W hole Numbers
WHOLE NUMBERS
T
PO R
PASS
www.mathletics.co.uk
It is important to be able to identify the different types of whole numbers and recognise their
properties so that we can apply the correct strategies needed when completing calculations.
1
1
1
1
1
2
1
1
3
4
3
6
1
4
1
Pascal's Triangle
Blaise Pascal developed this triangle
1 5 10 10 5 1
by simply adding two whole
numbers together each time
1 6 15 20 15 6 1
1
1
1
7
8
21
9
28
36
35
56
35
70
21
56
7
28
84 126 126 84
is
Give th
Q
1
8
36
1
9
1
a go!
The ant nest below has a tunnel system that leads down to a main chamber. After one ant enters
the tunnel from the top, how many different ways can it get to the main chamber if it only travels
downwards the entire way?
Main Chamber
Work through the book for a great way to solve this
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How does it work?
Whole Numbers
Place values
142
M
illi
o
H u ns
nd
Te red
ns s o
f
o
T h f t h t ho
ou ou us
a
s
Hu and san nds
nd s ds
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On
es
Numbers can be separated into columns that represent different multiples of 10.
The column where a number is found determines the place value of that number.
×1
× 10
× 100
× 1000
× 10 000
× 100 000
× 1 000 000
NU MB E R S
Remember: When multiplying by multiples of 10, just add the same number of zeros to the end
2 # 100 = 200
5 # 100000 = 500000
11 # 10000 = 110000
What is the place value of 4 in the number 34 250?
Method 1: Multiplying the number by the multiple of 10 matching its position in the number
34 250
4 is in the thousands position
Identify the position of the 4 in the number
Multiply 4 by the place value
` 4 # 1000
` place value of 4 is 4000
Method 2: The place value of a number can also be found by changing all the other numbers to a 0
34 250
04 000
Change all the other numbers to a zero
` place value of 4 is 4000
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Ignore all zeros in front of the 4
How does it work?
Whole Numbers
Multiplying the number by the multiple of 10 matching its position in the number.
Hu
nd
Te reds
ns
o
o f
T h f t h t ho
ou ou us
a
Hu sand san nds
nd s ds
Te reds
ns
On
es
(i) Write 631 405 using words
(ii) Write 631 405 in expanded form
×1
× 10
× 100
× 1000
× 10 000
× 100 000
631405
(i) Using words: Six hundred and thirty one thousand, four hundred and five
Name using groups of three
(ii) Expanded form:^6 # 100 000h + ^3 # 10 000h + ^1 # 1000h + ^4 # 100h + ^0 # 10h + ^5 # 1h
Multiply each number by the place value and add together
Here is another example.
1
0
7
2
1
3
8
× 1 000 000
× 100 000
× 10 000
× 1000
× 100
× 10
×1
M
illi
o
Hu ns
nd
Te red
ns s o
f
o
Th f th tho
ou ou us
a
s
Hu and san nds
d
nd s
s
Te red
ns s
On
es
(i) Write 1 072 138 using words
(ii) Write 1 072 138 in expanded form
(i) Using words: One million, seventy two thousand, one hundred and thirty eight
Name using groups of three
(ii) Expanded form: ^1 # 1 000 000h + ^0 # 100 000h + ^7 # 10 000h + ^2 # 1000h + ^1 # 100h + ^3 # 10h + ^8 # 1h
Multiply each number by the place value and add together
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How does it work?
Your Turn
Whole Numbers
Place value
1
Write down the place values for each of these numbers
a
1426 b
42 603
Place value of 1: Place value of 3:
Place value of 2: Place value of 4:
d 7 380 261
c 560 142 Place value of 5: Place value of 7:
Place value of 6: Place value of 8:
Write each of these ordinary numbers in:
(i) worded form
(ii) expanded form
2
a
2560 (i)
Two thousand, five hundred and sixty
(ii)
(2 # 1000) + (5
b
#
100) + (6 # 10)
1 306 211 (i)
(ii)
c
891 026 (i)
(ii)
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Whole Numbers
Place value
d
E NUMB
OL
708 002 (i)
Place Value
..../...../20...
E NUM
OL
(ii)
e
9 011 060
S * WH
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B
(i)
(ii)
3
Write the ordinary number for each of these:
a
Four hundred and thirty nine thousand, two hundred and six
b
^4 # 1000000h + ^2 # 100000h + ^0 # 10000h + ^1 # 1000h + ^0 # 100h + ^3 # 10h + ^0 # 1h
c
Eighty one thousand and five
d
^9 # 10000h + ^8 # 1000h + ^9 # 100h + ^9 # 10h + ^8 # 1h
e
Any number whose place values for 4, 5 and 2 are 4000, 5 and 200
f
Three million, thirty thousand and thirty
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How does it work?
Whole Numbers
Adding and subtracting large numbers
When adding or subtracting large numbers, make sure the place values are lined up correctly.
