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59
Chapter 2 Multiples, factors and primes
WORKED Example 10
a Find the prime factors of 50 by drawing a factor tree.
b Write 50 as a product of its prime factors.
THINK
WRITE
a
a
1
Find a factor pair of the given
number and begin the factor tree
(50 = 5 × 10).
2
If a branch is prime, no other factors
can be found (5 is prime). If a
branch is composite, find factors of
that number; 10 is composite so
10 = 5 × 2.
Continue until all branches end in a
prime number then stop.
Write the prime factors.
50
5
3
4
b Write 50 as a product of prime factors
found in part (a).
10
50
5
10
5
2
2 and 5 are prime factors of 50.
b 50 = 5 × 5 × 2
WORKED Example 11
Find the prime factors of 56.
THINK
1
WRITE
Draw a factor tree. When all factors are
prime numbers you have found the
prime factors.
56
8
2
4
2
2
Write the prime factors.
7
2
The prime factors of 56 are 7 and 2.
remember
remember
1. A factor tree shows the prime factors of a composite number.
2. The last numbers in the factor tree are all prime numbers,
therefore they are prime factors of the original number.
3. Every composite number can be written as a product of prime
factors.
For example, 20 = 2 × 2 × 5.
20
4
2
5
2
60
Maths Quest 7 for Victoria
2D
WORKED
Example
Prime
factors
Prime
factors
1
Prime factors and factor
trees
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.
10
a
d
g
j
2
b
e
h
k
15
100
18
84
c
f
i
l
30
49
56
98
24
72
45
112
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
d
g
j
Prime
factors
b
e
h
k
40
121
3000
196
c
f
i
l
35
110
64
90
32
150
96
75
3 multiple choice
a A factor tree for 21 is:
A
21
7
D
B
3
C
21
1
21
3
21
E
b A factor tree for 36 is:
A
36
1
B
36
1
7
1
2
C
18
36
9
4
36
9
9
7
36
18
2
3
E
36
2
c
7
7×1
1
D
1
21
3
3×1
21
3
4
3
The prime factors of 16 are:
A 1, 2
B 1, 2, 4
C 2, 4, 8
d The prime factors of 28 are:
A 1, 28
B 2, 7
C 1, 2, 14
2
2
D 2
E 1, 2, 4, 8, 16
D 1, 2, 7
E 2, 7, 14
Chapter 2 Multiples, factors and primes
4 Find the prime factors of each of the following numbers.
a 48
b 200
11
d 81
e 18
g 27
h 300
j 120
k 50
61
WORKED
Example
c
f
i
l
42
39
60
80
QUEST
GE
S
EN
MAT H
5 State whether each of the following is true (T) or false (F).
a The number 3 is the only prime factor of 9.
b No two numbers can have the same prime factors.
c The numbers 2, 3, 5 and 7 are the prime factors of 210.
d The numbers 1, 2 and 5 are the prime factors of 40.
e The prime factors of 220 are 2, 5 and 11.
f All numbers have exactly 2 prime factors.
CH
AL
L
1 A whole number is ‘perfect’ if it equals the sum of all its proper factors.
The proper factors of a number are all factors smaller than the
number. So 6 is perfect since its proper factors are 1, 2 and 3 and
6 = 1 + 2 + 3.
a Find all the proper factors of 28 and show that 28 is a perfect
number.
b Do the same for 496.
2 Find the 7 proper factors of the number 999. Is 999 a perfect number?
3 Use a calculator to find the 13 proper factors of 8128. Is 8128 a perfect
number?
Index notation
The product of factors can be written in a shorter form by using index notation or index
form. A number in index form has two parts, the base and the index.
The product of factors 3 × 3, can be written as 32. The 3 is the base and the 2 is the
index. Two other examples are given below:
4 × 4 × 4 = 43
43 has a base of 4 and an index of 3.
5
2×2×2×2×2=2
25 is a number in which 2 is the base and 5 is the index.
5
2 is in index notation or index form.
2 × 2 × 2 × 2 × 2 is in expanded form.
A composite number written as a product of prime factors can be written using index
notation. So 50 = 2 × 5 × 5 = 2 × 52 and 56 = 2 × 2 × 2 × 7 = 23 × 7.
62
Maths Quest 7 for Victoria
WORKED Example 12
Write the following using index notation.
a 5×5
b 2×2×6×6×6
THINK
WRITE
a
Write the multiplication.
Write the number being multiplied
as the base and the number of times
it is multiplied as the index.
a 5×5
= 52
Write the multiplication.
Write the number being multiplied
as the base and the number of times
it is multiplied as the index.
b 2×2×6×6×6
= 2 2 × 63
1
2
b
1
2
WORKED Example 13
Write 120 as a product of prime factors using index notation.
THINK
1
WRITE
Find a factor pair and begin the factor
tree. If the number on the branch is a
prime number, stop. If not, continue
until a prime number is reached.
120
12
4
2
2
3
Write the number as a product of prime
factors.
Write your answer using index notation.
10
3
5
2
120 = 2 × 2 × 2 × 3 × 5
120 = 23 × 3 × 5
WORKED Example 14
Write 53 in expanded form and then find the answer.
THINK
1
2
3
4
Write the question in expanded form.
Multiply the first 2 numbers.
Multiply the answer by the next number
and continue until all numbers have
been multiplied.
Write the answer.
2
WRITE
53 = 5 × 5 × 5
= 25 × 5
= 125
53 = 125
Chapter 2 Multiples, factors and primes
Graphics Calculator tip!
63
Calculating numbers
in index form
To calculate a number in index form, first type in the
base, then ^ , then the index and press ENTER .
The screen opposite shows the calculation for 53 (from
worked example 14).
remember
remember
1. The product of prime factors can be written in a shorter form by using index
notation. For example, 40 = 23 × 5. Using index notation, 23 is a number with
base 2 and index of 3.
2. The 2 (or base number) is the number being multiplied and the 3 (or index)
indicates how many times the base is being multiplied 23 = 2 × 2 × 2.
2E
WORKED
Example
12a
WORKED
Example
12b
WORKED
Example
13
WORKED
Example
14
Index notation
1 Write the following using index notation.
a 4×4×4
b 8×8
c 7×7×7×7
d 12 × 12 × 12 × 12 × 12
e 2×2×2×2×2×2×2
f 13 × 13 × 13
g 5×5×5
h 9×9×9×9×9×9×9×9
2 Write the following using index notation.
a 2×2×3
b 3×3×3×3×2×2
c 5×5×2×2×2×2
d 7×2×2×2
e 5 × 11 × 11 × 3 × 3 × 3
f 13 × 5 × 5 × 5 × 7 × 7
g 2×2×2×3×3×5
h 3×3×2×2×5×5×5
3 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
4 Write the following in expanded form and then find the answer.
b 112
c 53
d 24
a 32
2
6
3
2
e 2×7
f 3×2
g 9 ×3
h 2 5 × 42
4
3
2
2
3
i 3 +7
j 2 +3
k 10 − 3
l 5 3 − 24
4
2
5
3
4
2
m 2 ÷2
n 3 ÷ 3 (Hint: Write 2 ÷ 2 as a fraction and simplify.)
5 Write one million using index notation. Use 10 as the base number.
6 multiple choice
The largest number listed here is:
A 150
B 26
C 34
D 53
E 72
7 multiple choice
The smallest number listed here is:
B 102
C 115
A 43
D 025
E 44
Index
notation