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Prime and Composite Numbers
The positive integers (excluding 1) can be divided into two sets.
1 2 3 4 5 6 7 8 9 10
primes
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
composites
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50 All composite numbers can
be expressed as a product
51 52 53 54 55 56 57 58 59 60 of primes. For example:
61 62 63 64 65 66 67 68 69 70 55 = 5 x 11
71 72 73 74 75 76 77 78 79 80
70 = 2x5x7
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100 90 = 2 x 32 x 5
You may be familiar with these from the Sieve of Eratosthenes.…read more

Slide 3

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M1
The Fundamental Theorem of Arithmetic
Every positive integer (excluding 1) can be expressed as a product of primes and
this factorisation is unique (Euclid IX.14).
To write a number as a product of primes first write it as a product of
any two convenient factors.
Example 1: Write 180 as a product of primes.
None of these factors are prime so re-write them
180 = 10 x 18 as a product of smaller factors and keep
repeating if necessary until all factors are prime.
180 = 2 x 5 x 3 x 6
All factors are now prime so re-write
180 = 2 x 5 x 3 x 2 x 3 in ascending order as powers.
180 = 22 x 32 x 5 When written in this way we say that
it is expressed in canonical form.…read more

Slide 4

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The Fundamental Theorem of Arithmetic
Every positive integer (excluding 1) can be expressed as a product of primes and
this factorisation is unique (Euclid IX.14).
To write a number as a product of primes first write it as a product of
any two convenient factors.
Example 2: Write 200 as a product of primes.
None of these factors are prime so re-write them
200 = 10 x 20 as a product of smaller factors and keep
repeating if necessary until all factors are prime.
200 = 2 x 5 x 4 x 5
2 All factors are now prime so re-write
200 = 2 x 5 x 2 x 5 in canonical form.
200 = 23 x 52…read more

Slide 5

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The Fundamental Theorem of Arithmetic
Every positive integer (excluding 1) can be expressed as a product of primes and
this factorisation is unique (Euclid IX.14).
To write a number as a product of primes first write it as a product of
any two convenient factors.
Example 3: Write 84 as a product of primes.
84 = 7 x 12
84 = 7 x 3 x 4
84 = 7 x 3 x 22
84 = 22 x 3 x 7…read more

Slide 6

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The Fundamental Theorem of Arithmetic
Every positive integer (excluding 1) can be expressed as a product of primes and
this factorisation is unique (Euclid IX.14).
To write a number as a product of primes first write it as a product of
any two convenient factors.
Example 4: Write 144 as a product of primes.
144 = 12 x 12
144 = 3 x 4 x 3 x 4
144 = 3 x 22 x 3 x 22
144 = 24 x 32…read more

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Comments

daviesg

Prime factors (by factor trees and also by division) and LCM/HCF, good ppt

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