Transcript Document

KS3 Mathematics
N3 Multiples, factors and
primes
1 of 53
© Boardworks Ltd 2004
Contents
N3 Multiples, factors and primes
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
N3.5 HCF and LCM
2 of 53
© Boardworks Ltd 2004
Divisibility by 2
Is 367 908 divisible by 2?
367 908 is divisible by 2 if 2 divides into it exactly without
leaving any remainders.
We can tell if a number is divisible by 2 by looking at the
last digit in the number.
A number is divisible by 2 if the last digit is
0, 2, 4, 6 or 8.
Yes, 367 908 is divisible by 2. It is an even number.
3 of 53
© Boardworks Ltd 2004
Divisibility by 3
Is 35 841 divisible by 3?
We can tell if a number is divisible by 3 by adding together its
digits. This is called finding the digit sum.
A number is divisible by 3 if the digit sum is
divisible by 3.
The digit sum of 35 841 is 3 + 5 + 8 + 4 + 1 = 21
21 is divisible by 3.
Yes, 35 841 is divisible by 3.
4 of 53
© Boardworks Ltd 2004
Divisibility by 4
Is 1 934 274 divisible by 4?
We can tell if a number is divisible by 4 by looking at the last
two digits in the number.
A number is divisible by 4 if the last two
digits are divisible by 4.
A two-digit number is divisible by 4:
if it ends in 0, 4 or 8 and the first digit is even
if it ends in 2 or 6 and the first digit is odd.
No, 1 934 274 is not divisible by 4.
5 of 53
© Boardworks Ltd 2004
Divisibility by 5
Is 231 030 divisible by 5?
We can tell if a number is divisible by 5 by looking at the
last digit in the number.
A number is divisible by 5 if the last digit is 0 or 5.
Yes, 231 030 is divisible by 5.
6 of 53
© Boardworks Ltd 2004
Divisibility by 6
Is 53 702 divisible by 6?
We can tell if a number is divisible by 6 by looking at its last
digit and finding its digit sum.
A number is divisible by 6 if it is divisible by
both 2 and 3.
The last digit in 53 702 is a 2 and so it is divisible by 2.
The digit sum of 53 702 is 5 + 3 + 7 + 0 + 2 = 17
17 is not divisible by 3 and so 53 702 is not divisible by 3.
53 703
No,
53 702
is divisible
is not divisible
by 6 only
byif 6.
it is divisible by both 2 and 3.
7 of 53
© Boardworks Ltd 2004
Divisibility by 8
Is 63 072 divisible by 8?
We can tell if a number is divisible by 8 by halving it and
checking for divisibility by 4.
A number is divisible by 8 if half of it is
divisible by 4.
Half of 63 072 is 31 536
Look at the last two digits.
36 is divisible by 4.
Yes, 63 072 is divisible by 8.
8 of 53
© Boardworks Ltd 2004
Divisibility by 9
Is 6 874 275 divisible by 9?
We can tell if a number is divisible by 9 by finding its digit sum.
A number is divisible by 9 if its digit sum is
divisible by 9.
The digit sum of 6 873 975 is 6 + 8 + 7 + 3 + 9 + 7 + 5 = 45
45 is divisible by 9.
Yes, 6 873 975 is divisible by 9.
9 of 53
© Boardworks Ltd 2004
Divisibility by 10
Is 6 311 289 divisible by 10?
We can tell if a number is divisible by 10 by looking at the
last digit in the number.
A number is divisible by 10 if the last digit is 0.
No, 6 311 289 is not divisible by 10.
10 of 53
© Boardworks Ltd 2004
Divisibility by larger numbers
Is 214 875 divisible by 15?
We test for divisibility by some larger numbers by using two
simpler tests.
A number is divisible by 15 if it is divisible
by both 3 and 5.
The digit sum of 214 875 is 2 + 1 + 4 + 8 + 7 + 5 = 27
27 is divisible by 3 and so 214 875 is divisible by 3.
The last digit in 214 875 is a 5 and so it is also divisible by 5.
Yes, 214 875 is divisible by 15.
11 of 53
© Boardworks Ltd 2004
Divisibility by larger numbers
We can check for divisibility by other larger numbers by
testing for divisibility by factors of the number.
For example,
A number is divisible by 12 if it is divisible by both 3 and 4.
A number is divisible by 15 if it is divisible by both 3 and 5.
A number is divisible by 18 if it is divisible by both 2 and 9.
A number is divisible by 20 if it is divisible by both 4 and 5.
A number is divisible by 24 if it is divisible by both 3 and 8.
A number is divisible by 30 if it is divisible by both 3 and 10.
12 of 53
© Boardworks Ltd 2004
Using divisibility to find multiples
13 of 53
© Boardworks Ltd 2004
Contents
N3 Multiples, factors and primes
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
N3.5 HCF and LCM
14 of 53
© Boardworks Ltd 2004
Multiples
A multiple of a number is found by multiplying the number
by any whole number.
