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**Prime Factor Decomposition**

What does this mean? Prime Factor Decomposition Prime Numbers A number that divides exactly into another number (no remainder), Break down into smaller parts So it is the process of breaking numbers down into the prime factors that make it up. This allows us to write a number as a Product of its Prime Factors A ‘product’ is the answer to a multiplication so 10 is the product of 2 and 5 because 2 x 5 = 10

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**Prime Factor Decomposition**

Write the number 15 as a product of its prime factors: What are the factors of 15? 1 x 15 = 15 3 x 5 = 15 1 and 15 are not prime numbers 3 and 5 are prime numbers So writing 15 as a product of its Prime factors means writing 3 x 5 = 15 However: 8 can be written as the product of 2 and 4 2 x 4 = 8 but 4 is not a prime number so 2 x 4 is not writing 8 as a product of its prime factors. 4 can be written as the product of 2 and 2 2 x 2 = 4 Therefore 8 can be written as 2 x 2 x 2 = 8 This is now writing 8 as a product of its prime factors

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**Prime Factor Decomposition**

How do we find all the prime factors of a number: Start with the smallest prime factor Find the prime factors of 24 What is the smallest prime factor of 24? 2 24 What do we multiply by 2 to get 24? 12 2 12 What is the smallest prime factor of 12? 2 What do we multiply by 2 to get 12? 6 2 6 What is the smallest prime factor of 6? 2 What do we multiply by 2 to get 6? 3 2 3 We have arrived at another prime number therefore the factor tree is finished. We can now write 24 as a product of its prime factors x x x = 24 Or more simply in index notation 23 x 3 = 24

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**Prime Factor Decomposition**

Write 28 as a product of its prime factors 28 2 14 Notice how the factors Are written in order of size 2 7 2 x 2 x 7 = 28 Or more simply in index notation 22 x 7 = 28 Write 27 as a product of its prime factors 27 3 x 3 x 3 = 27 3 9 Or more simply in index notation 33 = 27 3 3

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**Prime Factor Decomposition**

Now answer these: Write the following as a product of its prime factors Leave your answer in index form: 20 32 63 96 144 720 450 624 8820 If 1080 = 2x x 3y x 5z what are the values of x,y and z? 22 x 5 = 20 25 = 32 32 x 7 = 63 25 x 3 = 96 24 x 32 = 144 24 x 32 x 5= 720 2 x 32 x 52= 450 25 x 39 = 624 22 x 32 x 5 x 72= 8820 x = 3, y = 3, z = 1

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**Finding the LCM and HCF using Prime Factor Decomposition**

We found that: x 5 = 20 and 25 = 32 The highest common factor is found by using the prime factors that are common in both numbers 2 2 x 2 x 5 = x 2 x 2 x 2 x 2 = 32 2 2 2 x = 4 So the HCF of 20 and 32 is 4 The Lowest Common Multiple is found by using HCF, then using all the numbers that are different 2 x 2 x 5 = x 2 x 2 x 2 x 2 = 32 2 2 5 2 2 2 2 2 HCF x x x x x = 160 So the LCM of 20 and 32 is 160

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**Prime Factor Decomposition**

Now answer these by using Prime Factor Decomposition: Find the HCF and LCM of these pairs of numbers 24 and 36 27 and 36 32 and 48 56 and 152 2 x 2 x 2 x 3 = 24 2 x 2 x 3 x 3 = 36 HCF = 2 x 2 x 3 = 12 LCM = 12 x 2 x 3 = 72 3 x 3 x = 27 2 x 2 x 3 x 3 = 36 HCF = 3 x = 9 LCM = 9 x 2 x 2 x 3 = 108 2 x 2 x 2 x 2 x 2 = 32 2 x 2 x 2 x 2 x 3 = 48 HCF = 2 x 2 x 2 x 2 = 16 LCM = 16 x 2 x = 96 2 x 2 x 2 x 7 = 56 2 x 2 x = 152 HCF = 2 x = 4 LCM = 4 x 7 x = 1092

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Number Theory: Prime and Composite Numbers

Number Theory: Prime and Composite Numbers

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