Here are some addition examples to refresh your memory.
Calculate 13 829 + 4271
13 829 + 4271 =
=
13 8 2 9 +
41 21 7 1 1 Ensure matching place values are aligned
13 8 2 9 +
Carry over the 'tens' value
41 21 7 1 1
18 1 0 0
` 13 829 + 4271 = 18100
You can check your answer by simply entering the sum into your calculator.
Always check that your answer makes sense if using a calculator.
It is easy to accidentally press a wrong button when entering numbers and operations into a calculator.
Calculate 317 293 + 20 011 + 102 356
317 293+
317 293 + 20 091 + 102 356 = 2 0 0 9 1
102 356
317 2 9 3+
= 20 0 9 1
1 0 2 32516
Ensure matching place values are aligned
Carry over the 'tens' value
439 7 4 0
` 317293 + 20091 + 102356 = 439740
3 1 7 2 9 3
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2 0 0 9 1
+
1 0 2 3 5 6
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=
439 740
How does it work?
Whole Numbers
There are many different accurate ways to subtract large numbers. You should always use the method
that you were taught or know best.
Here is an example using one way.
Calculate 7635 – 4829
Step 1:
7 6 3 5 –
1
1
4 8 2 9
Line up matching place values
Step 2:
7 6 3 5 –
1
1
4 8 2 9
Since 5 1 9 we cannot subtract, ` place a 1 between 5 and 2
The 1 is in front of the 5 (in the tens position), making it 15
The 1 is added to the 2 to make it 3 as ‘payback’ for using the 1 to make 15
Step 3:
7 6 3 5 –
1
1
4 8 2 9
0 6
15 – 9 equals 6 and 3 – 3 equals 0
Step 4:
7 6 3 5 –
1
1
4 8 2 9
0 6
Since 6 1 8 we cannot subtract, ` place a 1 between 4 and 6
7 6 3 5 –
1
1
4 8 2 9
2 8 0 6
16 – 8 equals 8 and 7 – 5 equals 2
Step 5:
The 1 is in front of the 6 (in the tens position), making it 16
The 1 is added to the 4 to make it 5 as ‘payback’ for using the 1 to make 16
` 7635 - 4829 = 2806
7 6 3 5
–
4 8 2 9
=
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Whole Numbers
Here is another example for subtraction.
Calculate 38 234 – 21 576
Step 1:
3 8 2 3 4 –
2 1 5 7 6
Line up matching place values
Step 2:
3 8 2 3 4 –
1
2 1 5 7 6
Since 4 1 6 we cannot subtract, ` place a 1 between 7 and 4
The 1 is in front of the 4 (in the tens position), making it 14
The 1 is added to the 7 to make it 8 as ‘payback’ for using the 1 to make 14
Step 3:
3 8 2 3 4 –
1
2 1 5 7 6
8
14 – 6 equals 8
Step 4:
3 8 2 3 4 –
1
1
2 1 5 7 6
8
Since 3 1 8 we cannot subtract, ` place a 1 between 3 and 5
The 1 is in front of the 6 (in the tens position), making it 13
The 1 is added to the 5 to make it 6 as ‘payback’ for using the 1 to make 13
Step 5:
3 8 2 3 4 –
1
1
2 1 5 7 6
5 8
13 – 8 equals 5
Step 6:
3 8 2 3 4 –
1
1
2 1 5 7 6
5 8
Since 2 1 6 we cannot subtract, ` place a 1 between 2 and 1
The 1 is in front of the 2 (in the tens position), making it 12
The 1 is added to the 1 to make it 2 as ‘payback’ for using the 1 to make 12
Step 7:
3 8 2 3 4 –
1
1
2 1 5 7 6
1 6 6 5 8
12 – 6 equals 6, 8 – 2 equals 6, 3 – 2 equals 1
Do all the subtractions since there are no more columns with the top 1 bottom
` 38234 - 21576 = 16658
3 8 2 3 4
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2 1 5 7 6
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=
16 658
How does it work?
Your Turn
Whole Numbers
Adding and subtracting large numbers
1
Calculate each of these addition questions showing all working.
a
5 6 2 1 0 +
8 8 3 5
3 0 6 1 4
b
7 1 4 0 0 +
1 0 8 0 9
4 2 0 1
Large Whole
Numbers
c
e
2
9 9 +
4 3 2 1
8 6 4 2
2 4 6 3
3 9 5
7 2
1
9
1
6
9
d
9+
1
0
3
f
..../...../20...
8 4 3 +
1 9 5
6 0 4
6 3
8 7 6 3
2
9 3 1 0
3 4 1
8 +
1
5
2
Combo Time!
Calculate the sum (+ ) of three hundred and forty five thousand, two hundred and nine and
eighteen thousand, seven hundred and ninety six.
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How does it work?
