1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 1.9 Prime Numbers and Prime Factorizations

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand the terms prime number and composite number. o Understand the Sieve of Eratosthenes. (Optional) o Determine whether a number is prime. o Find the prime factorization of a composite number. o Find all of the factors of a composite number.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Prime Number A prime number is a counting number greater than 1 that has exactly two different factors (or divisors) — itself and 1. Prime Numbers and Composite Numbers

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Composite Number A composite number is a counting number with more than two different factors (or divisors). Prime Numbers and Composite Numbers

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The following numbers are prime. 2 has exactly two different factors, 1 and 2. 7 has exactly two different factors, 1 and 7. 11 has exactly two different factors, 1 and 11. 29 has exactly two different factors, 1 and 29. Example 1: Prime and Composite Numbers 2 7 11 29

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Prime and Composite Numbers (cont.) The following numbers are composite. 1, 2, 3, 4, 6, and 12 are all factors of 12. Thus 12 has more than two different factors. So 1, 3, 11, and 33 are all factors of 33, and 33 has more than two different factors. 12 33

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Determining Whether a Number is Prime To Determine Whether a Number is Prime Divide the number by progressively larger prime numbers (2, 3, 5, 7, 11, and so forth) until: 1.The remainder is 0. This means that the prime number is a factor and the given number is composite; or 2.You find a quotient smaller than the prime divisor. This means that the given number is prime because it has no smaller prime factors.

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Determining Whether a Number is Prime Is 605 a prime number? Solution The ones digit is 5. Therefore, 605 is divisible by 5 and is not prime. The number 605 is a composite number and

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Determining Whether a Number is Prime Is 103 a prime number? Solution Tests for 2, 3, and 5 fail. (The number 103 is not even; 1 + 0 + 3 = 4 and 4 is not divisible by 3; and the last digit is not 0 or 5.)

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Determine Whether a Number is Prime Divide by 7:Divide by 11: The quotient is greater than the divisor. The remainder is not 0. The quotient is less than the divisor, so we are done. The remainder is not 0. The number 103 is prime.

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Determine Whether a Number is Prime Is 221 prime or composite? Solution Tests for 2, 3, and 5 fail. Divide by 7: The quotient is greater than the divisor. The remainder is not 0.

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Determine Whether a Number is Prime (cont.) Divide by 11:Divide by 13: The quotient is greater than the divisor. The remainder is not 0. The remainder is 0. The number 221 is composite. Note: that is 13 and 17 are factors of 221.

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Application of Factors of Counting Numbers One interesting application of factors of counting numbers (very useful in beginning algebra) involves finding two factors whose sum is some specified number. For example, find two factors of 70 such that their product is 70 and their sum is 19. Solution The numbers we are looking for are 5 and 14 because

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. The Fundamental Theorem of Arithmetic Every composite number has exactly one prime factorization. The Prime Factorization of a Composite Number

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Find the Prime Factorization of a Composite Number 1.Factor the composite number into any two factors. 2.Factor each factor that is not prime. 3.Continue this process until all factors are prime. The prime factorization is the product of all of the prime factors. The Prime Factorization of a Composite Number

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Finding the Prime Factorization of a Composite Number Find the prime factorization of 60. Solution 60 = 2  3  2  5 Since the last digit is 0, we know 10 is a factor. 6 and 10 can both be factored so that each factor is a prime number. This is the prime factorization of 60. We can also start with a different set of factors. = 6  10

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Finding the Prime Factorization of a Composite Number Writing the factors in order, we see the prime factorization of 60 is or, using exponents, 60 = 3  2  2  5 3 is prime, but 20 is not. 4 is not prime. All factors are prime. = 3  4  5 = 3  20

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Finding the Prime Factorization of a Composite Number Find the prime factorization of 72. Solution 72 = 2 3  3 2 = 2  4  3  3 = 2  2  2  3  3 72 = 2 3  3 2 = 2  6  6 = 2  2  3  2  3 or = 8  9= 2  36

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 8: Finding the Prime Factorization of a Composite Number Find the prime factorization of 196. Solution 196 = 2  98 = ______ = 2  __  __ = 2  __  __  __ 7 14 7 2 7 using exponents 2 49 2 7 7 22  7222  72

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Factors of a Composite Number The only factors (or divisors) of a composite number are: 1.1 and the number itself; 2.each prime factor; and 3.products formed by all combinations of the prime factors (including repeated factors). Finding All of the Factors of a Composite Number

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Finding the Factors of a Composite Number Find all the factors of 30. Solution Because the factors are a.1 and the number itself, 30. b.Each prime factor: 2, 3, 5. c.Products of all combinations of the prime factors: Therefore, the only factors are 1, 2, 3, 5, 6, 10, 15, and 30.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Finding the Factors of a Composite Number Find all the factors of 140. Solution The prime factorization of 140 is as follows

23. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Finding the Factors of a Composite Number (cont.) The factors of 140 are a. 1 and the number itself: 1 and 140. b. Each prime factor: 2, 5, 7. c. Products of all combinations of the prime factors: The factors are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. There are no other factors (or divisors) of 140.

24. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Determine whether each number is prime or composite. 1.392. 793. 1514. 143 5.Find two factors of 80 such that their product is 80 and their sum is 21. Find the prime factorization of each number. 6.427. 568. 230 9.Using the prime factorization of 63, find all of the factors of 63.

25. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.composite 2.prime 3.prime 4.composite 5.5 and 16 6.2 · 3 · 7 7.2 3 · 7 8.2 · 5 · 23 9.1, 3, 7, 9, 21, 63