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g
Comin
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Subject links
Multiples and
factors
G design and technology
up …
G using divisibility tests
G finding multiples of a number
G finding factors of a number
G the prime numbers between 1 and 100
Do you
remember?
G why a number is not prime
G finding the lowest common multiple of a set of numbers
G
G finding the highest common factor of a set of numbers
G solving problems involving the lowest common multiple and the
highest common factor
G
G
G
multiplication tables up
to 10 × 10
sequences
inverse operations
square numbers
Chapter starter
A Christmas tree has flashing lights.
The red lights flash every 4 seconds.
The blue lights flash every 6 seconds.
The green lights flash every 8
seconds.
The red, blue and green lights all
flash at the same time.
How long will it be before they all
flash at the same time again?
Key words
integer
divisibility
divisible
digit
multiple
sequence
term
factor
factor pair
prime
common multiple
lowest common multiple
common factor
highest common factor
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5.1 Tests of divisibility
N
All th ote
e
in this numbers
c
are p hapter
ositiv
e
integ
ers.
You can test whether an integer (a whole number) is exactly divisible by
another integer without actually doing the division.
Numbers are …
divisible by 2 if the last digit is an even number
Not
The e
also re are
divis
tests ibility
7 and for
11.
divisible by 3 if the sum of the digits is divisible by 3
divisible by 4 if the last two digits are divisible by 4
divisible by 5 if the last digit is 5 or 0
divisible by 6 if half of it is divisible by 3
divisible by 8 if half of it is divisible by 4
divisible by 9 if the sum of the digits is divisible by 9
divisible by 10 if the last digit is 0
A number is divisible by 3 if the sum of
the digits is divisible by 3.
Example
Solution
How can you tell if
123 456 789 is exactly
divisible by 3?
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 and 45 is divisible by 3.
Because 4 + 5 = 9, which is divisible by 3.
So 123 456 789 is divisible by 3.
A number is divisible by 5 if the last digit is 5 or 0.
Example
Solution
What is the nearest number
to 444 444 444 444 that is
divisible by 5?
444 444 444 445 is the nearest number which is divisible by 5.
Example
Solution
Explain why 3464 is exactly
divisible by 8.
A number is divisible by 8 if half of it is divisible by 4.
444 444 444 444 is closer to 444 444 444 445 than it is to 444 444 444 440.
Half of 3464 is 1732 and 1732 is divisible by 4 because the number
made by the last two digits (32) is divisible by 4.
Now try these 5.1
904
5536
5029
11 298
2 Which of these numbers are exactly divisible by 4?
2024
2012
1150
4442
divisible by 8?
345 111
4 Which of these numbers are exactly divisible by 9?
252
2140
5
3078
532
38 196
90 899
Explain why 4311 cannot be exactly divisible by 6.
nu
Conti
5 Multiples and factors
45
.
267
3 Which of the numbers in question 2 are exactly
..
1 Which of these numbers are exactly divisible by 3?
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9 Alisha says
Puzzle
Write a digit on the end of 541 to make a
number that is exactly divisible by
(a) 10
(b) 2
(c) 3
(d) 4
(e) 5
(f) 6
(g) 8
(h) 9.
7
Puzzle
Write a digit on the end of 772 06 to make a
number that is exactly divisible by
(a) 3
(b) 4
(c) 6
(d) 8
(e) 9.
8
If a number
is exactly divisible by 9
then it must be exactly
divisible by 3.
Is Alisha correct?
Give a reason for your answer.
Investigation
From this list of numbers
90 440
10 830
2431
6630
8303
383 332
59 049
27 528
Kieran says that if 900 018 is exactly
divisible by both 2 and 9 then it is
also divisible by 18 which is 2 × 9.
find a number that is exactly divisible by both
Is Kieran correct?
(a) 3 and 10
(d) 3 and 9
Does this rule always work?
(b) 3 and 4
(e) 4 and 6.
(c) 5 and 8
Investigate further.
5.2 Multiples
N
These are the multiples of 7.
A multiple of a number is found by multiplying
that number by any integer.
7, 14, 21, 28, 35, 42, 49, 56, ...
Here are some multiples of 3.
15
24
5 × 3 = 15
3000
8 × 3 = 24
All th ote
e
in this numbers
c
are p hapter
ositiv
e
integ
ers.
1000 × 3 = 3000
You can often use divisibility tests to answer
questions on multiples.
Example
Is 65 348 a multiple of 9?
These are the multiples of 3.
3,
6,
9,
12,
15,
18,
21,
24,
27,
The three dots show that the sequence
continues in the same way.
A sequence that goes on forever is called an
infinite sequence.
Here are some multiples of 7.