Your Turn
Whole Numbers
Adding and subtracting large numbers
3
4
Calculate each of these substraction questions showing all working.
a
5 2 6 8 –
2 3 5 2
b
2 5 2 7 2 –
5 6 4 0
c
3 6 5 2 6 8 –
1 0 4 8 2
d
5 4 3 2 1 –
1 2 3 4 5
e
2 0 3 0 4 0 –
1 0 2 0 3
f
7 0 0 0 0 –
2 6 7 8 9
Combo Time!
Calculate the difference (–) between:
five hundred and seventy thousand, two hundred and seventeen
and
ninety eight thousand, four hundred and twenty one
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Whole Numbers
Long multiplication
As you did when adding and subtracting, keep your place value columns lined up neatly.
You need to be aware of the place value of the number you are multiplying by.
Calculate 1429
#
32
1 4 2 9
3 2
+1
2 8 4 8
#
1 4 2
3
2 8 5
+1
+2
3 2 6 7
9
2
8
0
1 4 2
3
2 8 5
4 2 8 7
4 5 7 2
9 #
2
8 +
0
8
Line up matching place values
Multiply the 1429 by 2
Carry over any ‘tens’ values after multiplying
#
For 1429 # 3 tens, put a 0 in the ones column and multiply by 3
Carry over any ‘tens’ values after multiplying
Add the two new numbers together
` 1429
#
32 = 45728
Here is another example. Be careful to line up the columns correctly.
Calculate 423
#
506
4 2 3
5 0 6
+1 +1
2 4 2 8
#
4
5
2 5
0 0 0
2
0
3
0
3
6
8
0
4
5
2 5
0 0 0
+1 +1
2 0 0 5
2
0
3
0
0
3
6
8
0
0
4
5
2 5
0 0 0
+1
2 1 1 5
2 1 4 0
2
0
3
0
0
3
3 #
6
8 +
0
0
8
Line up matching place values
Multiply the 423 by 6
Carry over any ‘tens’ values after multiplying
#
For 423
#
0 tens, put a 0 in the ones column first and multiply by 0
#
For 423 # 5 hundreds, put a 0 in the ones and tens columns and multiply by 5
Carry over the ‘tens’ value after multiplying
Add the two new numbers together
` 423
#
506 = 214038
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Your Turn
Whole Numbers
LE NUM
HO
Long multiplication
RS * W
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B
MultipLlong
ication
1 0 1 2
3 7
e
12
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2 0 2 0 2
1 5
f
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#
3 8 7 6
4 5
d
#
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/20...
LE NUM
HO
9 5 7 0
6 3
c
..../.....
RS * W
BE
Calculate each of these multiplication questions showing all working.
Check your answers on the calculator.
a
3 0 1 6 #
2 5 8 1
b
2 1
1 9
1
#
#
How does it work?
Your Turn
Whole Numbers
Long multiplication
2
Calculate each of these multiplication questions showing all working.
Check your answers on the calculator.
a
c e
2 1 2×
1 2 1
b
9 0 8×
2 0 9
8 6 4×
3 4 5
6 4 8 5×
1 2 3
d
1 3 2 5×
4 3 7
f
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4 0 5
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How does it work?
Whole Numbers
Short and long division
Short and long division are only different due to the size of the number you are dividing by.
Here is a short division question.
Calculate 75 408 ' 6
1
6g 7 1 5 4 0 8
Step 1:
Divide 7 by 6
Put the whole number answer (1) above the 7
Make the remainder (1) the ‘tens’ digit for the next number
1 2
6g 7 1 5 3 4 0 8
Step 2:
Divide 15 by 6
Put the whole number answer (2) above the 5
Make the remainder (3) the ‘tens’ digit for the next number
1 2 5
Step 3:
6g 7 5 4 0 8
Divide 34 by 6
Put the whole number answer (5) above the 4
Make the remainder (4) the ‘tens’ digit for the next number
1 2 5 6
Divide 40 by 6
Put the whole number answer (6) above the 0
Make the remainder (4) the ‘tens’ digit for the next number
1
4
3
Step 4:
6g 7 1 5 3 4 4 0 4 8
Step 5:
1 2 5 6 8
6g 7 5 4 0 8
1
3
4
Divide 48 by 6
Put the answer (8) above the 8
4
` 75 408 ' 6 = 12 568
remainder fraction =
If there is a remainder at the end, always write it as a fraction.
the amount left over
the divisor
Calculate 518 ' 3
Step 1:
111
2
333ggg555 22111888
Divide 5 by 3
Put the whole number answer (1) above the 5
Make the remainder (2) the ‘tens’ digit for the next number
Step 2:
111 777
2
333ggg555 22111888
Divide 21 by 3
Put the whole number answer (7) above the 1
There is no remainder this time
Step 3:
111 777 222 323232
2
333ggg555 22111888
divisor
amount
left over
Divide 8 by 3
Put the whole number answer (2) above the 8
Write the remainder as a fraction ( 23 ) to the right
` 518 ' 3 = 172 23
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Whole Numbers
Here is a long division question.
