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More work on numbers
SLIDE 1
To partition means to divide a collection into two.
? Partition the following numbers into odd and even numbers.
? What is the number of numbers in each partition?
SLIDE 2
? Odd
Even
? There are 4 odd numbers and 3 even numbers
SLIDE 3
A prime number is a number into which only itself and the number 1 divide. A prime number has no factors other than itself and the number 1. All other numbers are composite numbers. Composite numbers have factors that are not the number 1 and the number itself. These are called non-trivial factors.
? Partition the following numbers into prime and composite numbers.
?
Find and for each of the partitions in the previous question.
? Is divided by an odd or an even number?
SLIDE 4
?
Prime
Composite
?
?
This number is odd and not even.
SLIDE 5
The digit sum of 3829 is
Find the digit sum of the following numbers.
? 4329
? 37256
? 93954
? 82783
SLIDE 6
? 4329
? 37256
? 93954
? 82783
SLIDE 7
What are the tests?
? Divisible by 2
? Divisible by 3
? Divisible by 4
? Divisible by 5
SLIDE 8
A number is
? divisible by 2 if it ends in a 0, 2, 4, 6 or 8
? divisible by 3 if the digit sum of the number is divisible by 3
? divisible by 4 if the last two digits of the number make a number divisible by 4
? divisible by 5 if it ends in 0 or 5
SLIDE 9
Divisible by
Number 2 3 4 5 10
9231 ? ? ? ? ?
5458
69323
71277
81980
63955
932640
SLIDE 10
Divisible by
Number 2 3 4 5 10
9231 ? ? ? ? ?
5458 ? ? ? ? ?
69323 ? ? ? ? ?
71277 ? ? ? ? ?
81980 ? ? ? ? ?
63955 ? ? ? ? ?
932640 ? ? ? ? ?
SLIDE 11
A number is prime if it has no factors other than itself and 1.
To test whether a number is prime, try dividing by smaller prime numbers.
To test any number up to 100, you only need to try dividing by 2, 3, 5, and 7.
Find all the prime numbers between 60 and 100.
SLIDE 12
Prime numbers between 60 and 100.
61 67 71 73 79 83 89 97
SLIDE 13
Every number can be written as the product of prime numbers.
This product is unique. It is called the prime factorisation of the number.
To find the prime factorisation, attempt to divide successively by prime numbers in order. The result can be displayed as a tree.
Example
Prime factorisation of 72
This factorisation can be written
The superscript is called an index or exponent
“72 is 2 cubed times 3 squared”
“72 is 2 to the power 3 times 3 to the power 2”
In the exponent of 2 is 3, and the exponent of 3 is 2.
SLIDE 14
Find the prime factorisation of 60 and 84. Write these in exponent form.
SLIDE 15
SLIDE 16
We often use a dot to show multiplication.
Now find the prime factorisation of all numbers between 60 and 100. Write these out using dots for multiplication, and exponents for multiples of factors. If a number is prime, mark it as prime.
SLIDE 17
60
75
90
61 prime 76
91
62
77
92
63
78
93
64
79 prime 94
65
80
95
66
81
96
67 prime 82
97 prime
68
83 prime 98
69
84
99
70
85
100
71 prime 86
72
87
73 prime 88
74
89 prime
SLIDE 18
Find the value of
? The square of 6
? The cube of 5
? The square of
? The cube of
SLIDE 19
?
?
?
?
SLIDE 20
We introduce brackets to make the meaning clear
An expression is ambiguous if it can have more than one meaning, or be interpreted in more than one way.
The expression is ambiguous.
SLIDE 21
Evaluate
?
?
?
?
SLIDE 22
?
?
?
?
SLIDE 23
Complete the tables
+ –
+
–
SLIDE 24
+ –
+ + –
– – +
SLIDE 25
O! A plus times a plus is a plus
A plus times a minus is a minus
A minus times a plus is a minus
And a minus times a minus … is a plus
Compose a tune to go with the above lyrics.
Sing the song!
SLIDE 26
“Two squared” “Two to the power two”
“Two cubed” “Two to the power three”
“Two to the power four”
“Two to the power five”
? Find the value of two to the power 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10
? Find the value of three to the power 1, 2, 3, 4 and 5
? Find the value of five to the power 1, 2, 3 and 4
? Find the value of seven to the power 1, 2 and 3
SLIDE 27
?
?
?
?
SLIDE 28
Evaluate
? ?
? ?
? ?
? ?
? ?
SLIDE 29
? ?
? ?
? ?
? ?
? ?
SLIDE 30
? Find the cube of 3 and evaluate
? Find the cube of 4 and evaluate
? Find the cube of 5 and evaluate
? If , what is
? Evaluate
? What is the fourth root of 16? This is written
? Evaluate
? Evaluate
? Evaluate
? What is ?
SLIDE 31
?
?
?
?
?
?
?
?
?
?
SLIDE 32
You do not need to work out first.
? Evaluate
? Evaluate
? Evaluate
? Find the square root of
? Evaluate
? Evaluate
SLIDE 33
?
?
?
? The square root of is
?
?
SLIDE 34
The word sum means add
The word product means multiply
Example
Find the sum and product of 8 and 9
Sum = Product =
? Find the sum of and
? Find the product of and
? Difference means take the smaller away from the larger
Find the difference between and
? Evaluate
? Find the sum of the following two products:
The product of and 2
The product of and 5
SLIDE 35
? The sum of and is
? The product of and is
? The difference between and
? Evaluate
? The product of and 2 is
The product of and 5 is
SLIDE 36
The exclamation mark is called the factorial
Find 5!, 6! and 7!
