Factors, Multiples and Primes

Prime Numbers

• A Prime Number is a number that can only be divided by two numbers:– The number 1– The number itself

• 3 is a prime number because it can be divided by 1 and 3

• 4 is not a prime number because it can be divided by three numbers: 1, 2 and 4

• 1 is not a prime number because it can only be divided by one number: 1

Eratosthenes of Cyrene (275-194 B.C)

• Eratosthenes was a prominent Greek scholar who spent his early life in Athens.

• He was a friend of Archimedes and excelled in mathematics, astronomy, geography, history, poetry and athletics.

• He was a universal genius who was known to his friends as Beta, because he was regarded as the second best in almost all the fields he studied.

• He eventually went to Alexandria (Egypt) where he became the 3rd librarian at the great university as well as private tutor to the son of Ptolemy III.

• It was Eratosthenes who suggested a calendar (later adopted by the Romans) of 365 days with an additional day every 4th year. During old age he went blind and ended his life by drinking poison.

Sieve of Eratosthenes1 11 21 31 41 51 61 71 81 91

2 12 22 32 42 52 62 72 82 92

3 13 23 33 43 53 63 73 83 93

4 14 24 34 44 54 64 74 84 94

5 15 25 35 45 55 65 75 85 95

6 16 26 36 46 56 66 76 86 96

7 17 27 37 47 57 67 77 87 97

8 18 28 38 48 58 68 78 88 98

9 19 29 39 49 59 69 79 89 99

10 20 30 40 50 60 70 80 90 100

To find all the Prime Numbers from 1 to 100 remove:• the number 1• any number divisible by 2• any number divisible by 3• any number divisible by 4• any number divisible by 5• etc• The numbers you have left

are the prime numbers.

• Question:• How far did you have to

go?

Factors

• A Factor of a number is a number that divides exactly into it.

• A factor of 20 would be 4.• All the factors of 20 are: 1, 2, 4, 5, 10, 20

Prime Factors

• A Prime Factor is a number that is a factor and is also a prime number

• For Example:– the factors of 20 are 1, 2, 4, 5, 10 and 20– the prime factors of 20 are 2, and 5

• We can write 20 as a product of its prime factors:– 20 = 2 x 2 x 5

Finding the Prime Factors

• To find the prime factors of a number we divide the number by prime numbers until we reach the number 1.

2 20

2 10

5 5

1 ← STOP

Common Factors

• If two or more numbers can have the same factors – these are called common factors.

• Example:– Find the common factors of 12 and 18.

Highest Common Factor

• The Highest Common Factor is the largest number that is a factor of two or more numbers.

• Example: – What is the HCF of 12 and 18?

Venn Diagrams

• We can use a Venn Diagram to help find the HCF.

• Example: Find the HCF of 12 and 18.• We first find the Prime Factors of 12 and 18• 12 = 2 x 2 x 3• 18 = 2 x 3 x 3

Venn Diagram

• We put the common factors in the middle and the others in either the 12 or 18 circle.

• The HCF is found by multiplying the numbers in the middle.

12 18

Highest Common Factor

• Remember when finding the HCF we multiply the numbers where the circles overlap.

• HCF = 3 x 5 = 15

35

3 5

45 75

Highest Common Factor

• Find the HCF of 126, 140 and 150

• HCF = 2

126 140

150

2

3 2

5

53

7

Multiples

• A multiple of a number is a product of the number and a whole number.

• The multiples of 3 are:– 3, 6, 9, 12, 15, …– We find these by multiply 3 by: 1, 2, 3, 4, 5, …

Lowest Common Multiple

• The Lowest Common Multiple is the smallest number that two or more numbers will divide into.

• For example: the lowest common multiple of 4 and 6 is 12.

Finding the LCM

• We can find the lowest common multiple by writing all the multiples in a list and looking for the smallest number that is in both lists

• For example, find the lowest common multiple of 4 and 6– The multiples of 4 are 4, 8, 12, 16, 20 – The multiples of 6 are 6, 12, 18

• 12 is the LCM since it is the smallest number in both lists

Lowest Common Multiple

• To find the LCM for larger numbers we can use a Venn diagram

• We put the common factors in the middle and the others in either the 12 or 18 circle.

• The LCM is found by multiplying all the numbers in the circles.

12 18

Lowest Common Multiple

• Remember when finding the LCM we multiply all the numbers in the circles.

• LCM = 3 x 3 x 5 x 5 =225

35

3 5

45 75

Number Patterns

• Number patterns are sequences of numbers that are formed by adding, subtracting or multiplying by a fixed amount.

• For example:– 3, 5, 7, 9, …– 50, 40, 30, 20, …– 2, 4, 8, 16, …

Square Numbers

• A square number is a number that can be represented by a square pattern of dots.

• For example:– 16 is a square number

Triangular Numbers

• A Triangular number is a is a number that can be represented by a triangular pattern of dots.

• For example:– 6 is a triangular number