Calculate 6259 ' 14
Step 1:
4
14 g 6 2 5 9
5 6
Step 2:
Step 3:
Step 4:
Step 5:
4
14 g 6 2 5 9
5 6
6 5
4
14 g 6 2
5 6
6
5
4
5 9
4
14 g 6 2
5 6
6
5
4
5 9
5
6
5
6
9 9
4 4 7
14 g 6 2 5 9
5 6
6 5
5 6
9 9
9 8
Step 6:
4
14 g 6 2
5 6
6
5
4 7
5 9
5
6
9 9
9 8
1
Divide 62 by 14
Put the whole number answer (4) above the 2
Multiply 14 by the answer (4) and write this underneath the 62
Subtract 56 from 62
Drop the 5 down next to the answer
Divide 65 by 14
Put the whole number answer (4) above the 5
Multiply 14 by the answer (4) and write this underneath the 65
Subtract 56 from 65
Drop the 9 down next to the answer
Divide 99 by 14
Put the whole number answer (7) above the 9
Multiply 14 by the answer (7) and write this underneath the 99
Subtract 98 from 99
Write the remainder as a fraction
1
` 6 259 ' 14 = 447 14
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Whole Numbers
Short and long division
a
4767 ' 3 b
6180 ' 5
g
c
g
6912 ' 4 d
12 054 ' 6
g
2
g
Calculate each of these short division questions showing all working.
(psst: remember to write any remainders as a simplified fraction)
Check your answers on the calculator.
a
8965 ' 7 b
3879 ' 2
g
c
g
9263 ' 8 d
5801 ' 6
g
16
g
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Short &
DivisioLong
n
..../.....
RS * W
BE
Calculate each of these short division questions showing all working.
Check your answers on the calculator.
1
LE NUM
HO
RS * W
BE
Your Turn
/20...
LE NUM
HO
How does it work?
How does it work?
Your Turn
Whole Numbers
Short and long division
3
Calculate each of these long division questions showing all working.
Check your answers on the calculator.
15
g
3
8
5
5
c 24 5
1
8
5
a
g
b
23
g8
9
4
7
d
17
g2
5
7
8
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How does it work?
Whole Numbers
Divisibility tests
Divisibility tests are used to see if a small whole number will be a factor of a larger composite number.
A number is always divisible by 2 if the last digit is an even number (i.e. 0 , 2 , 4 , 6 or 8)
234 is divisible by 2 as the last digit (4) is even
A number is always divisible by 3 if the sum (+) of all its digits is divisible by 3
234 is divisible by 3 because 2 + 3 + 4 = 9 (which is divisible by 3)
A number is always divisible by 4 if the number formed by the last two digits is divisible by 4
1324 is divisible by 4 because the last two digits form the number 24 (which is divisible by 4)
A number is always divisible by 5 if the last digit of the number is a 0 or 5
265 is divisible by 5 because the last digit is a 5
A number is always divisible by 6 if it is divisible by both 2 and 3
234 is divisible by 6 because it is even (so divisible by 2) and 2 + 3 + 4 = 9 (which is divisible by 3)
A number is always divisible by 8 if the number formed by the last three digits is divisible by 8
1328 is divisible by 8 because the last three digits form the number 328 (which is divisible by 8)
A number is always divisible by 9 if the sum (+) of all its digits is divisible by 9
234 is divisible by 9 because 2 + 3 + 4 = 9 (which is divisible by 9)
A number is always divisible by 10 if the last digit of the number is a 0
1840 is divisible by 10 because the last digit is 0
Investigate the divisibility tests for 7 and 11. They are a little more involved but interesting!
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Your Turn
Whole Numbers
Divisibility tests
Use the divisibility tests to determine whether each of these numbers are divisible by the numbers listed
on the right hand side. Draw a line to all the numbers each one is divisible by.
The first number is completed for you.
2
620
136
3
96
4
1491
345
5
207
6
512
588
8
738
9
1 001 001
312 756
10
8640
..../.....
/20...
FOR NU
6030
S
Y TEST
IT
12 871
ISIBIL
IV
ERS * D
MB
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How does it work?
Whole Numbers
Index notation for numbers
Index notation uses a small number called a ‘power’, ‘index’ or ‘exponent’ to show how many times a
number is multiplied by itself.
Simplify these products by using index notation and then calculate:
(i) 4 # 4
4 # 4 = 42
= 16
Two 4s in the multiplication, so the index is 2
We say ‘4 squared’
When a number is multiplied by itself once, this is called squaring the number
(ii) 2 # 2 # 2
2 # 2 # 2 = 23
=8
Three 2s in the multiplication, so the index is 3
We say ‘2 cubed’
When a number is multiplied by itself twice, this is called cubing the number.