7
1×7=7
46
5.2 Multiples
35
5 × 7 = 35
56
8 × 7 = 56
...
Solution
If 65 348 is a multiple of 9 then 65 348 will be
exactly divisible by 9.
A number is divisible by 9 if the sum of the digits
is divisible by 9.
6 + 5 + 3 + 4 + 8 = 26 which is not exactly
divisible by 9.
So 65 348 is not a multiple of 9.
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Now try these 5.2
1
2 In each part, give a reason for your answer.
Write down
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
the multiples of 4 that are less than 10
the multiples of 9 that are less than 40
the multiples of 6 that are between 15 and 35
the multiples of 3 that are between 10 and 25
the smallest multiple of 17
the multiples of 10 that are between 105 and 205
the smallest multiple of 5 that has exactly three digits
the largest multiple of 2 that has exactly four digits.
(a)
(b)
(c)
(d)
Is 562 336 a multiple of 4?
Is 25 556 a multiple of 5?
Is 36 822 a multiple of 6?
Is 825 867 a multiple of 9?
5.3 Factors
All th Note
en
this c umbers
in
posit hapter ar
e
ive in
teger
s.
A factor of a number is an integer that will divide exactly into that number.
These are all the factors of 8.
1
because 8 ÷ 1 = 8
2
4
because 8 ÷ 2 = 4
because 8 ÷ 4 = 2
The factors of 8 are 1, 2, 4 and 8.
Factors come in pairs.
8
1, 2, 4, 8
Using factor pairs is an easy way
to find factors.
because 8 ÷ 8 = 1
1 and 8 are a factor pair because 1 × 8 = 8.
2 and 4 are a factor pair because 2 × 4 = 8.
Hint
Example
Solution
Find all of the factors of 24.
Try 1
1 × 24 = 24
So 1 and 24 are factors.
Try 2
2 × 12 = 24
So 2 and 12 are factors.
Try 3
3 × 8 = 24
So 3 and 8 are factors.
Try 4
4 × 6 = 24
So 4 and 6 are factors.
Try 5
5 is not a factor.
Try 6
6 × 4 = 24
So 6 and 4 are factors.
Make sure you don’t
confuse factors and
multiples.
Multiples are always greater
than or equal to the number
itself.
Factors are always less than or
equal to the number itself.
Be
syste
matic
.
But you already have this factor pair. 6 × 4 is the same as 4 × 6.
So you have now finished finding factor pairs.
The factors of 24 are 1, 24, 2, 12, 3, 8, 4, 6.
In order they are 1, 2, 3, 4, 6, 8, 12, 24.
1, 2, 3, 4, 6, 8, 12, 24
5 Multiples and factors
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Now try these 5.3
(a)
(e)
(i)
(m)
(q)
10
16
40
42
6
(b)
(f)
(j)
(n)
(r)
7
11
2
50
54
(c)
(g)
(k)
(o)
(s)
15
28
35
3
32
(d)
(h)
(l)
(p)
2 Write down the factor pairs of
these numbers.
20
12
25
49
(a) 64
(c) 100
E
CE
FI
Investigation
Find all the numbers between 1 and 100 that
have an odd number of factors.
What is special about these numbers?
PE
TY
N
E
All the rows in a rectangle must have the
same number of plants in.
F
C
Draw diagrams to show all the ways he can
arrange them.
1
N
FIV
PE
E
5
19
N
10
2
He wants to plant them out so that they grow
in a rectangle shape.
PE
N
CE
2
C
TW
O PEN
O PEN
E
Puzzle
C
4
3 Charlie has 36 plants.
(b) 72
(d) 75
TE
1 Write down all the factors of each of these numbers.
TW
H
You c int
a
divisib n use the
il
to he ity tests
lp yo
u.
87
50
O
NE
POUN
D
(a) How many different ways can you make £1?
List the ways using just one type of coin.
(b) Using the same type or different types of
coins, how can you make £1 using
(i) one coin
(ii) two coins
(iii) three coins
Hin
Not a t
(iv) four coins
these ll of
(v) five coins
poss are
ible.
(vi) six coins
(vii) seven coins
(viii) eight coins
(ix) nine coins?
5.4 Prime numbers
A prime number has exactly two factors.
The factors of a prime number are 1 and itself.
5 is a prime number because it has exactly two factors.
13 is a prime number because it has exactly two factors.
1 is not a prime number because it only has one factor.
The factors of 5 are 1 and 5.
The factors of 13 are 1 and 13.
..
.
You can explain why a number is not prime by showing that it is divisible by a
number that is not 1 or itself.
Co
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5.4 Prime numbers
u
ntin
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H
Example
Explain why 49 is
not a prime number.
Solution
49 is a multiple of 7 so 7 is a factor of 49.
