1. http://www.mathsisfun.com/fraction s.html Year 8 Mathematics Multiples, Factors and Primes

2. Learning Intentions – Understand what is meant by the terms Factor and Multiple – Be able to find the factors of a number – Be able to find the prime factors of a number – Be able to write a number as a product of its prime factors – Be able to find the common factors of two or more numbers – Be able to find the highest common factor (HCF) between two or more numbers – Be able to find the multiples of a number – Be able to find the common multiples of two or more numbers – Be able to find the lowest common multiple (LCM) between two or more numbers

3. Factors of a Number A factor of a number is a number that divides evenly into it. For example, the factors of 6 are 1, 2, 3 and 6. Each of the numbers 1, 2, 3 and 6 divide evenly into 6. We can express 6 as a product of its factors  6 = 1 x 6  6 = 2 x 3

4. Prime Factors The prime factors of a number are the factors of a number that are also prime numbers. For example: – The factors of 6 are 1, 2, 3 and 6. – The prime factors of 6 are 2 and 3.

5. Eratosthenes of Cyrene (275-194 B.C) Eratosthenes was a prominent Greek scholar who spent his early life in Athens. He excelled in many areas, notably 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.

6. Eratosthenes Eratosthenes is best remembered today for two notable achievements: The use of his “Sieve” to isolate prime numbers His ingenious method for determining the distance around the Earth with a high degree of accuracy.

7. Strike out the twos, threes, fives and sevens from the Sieve of Eratosthenes 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899100

8. Finding the Prime Factors We can find the prime factors of a number using the following method. We divide the number by the smallest prime number that is a factor We divide the answer again by the smallest prime factor We repeat this process until we end up at the number 1 2120 260 30

9. Product of Prime Factors Once we have divided the number by the prime factors we can write the number as a product of the prime factors For example  6 = 2 x 3  120 = 2 x 2 x 2 x 3 x 5 Since 2 is multiplied by itself in the final sum, we can simplify this to:  120 = 2 3 x 3 x 5 This is referred to as INDEX NOTATION.

10. Common Factors Sometimes numbers have factors in common For example, the numbers 12 and 24 both have 2 as a common factor. However, they also have 3, 4, 8 and 12 as common factors. 12 is the Highest Common Factor since it is the largest factor that both numbers have.

11. Finding the HCF We can use the previous technique for finding the HCF. For example, find the HCF of 16 and 36 We begin by finding the prime factors of both numbers Next, we find the common prime factors Finally we multiply one set of the numbers together The HCF is 2 x 2 = 4 216 28 242 21 236 218 393 31

12. Multiples A multiple of a number is a number that the first number divides into. For example, some of the multiples of 6 are: – 6, 12, 18, 24, 30, …

13. Common Multiples Common multiples are multiples that are common to two or more numbers. For example, common multiples of 6 and 8 are: – 24, 48, 72, 96, …

14. Finding Common Multiples To find the common multiples of two or more numbers we list all the multiples and look for the multiples that are in common. For example: find the common multiples of 4 and 6.  4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, …  6: 6, 12, 18, 24, 30, 36, 42, 48, …

15. LCM The Lowest Common Multiple (LCM) is the smallest multiple of two or more numbers. Fore example, find the LCM of 12 and 30

16. Venn Diagram Venn diagrams were invented by John Venn in 1880. They consist of a rectangular box with circles inside it. Each circle represent the values in one set and the overlap represents values that appear in both sets. For example: – One circle could represent the prime factors of the number 12 – Another circle could represent the prime factors of the number 30 – The overlap would represent the numbers that were prime factor of both 12 and 30.

17. Using a Venn Diagram We can use the Venn diagram to help. We place the common factors in the middle and the others in the appropriate circles. 1230 2 2 3 5

18. Finding the LCM We find the LCM by multiplying all the numbers in the diagram. LCM = 2 x 2 x 3 x 5 = 60