The same pattern continues for any number of multiplications
(iii) 3 # 3 # 3 # 3 # 3 # 3
3 # 3 # 3 # 3 # 3 # 3 = 36
= 729
Six 3s in the multiplication, so the index is 6
We say ‘3 to the power of 6’
A mixture of numbers multiplied together can also be simplified using index notation
(iv) 4 # 5 # 5 # 5 # 4 # 5
Group identical numbers
4#5#5#5#4#5 = 4#4#5#5#5#5
= 42 # 5 4
= 16 # 625
= 10 000
We say ‘4 squared times 5 to the power of 4’
Doing the reverse to simplifying is called expanding.
Write these in expanded form:
(i) 74
74 = 7 # 7 # 7 # 7
The index is 4, so four 7s multiplied together
(ii) 97
97 = 9 # 9 # 9 # 9 # 9 # 9 # 9
The index is 7, so seven 9s multiplied together
Be careful: A lot of people make this mistake: 74 = 7 # 4 , which is NOT true.
74 ! 7 # 4
Make sure you can see the difference.
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Your Turn
Whole Numbers
a
5
#
5
c
2
#
2
#
2
#
2
#
2
e
7
#
7
#
7
#
7
#
7
2
b
#
7
4
#
d
11
f
3
4
#
#
..../.....
#
4
11
#
11
3
#
3
#
/20...
#
3
#
11
3
#
3×3
#
3
#
2
Write each of the mixed products using index notation and then calculate.
a
2
c
6
e
2
3
ION * I
T
A
Write each of these products using index notation.
EX NOT
ND
1
ION * I
AT
EX NOT
ND
Index notation for numbers
#
#
#
2
6
8
#
#
#
2
6
8
#
#
#
3
6
2
#
#
#
3
7
8
#
#
7
8
#
#
7
8
b
5
#
d
2
#
1
#
2
#
1
#
2
4
#
3
#
3
#
4
#
3
f 5
#
4
#
4
#
2
#
2
Change each of these to expanded form.
4
8
a
33 b
c
65 d 127
e
53 # 72 f 2 4 # 32
g
75 # 2 4 h
2
4
2
2 # 3 # 5
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How does it work?
Your Turn
Whole Numbers
Puzzle Time
2 units
The area of a square can be written using index notation:
2 units
Area = 2 # 2
= 22 units 2
= 4 units 2
62
Using each of the different grey squares below twice and the black square only once, form a rectangle
on the grid above. You can do this by shading in the squares using a pencil or cut some similar-sized
squares out of another sheet of paper and try to complete like a jigsaw.
The top left-hand corner of the rectangle is already completed for you, so only one more 62 grey square
can be used.
12
32
use twice
42
52
62
use once
use twice
use twice
use twice
When finished, have a go at writing two different expressions for the total area of the rectangle using
index notation.
Hint: For one expression multiply the side lengths together.
Area expression 1
22
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How does it work?
Whole Numbers
Square roots and cube roots
Finding the square root or cube root of a number is the opposite operation to squaring or cubing.
The radical symbol (
) is used for roots, with
used for square root and 3
for cube root.
The square root sign is asking: What number multiplied by itself once will get the number inside me?
Calculate the square root of these whole numbers
(i) 9
9
=3
Because 3 # 3 = 32
9 = 3#3
=9
32
=
=3
or
(ii) 36
9 written as a product of its prime factors
36 = 6
36 = 6 # 6
Because 6 # 6 = 62 = 36
36 written as a product of its prime factors
62
=
We look closely at
prime factors next
= 6
The cube root sign is asking: What number multiplied by itself twice will get the number inside me?
Calculate the cube root of these whole numbers
(i) 8
3
or
8 = 2
3
8 =3 2 # 2 # 2
Because 2 # 2 # 2 = 23 = 8
=
3
8 written as a product of its prime factors
23
= 2
33
The little ‘root’ number indicates how many times the same number appears
in the multiplication.
(ii) 343
3
or
343 = 7
3
343 = 3 7 # 7 # 7
Because 7 # 7 # 7 = 73 = 343
=
3
343 written as a product of its prime factors
73
= 7
You could be asked to write a value using square or cube root notation.
Rewrite these numbers
(i) 4 as a square root
42 = 4 # 4 = 16
` 4 = 16
(ii) 3 as cube root
33 = 3 # 3 # 3 = 27
` 3 = 3 27
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Your Turn
Whole Numbers
LE NUM
HO
Square roots and cube roots
a
4
b
16
c
25
d
49
e
81 f
121
2
Calculate each of these cube roots
a
3
27
b
c
3
216 d
3
3
a
3
c
6
512
b
8
d
12
Write each of these values using cube root notation
a
1
b
2
c
5
d
7
24
64
Write each of these values using square root notation
4
3
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Calculate each of these square roots
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LE NUM
HO
How does it work?