The factors of 49 are 1, 7 and 49.
So 49 is not a prime number.
Example
Solution
Explain why these numbers
cannot be prime numbers.
(a) 4 333 792 is divisible by 2 because the last digit is an even number.
(a) 4 333 792
(b) 98 765
(c) 111 111 111
Hin
Use t t
divisib he
ilit
rules y
.
Did you know?
Some people have a fear of the
number 13 but not because it is
prime.
These people have triskaidekaphobia
(a good word to make at Scrabble).
People with a fear of Friday 13th have
paraskavedekatriaphobia (an even
better word to make at Scrabble).
So 4 333 792 is not a
prime number.
It has at least 3 factors: 1, 2 and
4 333 792.
(b) 98 765 is divisible by 5 because the last digit is 5.
So 98 765 is not a
prime number.
It has at least 3 factors: 1, 5 and
98 765.
Adding the digits: 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9.
(c) 111 111 111 is divisible by 9 because the sum of the digits is divisible
by 9.
It has at least 3 factors:
So 111 111 111 is not a
1, 9 and 111 111 111.
prime number.
It is h
Hin
elp
the p ful if you t
rime
numb can reme
If y
whet ou are no ers up to mber
her th
t
1
e num sure, che 00.
ck
b
e
b
r
y
is
2, 3,
If it
divisib
le
divisib is less tha 5 or 7.
n 100
le by
and n
those
o
n
prime umbers, t
it is
.
Now try these 5.4
1
Puzzle
Here is a way to find all the prime numbers up to 100.
(a) Draw a hundred square.
Colour the square containing the number 1.
(b) Colour all the squares containing a multiple of 2 except
1 × 2. The first few have been done for you.
(c) Colour all the white squares containing a multiple of 3
except 1 × 3. The first few have been done for you.
(d) What do you notice about the squares containing a
multiple of 4? Which other numbers will this be true for?
(e) Colour all the white squares containing a multiple of 5
except 1 × 5.
(f) Colour all the white squares containing a multiple of 7
except 1 × 7.
The white squares that remain show all the prime numbers
up to 100.
(g) What do you notice about the numbers you coloured
multiples of?
(h) Why don’t you need to colour multiples of 11?
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
.
You don’t have to use different colours.
Co
..
A nu int
has mmber tha
ore th t
tw
a
cann o factors n
ot be
prime
.
u
ntin
5 Multiples and factors
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2 Explain why 266 is not a prime number.
5
3 Show that 477 is not a prime number.
6 Is your home telephone number a prime number?
Explain why 235 568 321 607 cannot be prime.
Is your mobile number a prime number?
4 Karen says that 34 895 is a prime number.
Explain why she must be wrong to say this.
7
?
Brain strain
Research
Explain why the only even prime number is 2.
Mathematicians like prime numbers.
8
1 Look at the puzzle in question 1.
Find the name of this method of finding
prime numbers.
Is it a new method?
Brain strain
Which prime numbers less than 100 can
be written as the sum of two square
numbers?
The first sum of two square numbers
that is a prime number is 5.
2 What work has been done to try to find
the largest prime number?
H
int
There
a
prime re ten m
ore
numb
ers to
Work find.
sy
to fin stematica
d the
ll
m all y
.
12 + 22 = 1 + 4 = 5
5.5 Lowest common multiple
The lowest common multiple of a set of numbers is the smallest number that is a
multiple of all the numbers in the set.
You can use the abbreviation LCM for lowest common multiple.
Here are some multiples of 4.
4,
8,
12,
16,
20,
24,
28,
32,
36,
N
ot
Anot
her n e
ame
the
for
comm lowest
on m
ultip
is
comm the least le
on m
ultiple
.
…
Some multiples of 6 are
6,
12,
0
18,
4
24,
6
30,
8
12
36,
...
16
18
20
24
28
30
32
36
These are some common multiples of 4 and 6.
36,
…
.
24,
..
12,
They are in both lists.
Co
50
5.5 Lowest common multiple
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ntin
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12 is the smallest common multiple.
So 12 is the lowest common multiple
of 4 and 6.
List some multiples of 10 and of 14.
Start with the smallest each time.
Example
Solution
Find the lowest common
multiple of 10 and 14.
Multiples of 10:
10,
20,
30,
40,
50,
60,
Multiples of 14:
14,
28,
42,
56,
70,
…
70,
80,
…
Hint
Stop
a
the fir fter you’v
ew
st few
term ritten
You c sequenc s of the
an
e.
later add more
if you
t
need erms
to.
You can stop at 70 as this is the first multiple
of 14 that is in the multiples of 10 list.
The lowest common multiple of 10 and 14 is 70.