What are the first six multiples of 4?
To find the first six multiples of 4 multiply 4 by 1, 2, 3, 4, 5
and 6 in turn to get:
4,
8,
12,
16,
20
and 24.
Any given number has infinitely many multiples.
15 of 53
© Boardworks Ltd 2004
Multiples patterns on a hundred square
16 of 53
© Boardworks Ltd 2004
Rectangular arrangements
17 of 53
© Boardworks Ltd 2004
Finding factors
A factor of a number is a whole number that divides exactly
into a given number.
Factors come in pairs.
For example, what are the factors of 30?
1 and 30, 2 and 15, 3 and 10, 5 and 6.
So, in order, the factors of 30 are:
1, 2, 3, 5, 6, 10, 15 and 30.
18 of 53
© Boardworks Ltd 2004
Factor finder
19 of 53
© Boardworks Ltd 2004
Circle and square puzzle
20 of 53
© Boardworks Ltd 2004
Contents
N3 Multiples, factors and primes
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
N3.5 HCF and LCM
21 of 53
© Boardworks Ltd 2004
Sieve of Eratosthenes
22 of 53
© Boardworks Ltd 2004
Prime numbers
If a whole number has two, and only two, factors it is called
a prime number.
For example, the number 17 has only two factors, 1 and 17.
Therefore, 17 is a prime number.
The number 1 has only one factor, 1.
Therefore, 1 is not a prime number.
There is only one even prime number. What is it?
2 is the only even prime number.
23 of 53
© Boardworks Ltd 2004
Prime numbers
The first 10 prime numbers are:
2
24 of 53
3
5
7
11
13
17
19
23
29
© Boardworks Ltd 2004
Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive numbers.
Is 107 divisible by 2?
The last digit is a 7 so, no.
Is 107 divisible by 3?
The digit sum is 8 so, no.
We don’t need to check for divisibility by 4 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
Is 107 divisible by 5?
The last digit is a 7 so, no.
We don’t need to check for divisibility by 6 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
25 of 53
© Boardworks Ltd 2004
Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive numbers.
Is 107 divisible by 7? Dividing by 7 leaves a remainder so no.
We don’t need to check for divisibility by 8 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
We don’t need to check for divisibility by 9 because if 3
doesn’t divide into 107, then no multiple of 2 can divide into it.
We don’t need to check for divisibility by 10 because if 2
doesn’t divide into 107, then no multiple of 2 can divide into it.
26 of 53
© Boardworks Ltd 2004
Testing for prime numbers
Is 107 a prime number?
We can check whether or not a number is prime by testing for
divisibility by successive prime numbers.
We don’t
Why
don’tneed
we need
to check
to check
for divisibility
for divisibility
by 11bybecause
11?
we have
found that no number below 10 divides into 107.
That means that any number that multiplied 11 would have to
be bigger than 10.
Since, 10 × 11 is bigger than 107 we can stop here.
107 is a prime number.
27 of 53
© Boardworks Ltd 2004
Testing for prime numbers
When we are testing whether or not a number is prime, we
only have to test for divisibility by prime numbers.
We don’t need to check for divisibility by any number bigger
than the square root of the number.
A number is prime if no prime number less than the square
root of the number divides into it.
Also, all prime numbers greater than 5 must end in a 1, 3, 7
or 9.
28 of 53
© Boardworks Ltd 2004
An amazing fact
29 of 53
© Boardworks Ltd 2004
Contents
N3 Multiples, factors and primes
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
N3.5 HCF and LCM
30 of 53
© Boardworks Ltd 2004
Prime factors
A prime factor is a factor that is a prime number.
For example,
What are the factors of 30?
The factors of 30 are:
1
2
3
5
6
10
15
30
The prime factors of 30 are 2, 3, and 5.
31 of 53
© Boardworks Ltd 2004
Products of prime factors
2 × 3 × 5 = 30
2 × 2 × 2 × 7 = 56
This can be written as 23 × 7 = 56
3 × 3 × 11 = 99
This can be written as 32 × 11 = 99
Every whole number greater than 1 is either a
prime number or can be written as a product of
two or more prime numbers.
32 of 53
© Boardworks Ltd 2004
The prime factor decomposition
When we write a number as a product of prime factors it is
called the prime factor decomposition.
For example,
The prime factor decomposition of 100 is:
100 = 2 × 2 × 5 × 5
= 22 × 52
There are 2 methods of finding the prime factor decomposition
of a number.