Where does it work?
Whole Numbers
Factor Trees
Composite numbers can be divided exactly (with no remainder), by other smaller or equal whole numbers
called factors.
Composite numbers:
15
9
1 , 3 , 5 , 15
Factors:
12
1 , 3 , 9 1 , 2 , 3 , 4 , 6 , 12
4
24
1 , 2 , 4
1 , 2 , 3 , 4 , 6 , 8 , 12 , 24
Prime numbers only have 1 and themself as factors.
2 3171131
Prime numbers:
1 , 2
Factors:
1 , 3
1 , 17
1 , 11
1 , 31
All composite numbers can be written as the product ( # ) of prime factors
(all the prime numbers that divide exactly into them). Let’s see how.
‘Express’ is a another way of saying ‘write’ in Mathematics.
Express 18 as a product of its prime factors
Split 18 into two smaller factors
18
Solid circle around prime numbers to stop that branch
3
6
2
Split 6 into two smaller factors
Solid circle around prime numbers to stop that branch
3
Once every branch has reached a prime number, multiply all the prime numbers together
` 18 = 2
=2
3 # 3
32
#
#
Simplify answer
ALWAYS
at the prime number.
3
Don’t ever do this
1
3
because 1 is NOT a prime number
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Remember:
A prime number has
two factors, itself and 1
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Where does it work?
Whole Numbers
Here are some more examples.
Express 38 as a product of its prime factors
38
Split 38 into two smaller factors
19
2
Solid circle around prime numbers to stop that branch
Once every branch has reached a prime number, multiply all the prime numbers together
` 38 = 19 # 2
There is often more than one way to create a factor tree for numbers with a lot of factors.
Express 48 as a product of its prime factors
Split 48 into two smaller factors
48
2
4
Split 6 and 8 into two smaller factors
6
8
3
2
2
2
Solid circle around prime numbers to stop that branch
Split 4 into two smaller factors
Solid circle around prime numbers to stop that branch
Once every branch has reached a prime number, multiply all the prime numbers together
` 48 = 2 # 2 # 2 # 3 # 2
= 24 # 3
26
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Simplify answer
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Where does it work?
Your Turn
Whole Numbers
Factor trees
Fill in the missing values on the following factor trees and write the number as a product of its primes.
12
b
18
4
2
2
3
` 12 =
c
` 18 =
32
d
56
14
4
2
4
2
2
` 56 =
2
e
84
f
128
12
..../...
../20...
TREES *
` 32 =
OR
IME FACT
PR
TREES *
OR
a
IME FACT
PR
1
2
4
2
3
` 84 =
` 12 8 =
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Where does it work?
Your Turn
Whole Numbers
Factor trees
2
Complete a factor tree for each number below and express them as a product of their prime factors.
a
8
b
20
` 8 = ` 20 =
c
24
d
` 24 = e
96
f
28
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` 60 =
` 96 = ` 144 =
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Where does it work?
Whole Numbers
Highest common factor (HCF)
The HCF is the largest number that divides exactly into two or more composite numbers.
Write all the factors of each number then circle the largest one which appears in both lists.
Find the highest common factor for these pairs of numbers
(i) 6 and 8
Factors of 6: 1 , 2 , 3 , 6
List all the factors for each number
Factors of 8: 1 , 2 , 4 , 8
Circle the largest number common to both lists
` The HCF for 6 and 8 is: 2
(ii) 18 and 12
Factors of 18: 1 , 2 , 3 , 6 , 9 , 18
List all the factors for each number
Factors of 12: 1 , 2 , 3 , 4 , 6 , 12
Circle the largest number common to both lists
` The HCF for 18 and 12 is: 6
We can use the list of prime factors for larger numbers to find the HCF.
Find the HCF for these pairs of larger numbers
(i) 72 and 96
Factors of 72: 2 , 2 , 2 , 3 , 3
List all the prime factors for each number
Factors of 96: 2 , 2 , 2 , 2 , 2 , 3
` The HCF for 72 and 96 is: 2 # 2 # 2 # 3 = 24
(ii) 528 and 624
Factors of 528: 2 , 2 , 2 , 2 , 3 , 11
Factors of 624: 2 , 2 , 2 , 2 , 3 , 13
` The HCF for 528 and 624 is: 2 # 2 # 2 # 2 # 3 = 48
List all the prime factors for each number
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Your Turn
Highest common factor (HCF)
RS * H
CTO
IG
Find the highest common factor for these pairs of numbers.
a
8 and 12
b
6 and 15
c
10 and 18
d
18 and 24
e
14 and 28
f
16 and 36
FA
HCFs
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O
ST C MMO N
HE
1
Whole Numbers
COMMO
ST
N
HE
Where does it work?
