and 12
d. 20 and 25 e. 24 and 32 f. 18 and 56
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Word Problems
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5. Find the LCM of the given numbers, using the prime factorisation
method.
a. 8, 16, and 24 b. 10, 15, and 25
c. 16, 48, and 96 d. 36, 60, and 72
e. 24, 39, and 48 f. 22, 132, 143
Solve the problems, writing complete statements.
Fo
1.What is the least number of children, who may be arranged in
rows of 12, 15 and 9?
2.Three bells toll at an interval of 4, 5, and 6 seconds. After how
much time will they toll together?
3.Saima goes to her friend every 15 days and Laiba goes to same
friend every 18 days. After how many days Saima and Laiba will
visit the friend together?
4.
2
hat is the least number which can be exactly divided by 36, 64,
W
and 72?
Factors and Multiples
46
1