1. OBJECTIVES 3.2 Factorizations Slide 1Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. aFind the factors of a number. bGiven a number from 1 to 100, tell whether it is prime, composite, or neither. cFind the prime factorization of a composite number.

2. 3.2 Factorizations FACTORS AND FACTORIZATIONS Slide 2Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

3. 3.2 Factorizations a Find the factors of a number. Slide 3Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

4. EXAMPLE 3.2 Factorizations a Find the factors of a number. 1 Slide 4Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find all the factors of 18. Beginning at 1, we check all positive integers to see if they are factors of 18. If they are, we write the factorization. We stop when we have already included the next integer in a factorization.

5. EXAMPLE 3.2 Factorizations a Find the factors of a number. 1 Slide 5Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

6. EXAMPLE 3.2 Factorizations a Find the factors of a number. 1 Slide 6Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. We need check no additional numbers, because any integer greater than 6 must be paired with a factor less than 6. We now write the factors of 18 beginning with 1, going down the list of factorizations writing the first factor, then up the list of factorizations writing the second factor: 1, 2, 3, 6, 9, 18.

7. 3.2 Factorizations PRIME AND COMPOSITE NUMBERS Slide 7Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. A natural number that has exactly two different factors, only itself and 1, is called a prime number. The number 1 is not prime. A natural number, other than 1, that is not prime is composite.

8. EXAMPLE 3.2 Factorizations b Given a number from 1 to 100, tell whether it is prime, composite, or neither. 2 Slide 8Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Determine which of the numbers 1, 2, 7, 8, 9, 11, 18, 27, 39, 43, 56, 59, and 77 are prime, which are composite, and which are neither.

9. EXAMPLE 3.2 Factorizations b Given a number from 1 to 100, tell whether it is prime, composite, or neither. 2 Slide 9Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. The number 1 is not prime. It does not have two different factors. The number 2 is prime. It has only the factors 2 and 1. The numbers 7, 11, 43, and 59 are prime. Each has only two factors, itself and 1. The number 8 is not prime. It has the factors 1, 2, 4, and 8 and is composite. The numbers 9, 18, 27, 39, 56, and 77 are composite. Each has more than two factors.

10. EXAMPLE 3.2 Factorizations b Given a number from 1 to 100, tell whether it is prime, composite, or neither. 2 Slide 10Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Thus, we have Prime: 2, 7, 11, 43, 59; Composite: 8, 9, 18, 27, 39, 56, 77; Neither: 1.

11. 3.2 Factorizations b Given a number from 1 to 100, tell whether it is prime, composite, or neither. Slide 11Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Because 0 is not a natural number, it is neither prime nor composite. The number 1 is not prime because it does not have two different factors. The number 2 is the smallest prime and the only even prime, since 2 is a factor of all even numbers.

12. 3.2 Factorizations b Given a number from 1 to 100, tell whether it is prime, composite, or neither. Slide 12Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. To determine whether an odd number is prime, check divisibility by prime numbers beginning with 3 and 5. If you reach a point where the quotient is less than the divisor and none of the primes up to that point are factors, the number you are checking is prime.

13. 3.2 Factorizations A TABLE OF PRIMES FROM 2 TO 157 Slide 13Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157

14. 3.2 Factorizations c Find the prime factorization of a composite number. Slide 14Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. When we factor a composite number into a product of primes, we find the prime factorization of the number. We can do this by making a series of successive divisions or by using a factor tree.

15. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 3 Slide 15Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the prime factorization of 39.

16. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 3 Slide 16Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.

17. Title 3.2 Factorizations Slide 17Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Every composite number has just one (unique) prime factorization.

18. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 5 Slide 18Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the prime factorization of 187. We check for divisibility by 2, 3, and 5, and find that 187 is not divisible by any of these numbers. The next prime number, 7, does not divide 187 evenly. However, when we divide by 11, the remainder is 0, so 11 is a factor of 187.

19. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 5 Slide 19Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Because 17 is prime, we can factor no further. The complete factorization is

20. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 6 Slide 20Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Find the prime factorization of 72.

21. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 6 Slide 21Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc. Another way to find the prime factorization of 72 is to use a factor tree as follows. Begin by determining any factorization you can, and then continue factoring. Each of the following trees gives the same prime factorization.

22. EXAMPLE 3.2 Factorizations c Find the prime factorization of a composite number. 6 Slide 22Copyright 2012, 2008, 2004, 2000 Pearson Education, Inc.