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10
Factors, powers
and roots
This chapter is about multiples, factors,
powers and roots.
Objectives
This chapter will show you how to
• use the terms positive and negative square
In the Chinese calendar two separate
root  D
cycles interact. There are 10
• calculate common factors, highest common factors
heavenly stems and 12 zodiac
and lowest common multiples  D   C
animals. You can use lowest
• use powers and roots to solve problems  D   C
common multiples to work out
• write a number as a product of prime factors   C
when a certain year will come
round again.
Before you start this chapter
2 Work out
a 62
Put your calculator away!
1 Which numbers from the cloud are
a multiples of 10
18
b multiples of 7?
70
52
60
21
26
3 List all the factors of
a 24
b 20
35
90
20
b (21)2
c (212)2 d 132
c 15
d 50
4 List all the prime numbers between 20 and 40.
10.1
Lowest common multiples
Why learn this?
Astronomers use lowest
common multiples to
calculate when the Sun
and Moon will be aligned
in an eclipse.
Keywords
lowest common multiple (LCM),
common multiple
Objectives
D   C   Find lowest common
multiples
Skills check
1 Write down the next two terms in each sequence.
a 4, 8, 12, 16, 20, …
b 27, 36, 45, 54, 63, …
2 Write down all the multiples of 7 between 40 and 60.
Lowest common multiples
The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.
The lowest common
multiple is also known as
the least common multiple.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, …
The multiples of 4 are 4, 8, 12, 16, 20, 24, …
The common multiples of 3 and 4 are 12, 24, …
Another way to find LCMs is
covered in Section 10.4.
So the lowest common multiple of 3 and 4 is 12.
Example 1
C
What is the lowest common multiple of 6 and 8?
Write down the multiples of both numbers and circle the common multiples.
Multiples of 6: 6, 12, 18, 24 , 30, 36, 42, 48 , …
Multiples of 8: 8, 16, 24 , 32, 40, 48 , …
6 3 8 5 48 is definitely a
common multiple of 6 and 8
so you can stop at 48.
The common multiples are 24, 48, …
The lowest common multiple is 24.
Exercise 10A
1 a Write down the first ten multiples of 6.
b Write down the first ten multiples of 9.
c What is the LCM of 6 and 9? LCM stands for lowest
common multiple.
D
2 Write down two common multiples of
a 2 and 3
b 7 and 5
c 2 and 10
d 9 and 3.
3 Work out the lowest common multiple of
a 5 and 6
b 8 and 10
c 2 and 5
d 10 and 15
e 12 and 15
f 20 and 30.
10.1 Lowest common multiples
C
157
C
4 Shazia says that you can find the LCM of two numbers by multiplying them
C
5 Carla and Guy have each got the same number of CDs.
together. Give an example to show that Shazia is wrong.
Carla has arranged her CDs into 8 equal piles.
Guy has arranged his CDs into 12 equal piles.
What is the smallest number of CDs they could each have?
6 Fred is building a wall. He uses red bricks which are 12 cm long and yellow bricks
which are 14 cm long. The bricks must line up at the start and end of the wall.
12 cm
14 cm
AO 2
Work out the shortest length of wall that Fred could build.
10.2
L
Highest common factors
Why learn this?
This topic often comes
up in the exam.
Keywords
highest common factor (HCF),
common factor
Objectives
D   C   Find highest common
factors
Skills check
1 Choose a number from the cloud which is
a a factor of 36
b a multiple of 7
c a factor of 40 and a multiple of 10
d a factor of 24 and a factor of 16.
16
8
6
28
20
30
Highest common factors
The highest common factor (HCF) of two numbers is the largest number that is a factor of
both numbers.
The factors of 12 are 1, 2, 3, 4, 6 and 12.
The factors of 16 are 1, 2, 4, 8 and 16.
The common factors of 12 and 16 are 1, 2 and 4.
So the highest common factor of 12 and 16 is 4.
158
Factors, powers and roots
Another way to find
HCFs is covered in
Section 10.4.
Example 2
C
What is the highest common factor of 20 and 30?
Write down the factors of both numbers and circle the common factors.
Remember to include 1
and the number itself in
your list of factors.
Factors of 20: 1,2, 4,5,10, 20
Factors of 30: 1,2, 3,5, 6, 10 , 15, 30
The common factors are 1, 2, 5 and 10.
The highest common factor is 10.
Exercise 10B
1 a Write down the factors of 12.
HCF stands for highest
common factor.
b Write down the factors of 8.
c What is the HCF of 12 and 8?
C
2 Work out the highest common factor of
a 15 and 25
b 14 and 12
c 21 and 15
d 24 and 20
e 8 and 10
f 8 and 16.
