3. multiples, factors and primes

© Boardworks Ltd 2004 1 of 53

1


Contents

Multiples, factors and primes

A

A

A

A

HCF and LCM

Multiples and factors

Prime numbers

Prime factor decomposition


Contents

Multiples, factors and primes

A

A

A

A

HCF and LCM

Multiples and factors

Prime numbers

Prime factor decomposition


MULTIPLESand

FACTORS


Multiples

A multiple of a number is found by multiplying the number by any whole number.

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.


Multiples patterns on a hundred square


Rectangular arrangements


Finding factors

A factor is a whole number that divides exactly into a given number.

A factor 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.


Factor finder


Circle and square puzzle


PRIME NUMBERS


Sieve of Eratosthenes


Prime numbers

If a whole number has two, and only two, factors it is called a prime number.

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.


The first 10 prime numbers are:

Prime numbers

2 3 5 7 11 13 17 19 23 29


Testing for prime numbers

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.


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 can check whether or not a number is prime by testing for divisibility by successive prime numbers.

Why don’t we need to check for divisibility by 11?We don’t need to check for divisibility by 11 because 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.




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.

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.


An amazing fact


PRIME FACTORS


A prime factor is a factor that is also a prime number.A prime factor is a factor that is also 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.

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.

Every whole number greater than 1 is either a prime number or can be written as a product of

two or more prime numbers.

Products of prime factors


The prime factor decomposition

When we write a number as a product of prime factors it is called 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:

There are 2 methods of finding the prime factor decomposition of a number.

100 = 2 × 2 × 5 × 5

= 22 × 52


36

4 9

2 2 3 3

36 = 2 × 2 × 3 × 3

= 22 × 32

Factor trees


36

3 12

4 3

2 2

36 = 2 × 2 × 3 × 3

= 22 × 32

Factor trees


2100

30 70

6 5

2 3

10 7

2 5

2100 = 2 × 2 × 3 × 5 × 5 × 7

= 22 × 3 × 52 × 7

Factor trees


780

78 10

392

3 13

25

780 = 2 × 2 × 3 × 5 × 13

= 22 × 3 × 5 × 13

Factor trees


962

482

242

122

62

33

1

2

2

2

2

2

3

96 = 2 × 2 × 2 × 2 × 2 × 3

= 25 × 3

Dividing by prime numbers


3153

1053

355

77

1

3

3

5

7

315 = 3 × 3 × 5 × 7

= 32 × 5 × 7



7022

3513

1173

393

1313

1

2

3

3

3

13

702 = 2 × 3 × 3 × 3 × 13

= 2 × 33 × 13



COMMON MULTIPLES

HCF LCM


Multiples of 6

Multiples of 8

24 48 72 968 16 32 40 56 64 80 88 104 …

612

18

24

3036

42

48

5460

66

72

7884

90

96

102…

Common multiples


Multiples on a hundred grid


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.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, 120, . . .

Multiples of 25 are : 25, 50, 75, 100, 125, . . .

The LCM of 20 and 25 is 100.


The first ten multiples of 8 are:

8 16 24 32 40 48 56 64 72 80 …

The first ten multiples of 10 are:

10 20 30 40 50 60 70 80 90 100 …

The lowest common multiple (LCM) of 8 and 10 is 40.

What is the lowest common multiple (LCM) of 8 and 10?



We use the lowest common multiple when adding and subtracting fractions.

For example,

Add together 4

9

5

12and .

The LCM of 9 and 12 is 36.

+4

9

5

12=

36

× 4

× 4

16+

36

× 3

× 3

15=

31

36



Common factor diagram


The highest common factor

The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.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, 9, 12, 18, 36.

Factors of 45 are :

1, 3, 5, 9, 15, 45.

The HCF of 36 and 45 is 9.


What is the highest common factor (HCF) of 24 and 30?


1 2 3 4 6 8 12 24


1 2 3 5 6 10 15 30

The highest common factor (HCF) of 24 and 30 is 6.



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.

36

48=

÷12

3

÷12

4



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 294.

602302153551

60 = 2 × 2 × 3 × 5

29421473497771

294 = 2 × 3 × 7 × 7

Using prime factors to find the HCF and LCM


60 294

60 = 2 × 2 × 3 × 5

294 = 2 × 3 × 7 × 7

22

35

7

7

HCF of 60 and 294 = 2 × 3 = 6

LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940





44

Do you find this slides were useful?

One second of your life , can bring a smile in a girl life

If Yes ,Join Dreams School “Campaign for Female Education”

Help us in bringing a change in a girl life, because “When someone takes away your pens you realize quite how important education is”.

Just Click on any advertisement on the page, your one click can make her smile. Eliminate Inequality “Not Women”

One second of your life , can bring a smile in her life!!

Do you find these slides were useful?

If Yes ,Join Dreams School “Campaign for Female Education”

Help us in bringing a change in a girl life, because “When someone takes away your pens you realize quite how important education is”.

Just Click on any advertisement on the page, your one click can make her smile. We our doing our part & u ?

Eliminate Inequality “Not Women”