NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
Lowest common multiple and highest common factors
using prime factorisation
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It can be difficult to find the lowest common multiple (LCM) or the highest common factor
(HCF) of large numbers by inspection alone.
By expressing large numbers as products of their prime factors, it can be easier to determine
their LCM or HCF.
The HCF of two numbers expressed as the products of their prime factors is the product of the
prime factors common to both numbers.
The LCM of two numbers expressed as the products of their prime factors is the product of
the prime factors that are factors of either number.
WORKED EXAMPLE 9
Determine the highest common factor and lowest common multiple of 270 and 900.
THINK
WRITE
1
Write 270 and 900 as products of their prime factors.
270 = 2 ì 3 ì 3 ì 3 ì 5
900 = 2 ì 2 ì 3 ì 3 ì 5 ì 5
2
Circle the prime factors common to both numbers.
270 =
2 ì 3 ì 3 ì3ì 5
900 = 2 ì 2 ì 3 ì 3 ì 5 ì 5
3
For the HCF, multiply the prime factors common to both
numbers.
2 ì 3 ì 3 ì 5 = 90
The HCF of 270 and 900 is 90.
4
For the LCM, multiply the prime factors that are factors of
either number.
2 ì 3 ì 3 ì 5 ì 2 ì 3 ì 5 = 2700
The LCM of 270 and 900 is 2700.
REMEMBER
1.
2.
3.
4.
5.
A prime number is a whole number which has exactly two factors, itself and 1.
A composite number is one which has more than two factors.
20
The number 1 is neither a prime number nor a composite number.
A factor tree shows the prime factors of a composite number.
The last numbers in the factor tree are all prime numbers, therefore
4
they are prime factors of the original number.
6. Every composite number can be written as a product of prime
2
2
factors; for example, 20 = 2 ì 2 ì 5.
7. The product of prime factors can be written in shorter form using index notation.
8. Lowest common multiples and highest common factors can be determined by
expressing numbers as the products of their prime factors.
EXERCISE
3B
Prime numbers and composite numbers
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
74
Maths Quest 7 for the Australian Curriculum
5
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
INDIVIDUAL
PATHWAYS
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Prime numbers and
composite numbers
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Activity 3-B-2
More prime numbers
and composite
numbers
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Activity 3-B-3
Advanced prime
numbers and
composite numbers
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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.
a 15
b 30
c 24
d 100
e 49
f 72
g 18
h 56
i 45
j 84
k 98
l 112
4 WE7 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.
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
d 81
e 18
g 27
h 300
j 120
k 50
42
39
60
80
c
f
i
l
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.
e 2 is the only even prime number
f The sum of two prime numbers is always even.
g The product of two prime numbers is always odd.
h There are no consecutive prime numbers.
10 MC a The number of primes less than 10 is:
A 4
B 3
C 5
b The first three prime numbers are:
A 1, 3, 5
B 2, 3, 4
C 2, 3, 5
D 2
E 1
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
D 2, 4, 6, 12
E 1, 2, 3, 4, 6, 12
C 2, 3
11 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.
Chapter 3
Indices and primes
75
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
12 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.
13 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
14 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 steps below to find all prime numbers between 1 and 100.
a Copy the numbers from 1 to 100 in a grid as shown below. Use 1-centimetre square grid
paper.
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
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
b Cross out 1 as shown. It is not a prime number.
c Circle the first prime number, 2. Then cross out all of the multiples of 2.
d Circle the next prime number, 3. Now cross out all of the multiples of 3 that have not
already been crossed out.
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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.
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
A
B
21
7
3
D
1
E
21
3
1
76
3
1
7
21
Maths Quest 7 for the Australian Curriculum
21
3
21
3ì1
21
1
7
C
21
7
7ì1
NUMBER AND ALGEBRA • NUMBER AND PLACE VALUE
b A factor tree for 36 is:
A
36
2
B
18
C
36
9
4
D
36
36
9
2
3
E
4
18
3
2
2
36
2
18
2
9
c The prime factors of 16 are:
A 1, 2
D 1, 2, 4, 8, 16
B 1, 2, 4
E 2, 4, 8
C 2
d The prime factors of 28 are:
A 1, 28
D 1, 2, 7
B 2, 7
E 2, 7, 14
C 1, 2, 14
REASONING
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16 What is the largest three-digit prime number in which each digit is a prime number?
17 Find a prime number greater than 10 where the sum
of the digits equals 11.
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?
REFLECTION
What strategies will you use to
help recall the difference between
prime and composite numbers?
Prime number sequence: 1–1000
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Prime number
game
1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,
97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181,
191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503,
509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619,
631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743,
751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863,
877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997
Chapter 3
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Indices and primes
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17/02/11 2:00 PM