Math 6 Unit 1 Lesson 5 Prime and Composite Numbers Numbers in Nature Did you know that the number of petals on a flower is almost always from a special set of numbers? Most often, the number of petals on a flower is one of these numbers: 3, 5, 8, 13, 21, 34 or 55 Here are a few specific examples: Flower Lily 3 Buttercups 5 Chicory Daisy Black-eyed Susan with 13 Petals Math 6 Number of Petals 21 21, 34 or 55 Shasta Daisy with 21 Petals 1-45 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers These are special numbers from a pattern called the Fibonacci sequence. Leonardo da Pisa, “Fibonacci” 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… There are two different types of numbers in this sequence. Some of them have only two factors: 1 and itself. Some have more factors. Leonardo da Pisa, who lived in Italy around the year 1200 was known as Fibonacci. He discovered this special sequence of numbers that often appear in nature. Reflection Which numbers of the Fibonacci sequence have factors that are not limited to 1 and itself? Objectives for this Lesson In this lesson you will explore the following concepts: • Provide an example of a prime number • Provide an example of a composite number • Sort a given set of numbers as prime and composite. 1-46 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers Prime and Composite Numbers You can use what you know about factors of numbers to classify whole numbers into two categories: prime numbers, and composite numbers. Term Definition Example Prime Number A counting number greater than 1 that has exactly two different whole number factors, 1 and itself. 5 = 1 x 5, and 5 has no other whole number factors Composite Number A counting number with more than two different whole number factors greater than 1. 8=1x8 8=2x4 The factors of 8 are 1, 2, 4 and 8. You can use these definitions to classify any whole number. Example 1 Write all of the factors for each number. Next, identify the number as prime or composite. a. 12 b. 17 Factors c. 36 d. 41 Prime or Composite? a. 1, 2, 3, 4, 6, 12 Composite b. 1, 17 Prime c. 1, 2, 3, 4, 6, 9, 12, 18, 36 Composite d. 1, 41 Prime In order for a number to be prime, it can only be written as a product of two unique factors. • ONE of the two factors is 1. • ONE of the two factors is a whole number, but is NOT 0 or 1. Math 6 1-47 Math 6 Unit 1 Factor Not 1 or 0 Lesson 5: Prime and Composite Numbers x Factor must be 1 = Product You can find factors easily enough if you know your multiplication facts. It is a bit harder to think backward when identifying the prime numbers. Use the following Exploration to identify all of the prime numbers up to 100. Let’s Explore Exploration 1: Can You Find the Primes? Materials: Unit 1, Lesson 5, Exploration 1 page in your Workbook, A Pencil Crayon, Pencil For 1 – 5: Use the Number Chart on the Exploration page in your Workbook. 1. Colour the 1 and circle the first four prime numbers: 2, 3, 5, 7 2. Colour the multiples of 2 but do not colour the 2. 3. Colour the multiples of 3 but do not colour the 3. 4. Colour the multiples of 5 but do not colour the 5. 5. Colour the multiples of 7 but do not colour the 7. 6.List the numbers that are circled and the numbers that are not coloured in order. 7.These are the prime numbers up to 100. Check each number to be sure that it has no factors other than 1 and itself. 8. Reflect: Why do you think you did not have to colour multiples of 6? 9. Reflect: Why do you think you did not have to colour multiples of 11? 10.What is the smallest prime number greater than 100? Describe how you found that number. 1-48 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers Let’s Practice • Turn in your Workbook to Unit 1, Lesson 5 and complete 1 to 18. Prime Number Factors For each composite number greater than one you can find a unique set of two or more prime numbers that are factors of the number. These primes will give you a product of the number. This is called the prime factorization of a number. Here is the prime factorization of each composite number up to 20. Number Prime Factorization 4 2x2 6 2x3 8 2x2x2 9 3x3 10 2x5 12 2x2x3 14 2x7 15 3x5 16 2x2x2x2 18 2x3x3 20 2x2x5 To find the prime factorization of a number you can use a factor tree. This is a way of reducing the number down to the primes, by using repeated division. Math 6 1-49 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers Example 2 Find the prime factorization of 68. 1. S tart by dividing by any number. Here you see how to start with dividing by 4: 2. N ow look at the factors you have so far. Can you find any factors of 4? If so, divide again 68 4 68 4 17 2 17 2 Now look at 17. Since 17 is prime and 2 is also prime you are finished. The prime factorization of 68 = 2 x 2 x 17 Example 3 Which number has a prime factorization of 2 x 3 x 3 x 5? Prime Factorization: Multiply the numbers in order from left to right: Now multiply 18 x 5: Finally, the answer: 2x3x3x5 6x3x5 18 x 5 90 • Go online to complete the Concept Capsule: Prime Factorization. 1-50 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers Let’s Practice • Turn in your Workbook to Unit 1, Lesson 5 and complete 19 to 26. What About 1 and 0? The definition of a prime number leaves out 1 and 0. These are also whole numbers. Why do you think they are not included in the prime number set? 0? Prime: 2, 3, 5, 7, 11,… Composite: 4, 6, 8, 9, 10,… 1? The definition tells you that a prime number is divisible by 1 and the number itself. It also says that they must be two different numbers. The number 1 can be expressed as a product in many ways: 1x1 1x1x1 1x1x1x1 …and so on. This means that by the definition of prime numbers, 1 cannot be a prime. The number 0 can also be expressed in many ways: 0x1 0x2 0x3 …and so on. Math 6 1-51 Math 6 Unit 1 Lesson 5: Prime and Composite Numbers By studying the definition you can see that 0 and 1 are not prime numbers. Since they are not considered prime, you may think, “Why are 0 and 1 not composite numbers?” A composite number must be the product of two factors that are • not the number itself. • different from 1 and the number. Look at 1 and 0: 1 only has one factor, not two different factors: 1 x 1 = 1 0 is not a product of two factors that are not 0. 0 is always a product of one or more factors that include 0. You should now be able to explain why 1 and 0 are not prime or composite. Let’s Practice • Turn in your Workbook to Unit 1, Lesson 5 and complete 27 to 28. 1-52