1. UNIT 3 DIVISIBILITY

2. ANDY WARHOL The man in the picture is Andy Warhol, an American artist. Look at the picture of your left. It's one of his paintings. The title is "Green Coca-Cola bottles". There are 196 bottles, and they form a rectangle. There are 7 rows and 28 columns. Can you make the rectangle just with 5 rows? Why?

3. MULTIPLES Imagine that you want to buy eggs. Eggs are normally sold by dozens, and a dozen is 12. So, how many eggs can you buy?

4. MULTIPLES Yes! The answer is: So the multiples of 12 are 12, 24, 36, 48, 60, 72, and so on.

5. MULTIPLES

6. FACTORS A teacher wants to form groups with the same number of people in a classroom with 18 students. How can she do it?

7. FACTORS Let's start dividing in order: 18 ÷ 1 = 18, so she can make 1 group with 18 students. 18 ÷ 2 = 9, so she can make 2 groups with 9 students. 18 ÷ 3 = 6, so he can make 3 groups with 6 students.

8. FACTORS 18 ÷ 4 is not an exact division. She can make 4 groups, but they don't have the same number of students in each group. 18 ÷ 5 is not an exact division. She can make 5 groups, but they don't have the same number of students. She can make 6 groups of 3.

9. FACTORS This table shows all the possibilities: Can you see any curious thing?

10. FACTORS

11. If a number "a" is a multiple of another number "b", then "b" is necessarily a factor of "a". For example, 20 is a multiple of 5, so 5 is a factor of 20.

12. EXERCISE Fill in the gaps with the words "multiple" or "factor": 7 is a __________ of 49 1 is a __________ of 17 25 is a __________ of 100 49 is a __________ of 7 17 is a __________ of 1 13 is a __________ of 13

13. EXERCISE What are the first five multiples of 6? Is 12 a factor of 144? Is 556 a multiple of 4? Find out all the factors of 24. Find out all the factors of 17.

14. PRIME NUMBERS AND COMPOSITE NUMBERS A prime number is a number that only has two factors: the number one and itself. For example, 11 and 23 are prime numbers: The factors of 11 are 1 and 11. The factors of 23 are 1 and 23

15. PRIME NUMBERS AND COMPOSITE NUMBERS A composite number has more than two factors. For example, 9 and 10 are composite numbers: The factors of 9 are 1, 3 and 9. The factors of 10 are 1, 2, 5 and 10.

16. PRIME NUMBERS AND COMPOSITE NUMBERS 1 is NOT a prime number and is NOT a composite number – it just isn't!!! 1 doesn't have exactly to factors, 1 has got just 1 factor!

17. SIEVE OF ERATHOSTENES A smart procedure to find the first prime numbers is the Sieve of Erathostenes. Eratosthenes of Cyrene was a Greek mathematician, geographer, poet, athlete, astronomer, and music theorist. He was the first person to use the word "geography" in Greek and invented a system of latitude and longitude.

18. SIEVE OF ERATHOSTENES

19. You have to cross out using a red pen and circle using a blue pen.

20. SIEVE OF ERATHOSTENES 1. Cross out 1, as 1 isn't a prime number by definition. 2. Circle 2, as 2 is the smallest prime number. 3. Cross out all the multiples of 2 after 2, this is, all the even numbers. 4. Circle 3. 5. Cross out all the multiples of 3. 6. Circle 5. 7. Cross out all the multiples of 5, this is, all numbers ending in 0 or 5. 8. Circle 7. 9. Cross out all the multiples of 7. 10.You have now all the primes numbers up to 100!!!!

21. SIEVE OF ERATHOSTENES Now check your answers!

22. DIVISIBILITY TESTS Divisibility rules help you check if one number is divisible by another, without doing too much calculation! Before we start, you need to know some new words: Even number: multiples of 2: 2, 4, 6, 8, 10, 12 … Odd number: not multiples of 2: 1, 3, 5, 7, 9, 11...

23. DIVISIBILITY TESTS Watch this video and write a short summary using the table.

24. PRIME FACTORIZATION The "prime factorization or decomposition of a number" is finding which prime numbers multiply together to make the original number. To find the prime factorization of a number, start by dividing the number by 2 as many times as you can, then by 3, then by 5, 7, 11... until you get a quotient of 1.

25. PRIME FACTORIZATION For example, what is the prime decomposition of 12? We start with 2, so let's check: 12 ÷ 2 = 6 The remainder is 0, so we have taken the first step! But the quotient isn't 1, so we need to continue. Let's try 2 again: 6 ÷ 2 = 3 Yes, the remainder is 0 again. And 3 is a prime number, so we divide by 3: 3 ÷ 3 = 1

26. PRIME FACTORIZATION The factorization can also be written using exponents: 12 = 2² · 3 · 1

27. EXERCISE Write the prime decomposition of these numbers:

28. GREATEST COMMON DIVISOR Imagine that there are two classrooms in a school. There are 12 students in the first classroom and 16 students in the second classroom. The teacher wants to make groups with the same number of students, but he doesn't want to mix students from different classes. The groups must be as big as possible.

29. GREATEST COMMON DIVISOR Let's calculate the divisors of the numbers: The number of students must be as big as possible, so 4 is the answer.

30. GREATEST COMMON DIVISOR The greatest common factor of two or more numbers is called the Greatest Common Factor (GCF) or the Greatest Common Divisor (GCD). For example, the GCF of 12 and 16 is 4.

31. LOWEST COMMON MULTIPLE My Social Sciences teacher sends homework every 6 days, and my English teacher sends homework every 4 days. I want to know in how many days I will have homework of English and Social Sciences at the same time.

32. LOWEST COMMON MULTIPLE Yes! I will have homework of English and Social Sciences every 12 days!

33. LOWEST COMMON MULTIPLE The smallest common multiple of two or more numbers is called the lowest common multiple (LCM). For example, the LCM of 4 and 6 is 12.

34. EXERCISES Circle the multiples of 2 and underline the multiples of 5: 232835445162 65677085101 136 240302455

35. EXERCISES Circle the multiples of 3 and cross out the numbers that aren't multiples of 3: 182431233242 464851555761 6388

36. EXERCISES Use the divisibility rules to complete (with true or false) the following table:

37. EXERCISES Anwser these questions: Is 729,064 a prime number? Why? Is 3 a factor of 2001? Which of these numbers are not prime? Show a divisor for evidence. 221 35 784 20 97

38. EXERCISES What is the Highest Common Factor (HCF) of these numbers? 3 and 66 and 920 and 30 8 and 2030 and 5015 and 18 What is the Lowest Common Multiple of these numbers? 2 and 42 and 68 and 12 3 and 95 and 64 and 6