Exercise 3.3 Prime numbers and composite numbers

NUMBER AND ALGEBRA
Exercise 3.3 Prime numbers and
composite numbers
INDIVIDUAL PATHWAYS
REFLECTION
What strategies will you use
to help recall the difference
between prime and composite
numbers?
⬛
⬛
PRACTISE
Questions:
1, 2a–f, 3a–f, 4a–f, 5a–f, 6a–d, 7
⬛
CONSOLIDATE
Questions:
1, 2d–i, 3d–i, 4d–i, 5d–i, 6c–f, 7
⬛ ⬛ ⬛ Individual pathway interactivity
MASTER
Questions:
1, 2g–l, 3g–l, 4g–l, 5g–l, 6e–h,
7, 8
int-4320
FLUENCY
1 WE4 Find four prime numbers that are between 20 and 40.
2 WE5 State whether each of the following numbers is prime or composite.
a 9
b 13
c 27
d 55
e 41
f 64
g 49
h 93
i 51
j 79
k 243
l 101
3 WE6 i Find the prime factors of each of the following numbers by drawing a
factor tree.
ii Write each one as a product of its prime factors in index form.
a 15
b 30
c 24
d 100
e 49
f 72
g 18
h 56
i 45
j 84
k 98
l 112
WE7
4
i Find the prime factors of the following numbers by drawing a factor tree.
ii Express the number as a product of its prime factors in index form.
a 40
b 35
c 32
d 121
e 110
f 150
g 3000
h 64
i 96
j 196
k 90
l 75
5 Find the prime factors of each of the following numbers.
a 48
b 200
c 42
d 81
e 18
f 39
g 27
h 300
i 60
j 120
k 50
l 80
6 WE8 Write the following as a product of prime factors using index notation.
a 60
b 50
c 75
d 220
e 192
f 72
g 124
h 200
UNDERSTANDING
7 WE9 By expressing
the following pairs of numbers as products of their prime factors,
determine their lowest common multiple and their highest common factor.
a 36 and 84
b 48 and 60
c 120 and 400
d 220 and 800
8 Can you find four prime numbers that are even? Explain.
9 Answer True (T) or False (F) for each of the following.
a All odd numbers are prime numbers.
b No even numbers are prime numbers.
c 1, 2, 3 and 5 are the first four prime numbers.
d A prime number has two factors only.
72
Maths Quest 7
NUMBER AND ALGEBRA
2 is the only even prime number
The sum of two prime numbers is always even.
The product of two prime numbers is always odd.
There are no consecutive prime numbers.
MC a The number of primes less than 10 is:
A 4
B 3
C 5
D 2
E 1
b The first three prime numbers are:
A 1, 3, 5
B 2, 3, 4
C 2, 3, 5
D 3, 5, 7
E 2, 5, 7
c The number 15 can be written as the sum of two prime numbers. These are:
A 3 + 12
B 1 + 14
C 13 + 2
D 7+8
E 9+6
d Factors of 12 that are prime numbers are:
A 1, 2, 3, 4
B 2, 3, 6
C 2, 3
D 2, 4, 6, 12
E 1, 2, 3, 4, 6,
12
Twin primes are pairs of primes that are separated from each other by one
even number. For example, 3 and 5 are twin primes. Find two more pairs of twin
primes.
a Which of the numbers 2, 3, 4, 5, 6 and 7 cannot be the difference between two
consecutive prime numbers? Explain.
b For each of the numbers that can be a difference between two consecutive primes,
give an example of a pair of primes less than 100 with such a difference.
The following numbers are not primes. Each of them is the product of two primes. Find
the two primes in each case.
a 365
b 187
An easy way to find prime numbers is to use the Sieve of Eratosthenes.
Eratosthenes discovered a simple method of sifting out all of the composite numbers
so that only prime numbers are left. Alternatively, you can use the Excel file on your
eBookPLUS.
You can follow the following steps to find all prime numbers between 1 and 100.
a Write the numbers from 1 to 100 in a grid as shown.
e
f
g
h
10
11
12
13
14
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
Digital doc
Sieve of Eratosthenes
doc-1689
10
20
30
40
50
60
70
80
90
100
Cross out 1 as shown. It is not a prime number.
Circle the first prime number, 2. Then cross out all of the multiples of 2.
Circle the next prime number, 3. Now cross out all of the multiples of 3 that have not
e The number 4 is already crossed out. Circle the next prime number, 5. Cross out all of
the multiples of 5 that are not already crossed out.
b
c
d
5PQJDt Indices and primes
73
NUMBER AND ALGEBRA
The next number that is not crossed out is 7. Circle 7 and cross out all of the
multiples of 7 that are not already crossed out.
g Do you need to check the multiples of any primes greater than 7? Why or
why not?
15 MC a A factor tree for 21 is:
f
21
A
7
3
1
b
3
1
7×1
21
1
7
3×1
21
E
3
21
C
1
21
D
21
B
21
3
7
7
A factor tree for 36 is:
A
B
36
2
18
9
36
D
E
9
3
4
3
2
4
36
2
18
36
2
18
2
The prime factors of 16 are:
A 1, 2
B 1, 2, 4
d The prime factors of 28 are:
A 1, 28
B 2, 7
C
36
2
9
c
C
2
D
1, 2, 4, 8, 16
E
2, 4, 8
C
1, 2, 14
D
1, 2, 7
E
2, 7, 14
REASONING
16 What is the
largest three‐digit prime number in which each digit is a prime number?
Prove that this number is a prime number.
17 Find a prime number greater than 10 where the sum of the digits equals 11. Show your
working.
18 My age is a prime number. I am older than 50. The sum of the digits in my age is
also a prime number. If you add a multiple of 13 to my age the result is 100. How
old am I?
PROBLEM SOLVING
19 Twin primes are pairs
Digital doc
WorkSHEET 3.1
doc-1687
74
Maths Quest 7
of prime numbers that differ by 2. Except for the pair of primes
2 and 3, this is the smallest difference between two prime numbers. The first twin
primes are 3 and 5, followed by 5 and 7, then 11 and 13. What other twin primes are
there below 100?
20 Find two prime numbers with a product of:
a 21
b 26
c 323.