0
10
14
20
2830
4042
50
56
60
70
You can find the lowest common multiple of three or more numbers by an
extension of this method.
The common multiples must be in all the lists.
Example
Solution
Find the lowest common
multiple of 2, 3 and 6.
Multiples of 2:
2,
4,
6,
8,
10,
Multiples of 3:
3,
6,
9,
12,
...
Multiples of 6:
6,
12,
...
...
0
2 3 4
6
8 9 10
12
The lowest common multiple of 2, 3 and 6 is 6.
Now try these 5.5
Find the lowest common multiple of these pairs of
numbers.
(b) 3 and 4
(e) 8 and 10
(h) 50 and 10
(c) 6 and 9
( f ) 6 and 10
2 Find the lowest common multiple of these sets of
numbers.
(a) 2, 3 and 4
(d) 5, 8 and 10
set out in rows of 8, 10 or 12.
The school caretaker uses a mixture
of rows of 8, 10 and 12 chairs.
What is the smallest number of chairs
he sets out that is a multiple of these
three numbers?
(b) 4, 5 and 10
(c) 2, 6 and 8
(e) 2, 3, 4 and 5 (f) 150, 200 and 300
.
(a) 2 and 5
(d) 7 and 14
(g) 12 and 8
3 The chairs in a school hall can be
..
1
Co
u
ntin
5 Multiples and factors
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4 A bus stop is used by three bus routes.
5
The number 64 leaves every 7 minutes.
The number 54 leaves every 10 minutes.
The number 92 leaves every 15 minutes.
Buses on all three routes leave together at 9 a.m.
(a) When do a number 64 and a number 54 next
leave at the same time?
(b) When do a number 54 and a number 92 next
leave at the same time?
(c) When do buses on all three routes next leave at
the same time?
Puzzle
Julie’s watch gains 1 minute every hour.
Kate’s watch loses 1 minute every hour.
Adam’s watch has stopped altogether.
Whose watch keeps the best time?
Don’t forget there are
60 minutes in an hour.
5.6 Highest common factor
The highest common factor of a set of numbers is the largest number that is a factor of all the
numbers in the set.
You can use the abbreviation HCF for highest common factor.
The factors of 12 are
In order:
1,
2,
3,
The factors of 30 are
In order:
1,
2,
3,
1, 12,
4,
2, 6,
6,
1, 30,
5,
Writing the factors in order makes it easier to
compare the factors of different numbers.
12
2, 15,
6,
1 2 3 4 5 6
10
Work out the factor pairs.
3, 4
10,
3, 10,
15,
5, 6
30
12
15
30
These are the common factors of 12 and 30.
1, 2, 3 and 6
They are in both lists.
6 is the largest common factor.
So 6 is the highest common factor of 12 and 30.
..
.
You can find the highest common factor of three or more numbers by an extension of this method.
The common factors must be in all the lists.
nu
Conti
52
5.6 Highest common factor
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Example
Solution
Find the highest common
factor of 16, 20 and 36.
Factors of 16:
Don’t mix up lowest
common multiple
and highest
common factor.
The lowest common multiple is
always greater than (or equal
to) the largest number. The
highest common factor is
always less than (or equal to)
the smallest number.
Factors of 20:
Factors of 36:
1,
2,
1,
1,
2,
2,
4,
8,
4,
16
5,
3,
1 × 16
2×8
4×4
4,
10,
6,
1 × 20
2 × 10
4×5
20
9,
12,
18,
36
1 × 36
2 × 18
3 × 12
4×9
6×6
The highest common factor of 16, 20 and 36 is 4.
1 2 3 4 5 6
8 9 10
12
16
18
20
36
Now try these 5.6
1 Find the highest common factor of these pairs of
numbers.
(a) 6 and 8
(d) 14 and 21
(g) 30 and 50
(b) 9 and 12
(e) 3 and 27
(h) 25 and 45
(c) 10 and 15
( f ) 18 and 24
2 Find the highest common factor of these sets of
numbers.
(a) 4, 6 and 8
(c) 16, 32 and 40
(e) 24, 28, and 40
(b) 12, 16 and 20
(d) 9, 30 and 60
( f ) 300, 400 and 500
3 What is the largest number of children who
can share equally 96 chocolates and 64
toffees?
4
Puzzle
When books are printed a certain number of
pages are printed onto a large sheet of paper.
The large sheets are then folded and glued
together to make the book. The number of large
sheets used for each book depends on how
many pages the finished book will have, as the
number of pages printed on to each large sheet
is always the same.
Jane looks at two of her books.
One has 96 pages.
The other has 160 pages.
How many pages do you think were printed on
the large sheets used to make these books?
5 Multiples and factors
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