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15 .1 Prime factors

Prime Factors

If a number has only two factors (1 and itself) it is a prime number .

! Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17,

All numbers can be written as a product of their prime factors .

Example 1: Write in terms of their prime factors:

a) 14 b) 147c) 50d) 81

a) 14

It is best to start working from the smallest prime number, which is 2, so let's check2 147 7

1

As you can see, every factor is a prime number , so the answer must be right.

! The prime factorisation of 14 is 2 ! 7.

b) 147

Can we divide 147 evenly by 2? No, so we should try the next prime number, 3.

3 1477 497 7

1

But 49 is not a prime number, so we need to factor it further

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And that is as far as we need to go.

147 = 3 ! 7 ! 7 = 3 ! 72

c) 50

d) 81

15 .2 ! LCM (Lowest common multiple)What is a multiple?

The multiples of a number are what you get when you multiply it by other numbers (such as ifyou multiply it by 1,2,3,4,5, etc). Just like the multiplication table.

Here are some examples:

The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, etc ...The multiples of 12 are 12, 24, 36, 48, 60, 72, etc...

There are two methods to find the LCM.

Method 1

Example1: Find the LCM of 7 and 9.

Step 1: List all the multiples of 6 and 8.

multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63 , 70,

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multiples of 9: 9, 18, 27, 36, 45, 54, 63 , 72

Step 2: Choose the lowest common multiple.

63.

Example 2: Find the LCM of 4 and 9.

Method 2

Example 3: Find the LCM of 24 and 174.

2 24 1742 12 872 6 873 3 8729 1 29

1 1

LCM = 2 ! 2 ! 2 ! 3 ! 29 = 2 3 ! 3 ! 29 = 696

Example 4: Find the LCM of 50 and 14.

Example 5: Find the LCM of 108 and 801.

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15 .3 HCF (Highest common factor)What is a factor?

Factors are the numbers you multiply together to get another number:

Sometimes we want to find ALL the factors of a number:

The factors of 12 are 1,2,3,4,6 and 12 ...

... because 2 ! 6 = 12, or 4 ! 3 = 12, or 1 ! 12 = 12.

Example 1: Find the HCF of 24 and 36

Step 1: Find the Prime factors of each number

2 242 122 63 3

1

2 362 183 9

3 31

Step 2: Find common pairs of prime factors and multiply them.

The HCF is24 = 2 ! 2 ! 2 ! 3 36 = 2 ! 2 ! 3 ! 3 HCF = 2 ! 2 ! 3 = 24

Example 2: Find the HCF of 12 and 42.

Step 1: Find the Prime factors of each number

2 12 2 422 6 3 213 3 7 7

1 1

The prime factors of 24 are2 ! 2 ! 2 ! 3

The prime factors of 36 are2 ! 2 ! 3 ! 3

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Step 2: Find common pairs of prime factors and multiply them.

The HCF is:12 = 2 ! 2 ! 3 42 = 2 ! 3 ! 7

HCF = 2!

3 = 6

Example 3: Find the HCF of 180 and 56.

Example 4: Find the HCF OF 108 and 801.