D
3 Write down two numbers larger than 10 with an HCF of 8.
NA
U
NCTIO
NCTIO
L F
U
NA
C
72 cm
4 Lydia is making Christmas decorations. She needs to cut identical squares out of a rectangular sheet
of paper.
a What is the largest square size Lydia can use
without wasting any paper?
60 cm
AO 2
L F
b How many of these squares will Lydia be able to
cut from this sheet of paper?
10.3
L
Squares, cubes and roots
Why learn this?
You are expected to
recall certain squares,
cubes, square roots and
cube roots in the exam.
Keywords
square root, positive
square root, negative
Objectives
square root, cube root
D Understand the difference
between positive and negative square roots
C Evaluate expressions involving squares, cubes and roots
Skills check
1 Work out
b 53
a 22
2 Work out
a 23 3 23
c 21 3 21
c 72
d 33
____
e​ √121 ​ 
3
___
f​√  64 ​ 
b 26 3 26
d 212 3 212
10.3 Squares, cubes and roots
159
Squares and square roots
To square a number you multiply it by itself. The inverse of squaring is finding the square root.
square
5
25
square root
52 5 5 3 5 5 25
(25)25 25 3 25 5 25
Every positive number has two square roots. 5 is the positive
square root of 25 and 25 is the negative square root of 25.
__
The symbol √
​  0 ​  is used to___
represent the positive square root
√
of a number. You write ​  25 ​ 5 5.
You can write the negative
square root of 25 as 2√25.
You need to know the squares of integers up to 15 and their corresponding square roots.
Cubes and cube roots
The inverse operation of cubing is finding the cube root.
__
3
√
Every number has exactly one cube root. The symbol ​  0 ​  
is used to represent the cube root of a number.
43 5 64
3
√64 5 4
You need to know the cubes of 1, 2, 3, 4, 5 and 10 and their corresponding cube roots.
D
C
Example 3
3
______
a Write down the negative square root of 81. b Work out ​√  52 1 2 ​ 
.
___
a​ √81 ​ 5 9 so the negative square root of 81 is 29.
_______
3
√ 2
3
(29)2 5 81
________
b​   5 1 2 ​5
 ​√
  25 1 2 ​
 
___
3
5 ​√  27 ​ 
5 3
The cube root sign is like a bracket.
You have to work out the value
underneath the cube root first.
Exercise 10C
D
1 Work out the negative square root of
D
2 Write down two possible values that would make this statement true.
a 49
b 25
2
1 9 5 45
AO 2
160
Factors, powers and roots
c 4
d 9
3 Work out
______
______
a​ √22 1 5 ​ 
_______
b​ √62 1 82 ​ 
4 Work out
_______
c​ √52 2 42 ​ 
_______
a​ √ 53 2 52 ​ 
3
b​ √23 1 23 ​ 
________
c​√  112 1 22 ​ 
_________
d​ √132 2 122 ​ 
3
C
______
d​√  32 2 1 ​ 
5 Estimate the answers to these calculations by rounding each value to the nearest
whole number.
_________
b 5.043 2 3.283
a 6.7 2 1 3.12
c​ √19.8 2 4.1 ​ 
3
__________
d​√  2.12 1 1.7 2 ​ 
6 Amber and Simon have each made a 10 cm square pattern using 1 cm square tiles.
Amber gives some tiles to Simon. They are both able to arrange their tiles exactly
into square patterns.
How many tiles did Amber give to Simon?
10.4
L
C
AO 3
Prime factors
Keywords
prime factor
Why learn this?
You can use prime factors
to calculate lowest common
multiples and highest common
factors much more quickly.
Objectives
C Write a number as a product of prime factors using
index notation
C Use prime factors to find HCFs and LCMs
Skills check
1 Write down all the prime numbers from the cloud.
1
5
18
21
39
33
87
17
19
2 Write down a multiplication fact to show that 111 is not a prime number.
Writing a number as a product of prime factors
You can write any number as a product of prime factors.
You can use a factor tree to write a number as a product of prime factors.
60
2
30
2
15
3
5
• Split each number up into factor pairs.
• When you reach a prime number, draw
a circle around it. These are the ends of
the branches.
• The answer is the product of the prime
numbers on the branches.
60 5 2 3 2 3 3 3 5
You can write this using index notation as 60 5 22 3 3 3 5.
10.4 Prime factors
161
You can also use repeated division to write a number as a product of prime factors.
Divide by 2 as many times as possible.
2 60
2 30
3 15
5 5
1
You cannot divide 15 by 2.
Try the next prime number.
Divide by each prime number as many
times as possible. Stop when you reach 1.
2
60 5 2 3 3 3 5
When writing a number as a product of prime factors, write
the prime factors in order from smallest to largest.
C
Example 4
Write each number as a product of prime factors using index notation.
a 50 b 1960
You could also use
a
repeated division.