RS * H
CTO
IG
FA
2
Use the prime factors to find the HCF for these larger numbers.
a
42 and 84
b
92 and 72
c
280 and 490
d
256 and 640
30
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Where does it work?
Whole Numbers
Lowest common multiple (LCM)
The LCM is the smallest number that is common to the multiplication tables of two or more numbers.
Write down the multiples of the numbers and stop once you find the lowest common multiple.
Find the lowest common multiple for these pairs of numbers
(i) 2 and 5
2#2
6#2
Multiples of 2: 2 , 4 , 6 , 8 , 10 , 12 , 14 ,... List some multiples of the first number
1#2
4#2
3#2
7#2
5#2
2#5
Multiples of 5: 5 , 10 ,...
List the multiples of the second number until there is a match
1#5
` The LCM for 2 and 5 is: 10
(ii) 6 and 8
Multiples of 6: 6 , 12 , 18 , 24 , 30 ,...
List some multiples of the first number
Multiples of 8: 8 , 16 , 24 ,...
List the multiples of the second number until there is a match
` The LCM for 6 and 8 is: 24
We can use the list of prime factors for larger numbers to find the LCM by looking at the differences.
Find the LCM for these pairs of larger numbers
(i) 30 and 100
Prime factors of 30: 2 , 3 , 5
List all the prime factors for both numbers
Prime factors of 100: 2 , 2 , 5 , 5
Circle all the different factors in the smaller number
` The LCM for 30 and 100 is: 100 # 3 = 300
Multiply the larger number by the different factor
(ii) 24 and 388
Prime factors of 24: 2 , 2 , 2 , 3
List all the prime factors for both numbers
Prime factors of 388: 2 , 2 , 97
Circle all the different factors in the smaller number
` The LCM for 15 and 388 is: 388 # 2 # 3 = 2328
Multiply the larger number by the different factors
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Your Turn
Lowest common multiple (LCM)
Find the lowest common multiple for these pairs of numbers.
a
3 and 9
b
5 and 10
c
4 and 6
d
5 and 6
e
6 and 7
f
12 and 16
PLE * L
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O
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M
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s
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COMMON
ST
WE
1
Whole Numbers
COMMON
ST
WE
Where does it work?
IPLE * L
LT
O
MU
2
Use the prime factors to find the LCM for these larger numbers.
a
60 and 108
b
42 and 150
c
168 and 180
d
210 and 385
32
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Whatelse
elsecan
canyou
youdo?
do?
What
Whole
Numbers
Whole
Numbers
Pascal’s triangle
This amazing triangle developed in 1653 by French mathematician Blaise Pascal uses the addition of two
whole numbers to create it.
The number pattern forms the shape of a triangle and contains many mathematical applications.
To create Pascal’s triangle, each number on the line is obtained by adding the two numbers above it.
The first seven lines of Pascal’s Triangle
1
0+1
1+0
1
1
0+1
1+1
1+0
1
2
1
0+1
1+2
2+1
1+0
1
3
3
1
0+1
1+3
3+3
3+1
1+0
1
4
6
4
1
0+1
1+4
4+6
6+4
4+1
1+0
1
5
10
10
5
1
0+1
1+5
5 + 10
10 + 10
10 + 5
5+1
1+0
1
6
15
20
15
6
1
The pattern continues in the same fashion for each added row of numbers
The second diagonal of Pascal’s triangle contains all the counting numbers
Counting numbers
Counting numbers
1
1
1
2
1
3
1
5
1
1
6
3
4
1
1
15
4
6
10
1
5
10
20
1
15
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6
1
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What else can you do?
Whole Numbers
Here are some more patterns found within Pascal’s triangle.
The third diagonal of Pascal’s triangle contains triangular numbers
1
Triangular numbers
1
1
1
1
1
6
3
6
4
10
5
1
2
3
1
15
Triangular numbers
1
1
4
1
10
5
15
20
1
6
1
Triangular numbers are formed by creating equilateral triangles using dot diagrams starting from 1 dot
, ,,,
,...
1
36 10 15
A very well known number pattern which occurs frequently in nature is the Fibonacci Sequence.
The Fibonacci sequence is also within Pascal’s triangle and is found by adding terms along the lines shown
1
1
1+1= 2
1+2= 3
1+3+1= 5
3+4+1= 8
1 + 6 + 5 + 1 = 13
1
1
1
1
1
1
1
2
3
4
5
6
1
15
1
3
6
10
1
4
10
20
Sunflowers contain a
Fibonacci sequence
1
5
15
1
6
1
Each number in a Fibonacci Sequence is found mathematically by adding the two numbers before it
1
34
,1,2 ,
3
,5,8,13 ,21
,...
0 + 1 1 + 11 + 22 + 3 3 + 55 + 88 + 13
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Your Turn
Whole Numbers
Pascal’s triangle
Another special pattern is called the Sierpinkski Triangle. This is a special fractal pattern made using triangles.