SLIDE 37
SLIDE 38
We introduce the special symbols
zero factorial evaluates to 1
Any number to the power zero evaluates to 1
? What is ?
? Find the sum and product of and
? What is the difference of and ?
SLIDE 39
?
?
?
Difference =
SLIDE 40
From the collection of numbers 24, 29, 39, 45, 49 and 61 find
? an even number
? an odd number
? a prime number
? a multiple of 7
? a factor of 135
SLIDE 41
24, 29, 39, 45, 49 and 61
? an even number is 24
? an odd number is any of 29, 39, 45, 49 or 61
? a prime number is 61 or 41
? a multiple of 7 is 49
? a factor of 135 is 45
SLIDE 42
Using only the numbers 11, 21, 31, 77 and 126
? Find the number that is exactly half-way between two prime numbers.
? Find the factor that is common to three of these numbers.
SLIDE 43
11, 21, 31, 77 and 126
? The numbers 11 and 31 are prime. The number 21 is half-way between them.
? Factors
11 prime
21 1, 3, 7, 21
31 prime
77 1, 7, 11, 77
126 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
The number 7 is common to three of these numbers.
SLIDE 44
Since every number has positive and negative factors
However, it is customary to ignore the negative factors of a positive number
Example
Find the factors of 30
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30
We ignore the negative factors
Factors of a negative number
The negative factors of a negative number are listed as factors
Example
Find the factors of
Factors of are
The symbol
stands for
The above list of factors of is written
This is shorter
SLIDE 45
Using the symbol find all the factors of and
SLIDE 46
Factors of are
Factors of are
SLIDE 47
Using the special symbols and it is possible to write numbers using exactly four 4s.
Examples
Write all the numbers between 1 and 40 using exactly four 4s.
You may also use and
For each number there are many different solutions.
SLIDE 48
Only one possible solution is given in each case.
1
21
2
22
3
23
4
24
5
25
6
26
7
27
8
28
9
29
10
30
11
31
12
32
13
33
14
34
15
35
16
36
17
37
18
38
19
39
20
40
SLIDE 49
The common factors of 36 and 24 are
1, 2, 3, 4, 6 and 12
The highest common factor is 12
The highest common factor is written HCF
HCF is the abbreviation for highest common factor
SLIDE 50
Complete the factor wheels and find the HCF
SLIDE 51
The highest common factor of 100 and 45 is 5
SLIDE 52
Complete the factor wheels and find the common factors and HCF of 60 and 75
SLIDE 53
Common factors of 60 and 75 are 1, 3, 5 and 15
The HCF of 60 and 75 is 15
SLIDE 54
The expression “the highest common factor of 8 and 36 is 4” is written
Find
?
? the highest common factor of 36 and 48
?
? the highest common factor of 26, 39 and 52
? the highest common factor of 12, 18, 30 and 54
?
SLIDE 55
The expression “the highest common factor of 8 and 36 is 4” is written
?
?
?
?
?
?
SLIDE 56
Multiples of 6
Multiples of 10
The lowest number on both lists is called the lowest common multiple
The lowest common multiple of 6 and 10 is 30
is short for “the lowest common multiple of 6 and 10 is 30”
List multiples of 12 and 14 and find
SLIDE 57
Multiples of 12
Multiples of 14
SLIDE 58
The quickest way to find an LCM is to look at successive multiples of the largest of a pair of numbers
Example
Find
We find multiples of 18 and stop when we get to one divisible by 10
18 36 54 72 90
? ? ? ? ?
Advice: if you have three numbers, repeat this process in pairs.
Find
? the lowest common multiple of 8 and 12
?
? the lowest common multiple of 12, 42 and 70
?
SLIDE 59
?
?
? the lowest common multiple of 12, 42 and 70
42 84?
84 168 252 336 420?
?
SLIDE 60
A number is prime if only 1 and itself divide into it. Otherwise, it is composite.
The numbers 9 and 10 are both composite numbers, but they do not have any common factors.
When two numbers have no common factors, they are said to be relatively prime.
Which of these collections of numbers are relatively prime? If they are not relatively prime, state the highest common factor.
? 4 and 7
? 8 and 18
? 3, 4 and 5
? 9, 27 and 45
Answer this question
? Is the following statement true or false?
“Any two prime numbers are also relatively prime.”
SLIDE 61
? 4 and 7 are relatively prime
? 8 and 18 are not relatively prime.
? 3, 4 and 5 are relatively prime
? 9, 27 and 45 are not relatively prime.
? “Any two prime numbers are also relatively prime.”
This statement is true.
SLIDE 62
When two numbers are relatively prime, their lowest common multiple is found simply by multiplying the two numbers together.
Example
5 and 8 are relatively prime.
Find the LCM of
? 4 and 7
? 8 and 18
? 3, 4 and 5
? 9, 27 and 45
SLIDE 63
? 4 and 7 are relatively prime.
? 8 and 18 are not relatively prime.
? 3, 4 and 5 are relatively prime.
? 9, 27 and 45 are not relatively prime.
SLIDE 64
p q
8 12 96 4 24
10 15
24 36
18 24
14 21
16 20
? Complete the table
? What do you observe about the entries in the third and last columns? Do you think this is a coincidence?
SLIDE 65
p q
8 12 96 4 24
10 15 150 5 30
24 36 864 12 72
18 24 432 6 72
14 21 294 7 42
16 20 320 4 80
This is not a coincidence.
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