50
2
2 50
5 25
5 5
1
25
5
5
505 2 3 5 3 5
5 2 3 52
b
1960
10
2
You could also use
repeated division.
2 1960
2 980
2 490
5 245
7
49
7
7
1
196
5
14
2
14
7
2
7
19605 2 3 2 3 2 3 5 3 7 3 7
5 23 3 5 3 72
Exercise 10D
C
1 a Copy and complete this factor tree. 84
b Write 84 as a product of prime factors
using index notation.
42
6
2 Write each number as a product of prime factors using index notation.
a 20
b 63
c 64
d 45
e 110
f 81
3 Write each number as a product of prime factors using index notation.
a 156
162
b 1980
c 7700
Factors, powers and roots
d 608
e 2025
f 980
4 a Copy and complete these two 100
factor trees for 100.
10
100
10
2
50
b Jamie says that the prime factors will be different depending on
which factor pair you choose first.
Do you agree with him?
Demonstrate your answer using another number.
C
AO 2
Using prime factors to find the HCF
• Write each number as the product of prime factors.
• If a prime number is in both lists, circle the lowest power.
• Multiply these to find the HCF.
If a prime number is only
in one of the lists you can’t
include it in your HCF.
Using prime factors to find the LCM
• Write each number as the product of prime factors.
• Circle the highest power of each prime number.
• Multiply these to find the LCM.
C
Example 5
Work out
a the HCF of 180 and 168 b the LCM of 24 and 60.
The prime numbers in both lists
are 2 and 3. The lowest power of 2
is 22. The lowest power of 3 is 31.
a 180 5 22 3 32 3 5
168 5 23 3 3 3 7
The HCF of 180 and 168 is 22 3 3 5 12.
b 24 5 23 3 3 You only need to circle the highest power of 2.
60 5 22 3 3 3 5
3
The LCM of 24 and 60 is 2 3 3 3 5 5 120.
3 is a factor of 24 and
60. You only need to
circle it once.
Exercise 10E
1 a Write 90 as a product of prime factors.
b Write 165 as a product of prime factors.
c Find the HCF of 90 and 165.
C
2 a Write 42 as a product of prime factors.
b Write 30 as a product of prime factors.
c Find the LCM of 42 and 30.
3 Work out the highest common factor of each pair of numbers.
a 32 and 56
b 80 and 72
c 27 and 45
d 100 and 75
4 Work out the lowest common multiple of each pair of numbers.
a 18 and 20
b 6 and 32
c 27 and 15
d 9 and 75
10.4 Prime factors
163
C
5 Work out the highest common factor of 2016 and 1512.
6 Amy is investigating the relationship between the LCM and the HCF.
a Work out the HCF of 18 and 30.
AO 2
C
18 3 30
 ​ 
. 
b Amy says that she can find the LCM of 18 and 30 using the rule LCM 5 ​ _______
HCF
Show working to check that Amy’s rule works.
c Show that Amy’s rule will also work for 16 and 40.
7 David has a pack of playing cards with some missing. AO 3
He arranges his playing cards into 15 rows of equal length.
He then rearranges his cards into 9 rows of equal length.
How many cards are missing from David’s pack?
A normal pack
of playing cards
contains 52 cards.
Review exercise
D
1 Write down the negative square root of
D
2 Write down two different solutions to each equation.
a 144
a x2 5 64
b 81
b x2 1 1 5 50
c 225
c x2 2 10 5 90
d 16
[1 mark each]
d 2x2 5 8
[2 marks each]
AO 2
________
[2 marks]
4 Write 3300 as a product of prime factors using index notation.
[3 marks]
180 cm
[3 marks]
[3 marks]
8 A clock tower has three different bells. The largest bell rings once every 12 seconds.
AO 3
The middle bell rings once every 10 seconds. The smallest bell rings once every
8 seconds. The bells all ring at the same time.
a How long will it take before the bells all ring at the same time again?
[3 marks]
Give your answer in minutes.
b How many times has each bell rung?
[3 marks]
Chapter summary
In this chapter you have learned how to
• understand the difference between positive and
negative square roots  D
• find lowest common multiples  D   C
• find highest common factors  D   C
164
Factors, powers and roots
• evaluate expressions involving squares, cubes
and roots  C
• write a number as a product of prime factors
using index notation  C
• use prime factors to find HCFs and LCMs  C
NA
L F
NCTIO
C
7 Work out the HCF of 96, 120 and 150.
260 cm
U
AO 2
She wants to use identical square tiles
to completely cover the floor with no overlap.
Work out the largest size of square tile Daisy can use.
NCTIO
6 Daisy is tiling her bathroom floor. [3 marks each]
U
C
b the LCM of 45 and 54.
L F
5 Work out a the HCF of 80 and 96
NA
C
3 Work out the value of √
​  102 2 62 ​ 
.