Each dark equilateral triangle is split into four smaller equilateral triangles at every step.
This pattern can be reproduced using Pascal’s triangle by simply separating the odd and even numbers.
In Pascal’s triangle below, colour in all the odd numbered hexagons to see this pattern emerge!
1
14641
AL’S TRI
SC
1331
1
1
1 2 1
1
..../...
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GLE * PA
AN
121
AL’S TRI
SC
GLE * PA
AN
11
15101051
1615201561
1 721353521 7 1
1 82856705628 8 1
1 936841261268436 9 1
1 10 45120210252210120 45 10 1
1 11 55165330462462330165 55140 1
1 12 66220495792924792495220 66 12 1
1 13 782867151287171617161287715286 78 13 1
1 14 91 3641001200230033432300320021001 364 91 14 1
1 15 105 45513653003500564356435500530031365 455 105 15 1
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What else can you do?
Whole Numbers
Applications of Pascal’s triangle
Pascal’s triangle can often be useful when solving problems like the ones shown here.
Each number in Pascal’s triangle represents the number of paths that can be taken to get to that point.
Show all the different downward paths that can be taken to get to the circled number in the triangle
1
1
1
1
2
3
1
1
3
1
The number circled is 3, so there are 3 different downward paths leading to this point
1
1
1
1
1
1
2
3
1
1
3
1
1
1
1
1
2
3
1
1
3
1
1
Path 2
Path 1
1
1
2
3
1
3
1
Path 3
The total number of different paths to the bottom of a Pascal triangle is found by adding the numbers across.
For this four-line Pascal triangle:
(i) How many different paths can be taken to reach the bottom of the triangle below?
1
1
1
1
1
2
3
The total number of different paths = 1 + 3 + 3 + 1
=8
1
3
1
(ii) How many paths to reach the bottom if one more line was added?
1
1
1
1
1
36
1
2
3
4
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1
3
6
The total number of different paths = 1 + 4 + 6 + 4 + 1
= 16
1
4
1
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Whole Numbers
Applications of Pascal's triangle
1
Write down how many different downward paths there are to each of the points circled on this triangle.
A
B
C
D
F
E
Number of downward pathways to:
2
A
=
B
=
C
=
D
=
E
=
F
=
Show the six different downward paths that lead to the circled point on this triangle from the top.
Start
6
Start
Start
Start
Path 1
Path 2
Path 3
Start
Start
Start
Path 4
Path 5
Path 6
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What else can you do?
Your Turn
Whole Numbers
Applications of Pascal's triangle
3
The ant nest below has a tunnel system that leads down to a main chamber.
After one ant enters the tunnel from the top, how many different ways
can it get to the main chamber if it only travels downwards the entire way?
Hint: Fill in Pascal’s triangle values.
Main Chamber
38
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Rememb
er me?
Cheat Sheet
Whole Numbers
M
×1
× 10
× 100
× 1000
× 10 000
× 100 000
Adding and subtracting large numbers
When adding or subtracting large numbers, make sure the
place values are lined up correctly first.
NU MB E R S
× 1 000 000
Place value
• Writing numbers using words, name using groups of three digits.
• To write in expanded form, multiply each number by the place value
and add together.
• The place value of a numeral in a large number is found by
multiplying the numeral by the matching position value.
illi
o
H u ns
nd
Te red
ns s o
f
o
T h f t h t ho
ou ou us
a
s
Hu and san nds
d
nd s
s
r
e
Te
ns ds
On
es
Here is a summary of the important things to remember for whole numbers.
Long multiplication
• Make sure the place values are lined up correctly first.
• Add zeros on each line to match the place value of the number you are multiplying by.
• Add together the new numbers formed after multiplying.
Short and long division
• Keep all place values lined up neatly.
• Be careful and methodical with each step.
• Always write the remainder as a fraction.
Index notation for numbers
Index notation is used to show how many times a number is multiplied by itself.
3 # 3 # 3 # 3 # 3 = 35
Square and cube roots
• The square root or cube root of a number is the opposite operation to squaring or cubing.
for square root and 3 for cube root.
• The symbols used are
9 = 3 because 3 # 3 = 9 and 3 27 = 3 because 3 # 3 # 3 = 27
Factor trees
These are used to write any composite number as the product of prime number factors only.
Highest common factor (HCF)
The HCF is the largest number that divides exactly into two or more composite numbers.
Lowest common multiple (LCM)
The LCM is the smallest number that is common to the multiplication tables of two or more numbers.
Pascal’s triangle
• Each number in Pascal’s triangle is the sum of the two numbers above it.
• Each number is the number of different downward paths that can be taken to get to that point.
• The number of different downward paths to the bottom of a Pascal’s triangle is found by adding
together all the values across the bottom.
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Whole Numbers
40
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Notes
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Short a
Divisnido Long
n
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Squ
and Caurbe roots
e roots
NUMB ERS