33 of 53
© Boardworks Ltd 2004
Factor trees
36
4
2
9
2
3
3
36 = 2 × 2 × 3 × 3
= 22 × 32
34 of 53
© Boardworks Ltd 2004
Factor trees
36
3
12
4
2
3
2
36 = 2 × 2 × 3 × 3
= 22 × 32
35 of 53
© Boardworks Ltd 2004
Factor trees
210
30
6
2
70
5
3
10
2
7
5
210 = 2 × 2 × 3 × 5 × 5 × 7
= 22 × 3 × 52 × 7
36 of 53
© Boardworks Ltd 2004
Factor trees
780
78
2
10
39
3
5
2
13
780 = 2 × 2 × 3 × 5 × 13
= 22 × 3 × 5 × 13
37 of 53
© Boardworks Ltd 2004
Dividing by prime numbers
2
96
2
48
2
24
2
12
2
6
3
3
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3
1
38 of 53
© Boardworks Ltd 2004
Dividing by prime numbers
3
315
3
105
315 = 3 × 3 × 5 × 7
5
35
= 32 × 5 × 7
7
7
1
39 of 53
© Boardworks Ltd 2004
Dividing by prime numbers
2
702
3
351
3
117
3
39
13
13
702 = 2 × 3 × 3 × 3 × 13
= 2 × 33 × 13
1
40 of 53
© Boardworks Ltd 2004
Contents
N3 Multiples, factors and primes
N3.1 Divisibility
N3.2 Multiples and factors
N3.3 Prime numbers
N3.4 Prime factor decomposition
N3.5 HCF and LCM
41 of 53
© Boardworks Ltd 2004
Common multiples
Multiples of 6
12
6
8
16
Multiples of 8
18
54
24
32
40
30
48
42
36
42 of 53
60
56
64
66
102…
72
78
80
88
84
96
104 …
90
© Boardworks Ltd 2004
Multiples on a hundred grid
43 of 53
© Boardworks Ltd 2004
The lowest common multiple
The lowest common multiple (or LCM) of two numbers is the
smallest number that is a multiple of both the numbers.
We can find this by writing down the first few multiples for both
numbers until we find a number that is in both lists.
For example,
Multiples of 20 are :
20,
40,
60,
80,
100,
Multiples of 25 are :
25,
50,
75,
100,
125, . . .
120, . . .
The LCM of 20 and 25 is 100.
44 of 53
© Boardworks Ltd 2004
The lowest common multiple
What is the lowest common multiple (LCM) of 8 and 10?
The first ten multiples of 8 are:
8
16
24
32
40
48
56
64
70
80
72
80
…
100
…
The first ten multiples of 10 are:
10
20
30
40
50
60
90
The lowest common multiple (LCM) of 8 and 10 is 40
The lowest common multiple (LCM) of two numbers is
the smallest number that is a multiple of both.
45 of 53
© Boardworks Ltd 2004
The lowest common multiple
We use the lowest common multiple when adding and
subtracting fractions.
For example,
Add together 4
9
and 5
12
The LCM of 9 and 12 is 36.
4
9
+
×4
×3
5
16
12
×4
46 of 53
=
36
+
15
36
=
31
36
×3
© Boardworks Ltd 2004
Common factor diagram
47 of 53
© Boardworks Ltd 2004
The highest common factor
The highest common factor (or HCF) of two numbers is the
highest number that is a factor of both numbers.
We can find the highest common factor of two numbers by
writing down all their factors and finding the largest factor in
both lists.
For example,
Factors of 36 are : 1,
2,
3,
4,
6,
Factors of 45 are : 1,
3,
5,
9,
15,
9,
12,
18,
36.
45.
The HCF of 36 and 45 is 9.
48 of 53
© Boardworks Ltd 2004
The highest common factor
What is the highest common factor (HCF) of 24 and 30?
The factors of 24 are:
1
2
3
4
6
8
12
24
10
15
30
The factors of 30 are:
1
2
3
5
6
The highest common factor (HCF) of 24 and 30 is 6
The highest common factor (HCF) of two numbers is
the highest whole number which divides into both.
49 of 53
© Boardworks Ltd 2004
The highest common factor
We use the highest common factor when cancelling fractions.
For example,
Cancel the fraction 36
48
The HCF of 36 and 48 is 12, so we need to divide the
numerator and the denominator by 12.
÷12
36
48
=
3
4
÷12
50 of 53
© Boardworks Ltd 2004
Using prime factors to find the HCF and LCM
We can use the prime factor decomposition to find the HCF
and LCM of larger numbers.
For example,
Find the HCF and the LCM of 60 and 125.
2
2
3
5
60
30
15
5
1
60 = 2 × 2 × 3 × 5
51 of 53
2
3
7
7
294
147
49
7
1
294 = 2 × 3 × 7 × 7
© Boardworks Ltd 2004
Using prime factors to find the HCF and LCM
60 = 2 × 2 × 3 × 5
294 = 2 × 3 × 7 × 7
60
294
2
7
2
5
3
7
HCF of 60 and 294 = 2 × 3 = 6
LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940
52 of 53
© Boardworks Ltd 2004
Using prime factors to find the HCF and LCM
53 of 53
© Boardworks Ltd 2004