Factoring Whole Numbers

2.2 Factoring Whole Numbers 2.2 OBJECTIVES 1. Find the factors of a whole number 2. Find the prime factorization for any number 3. Find the greatest common factor (GCF) of two numbers 4. Find the GCF for a group of numbers To factor a number means to write the number as a product of its whole-number factors. Example 1 Factoring a Composite Number Factor the number 10. 10 2 5 The order in which you write the factors does not matter, so 10 5 2 would also be correct. Of course, 10 10 1 is also a correct statement. However, in this section we are interested in factors other than 1 and the given number. Factor the number 21. 21 3 7 CHECK YOURSELF 1 Factor 35. In writing composite numbers as a product of factors, there will be a number of different possible factorizations. Example 2 Factoring a Composite Number Find three ways to factor 72. NOTE There have to be at least two different factorizations, because a composite number has factors other than 1 and itself. 72 8 9 (1) 6 12 (2) 3 24 (3) CHECK YOURSELF 2 Find three ways to factor 42. We now want to write composite numbers as a product of their prime factors. Look again at the first factored line of Example 2. The process of factoring can be continued until all the factors are prime numbers. 137

138 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS Example 3 Factoring a Composite Number NOTE This is often called a factor tree. NOTE Finding the prime factorization of a number will be important in our later work in adding fractions. 72 8 9 2 4 3 3 2 2 2 3 3 4 is still not prime, and so we continue by factoring 4. 72 is now written as a product of prime factors. When we write 72 as 2 2 2 3 3, no further factorization is possible. This is called the prime factorization of 72. Now, what if we start with the second factored line from the same example, 72 6 12? Example 3 (Continued) Factoring a Composite Number 72 6 12 Continue to factor 6 and 12. 2 3 3 4 2 3 3 2 2 Continue again to factor 4. Other choices for the factors of 12 are possible. As we shall see, the end result will be the same. No matter which pair of factors you start with, you will find the same prime factorization. In this case, there are three factors of 2 and two factors of 3. Because multiplication is commutative, the order in which we write the factors does not matter. CHECK YOURSELF 3 We could also write 72 2 36 Continue the factorization. Rules and Properties: The Fundamental Theorem of Arithmetic There is exactly one prime factorization for any composite number. NOTE The prime factorization is then the product of all the prime divisors and the final quotient. The method of the previous example will always work. However, an easier method for factoring composite numbers exists. This method is particularly useful when numbers get large, in which case factoring with a number tree becomes unwieldy. Rules and Properties: Factoring by Division To find the prime factorization of a number, divide the number by a series of primes until the final quotient is a prime number.

FACTORING WHOLE NUMBERS SECTION 2.2 139 Example 4 Finding Prime Factors To write 60 as a product of prime factors, divide 2 into 60 for a quotient of 30. Continue to divide by 2 again for the quotient 15. Because 2 won t divide evenly into 15, we try 3. Because the quotient 5 is prime, we are done. NOTE Do you see how the divisibility tests are used here? 60 is divisible by 2, 30 is divisible by 2, and 15 is divisible by 3. 30 15 5 Prime 2 60 2 30 3 15 Our factors are the prime divisors and the final quotient. We have 60 2 2 3 5 CHECK YOURSELF 4 Complete the process to find the prime factorization of 90. 45? 2 90? 45 Remember to continue until the final quotient is prime. Writing composite numbers in their completely factored form can be simplified if we use a format called continued division. Example 5 Finding Prime Factors Using Continued Division NOTE In each short division, we write the quotient below rather than above the dividend. This is just a convenience for the next division. Use the continued-division method to divide 60 by a series of prime numbers. 2B60 Primes 2B30 3B15 5 Stop when the final quotient is prime. To write the factorization of 60, we list each divisor used and the final prime quotient. In our example, we have 60 2 2 3 5 CHECK YOURSELF 5 NOTE Again the factors of 20, other than 20 itself, are less than 20. Find the prime factorization of 234. We know that a factor or a divisor of a whole number divides that number exactly. The factors or divisors of 20 are 1, 2, 4, 5, 10, 20 Each of these numbers divides 20 exactly, that is, with no remainder. Our work in this section involves common factors or divisors. A common factor or divisor for two numbers is any factor that divides both the numbers exactly.

140 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS Example 6 Finding Common Factors Look at the numbers 20 and 30. Is there a common factor for the two numbers? First, we list the factors. Then we circle the ones that appear in both lists. Factors 20: 1, 2, 4, 5, 10, 20 30: 1, 2, 3, 5, 6, 10, 15, 30 We see that 1, 2, 5, and 10 are common factors of 20 and 30. Each of these numbers divides both 20 and 30 exactly. Our later work with fractions will require that we find the greatest common factor (GCF) of a group of numbers. Definitions: Greatest Common Factor The greatest common factor (GCF) of a group of numbers is the largest number that will divide each of the given numbers exactly. Example 6 (Continued) Finding Common Factors In the first part of Example 6, the common factors of the numbers 20 and 30 were listed as 1, 2, 5, 10 Common factors of 20 and 30 The greatest common factor of the two numbers is then 10, because 10 is the largest of the four common factors. CHECK YOURSELF 6 List the factors of 30 and 36, and then find the greatest common factor. The method of Example 6 will also work in finding the greatest common factor of a group of more than two numbers. NOTE Looking at the three lists, we see that 1, 2, 3, and 6 are common factors. Example 7 Finding the Greatest Common Factor (GCF) by Listing Factors Find the GCF of 24, 30, and 36. We list the factors of each of the three numbers. 24: 1, 2, 3, 4, 6, 8, 12, 24 30: 1, 2, 3, 5, 6, 10, 15, 30 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 6 is the greatest common factor of 24, 30, and 36.

FACTORING WHOLE NUMBERS SECTION 2.2 141 CHECK YOURSELF 7 Find the greatest common factor (GCF) of 16, 24, and 32. The process shown in Example 7 is very time-consuming when larger numbers are involved. A better approach to the problem of finding the GCF of a group of numbers uses the prime factorization of each number. Let s outline the process. Step by Step: Finding the Greatest Common Factor NOTE If there are no common prime factors, the GCF is 1. Step 1 Step 2 Step 3 Write the prime factorization for each of the numbers in the group. Locate the prime factors that are common to all the numbers. The greatest common factor (GCF) will be the product of all the common prime factors. Example 8 Finding the Greatest Common Factor (GCF) Find the GCF of 20 and 30. Step 1 Write the prime factorization of 20 and 30. 20 2 2 5 30 2 3 5 Step 2 Find the prime factors common to each number. 20 2 2 5 30 2 3 5 2 and 5 are the common prime factors. Step 3 Form the product of the common prime factors. 2 5 10 10 is the greatest common factor. CHECK YOURSELF 8 Find the GCF of 30 and 36. To find the greatest common factor of a group of more than two numbers, we use the same process. Example 9 Finding the Greatest Common Factor (GCF) Find the GCF of 24, 30, and 36. 24 2 2 2 3 30 2 3 5 36 2 2 3 3

142 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS 2 and 3 are the prime factors common to all three numbers. 2 3 6 is the GCF. CHECK YOURSELF 9 Find the GCF of 15, 30, and 45. NOTE If two numbers, such as 15 and 28, have no common factor other than 1, they are called relatively prime. Example 10 Finding the Greatest Common Factor (GCF) Find the greatest common factor of 15 and 28. 15 3 5 28 2 2 7 There are no common prime factors listed. But remember that 1 is a factor of every whole number. The greatest common factor of 15 and 28 is 1. CHECK YOURSELF 10 Find the greatest common factor of 30 and 49. CHECK YOURSELF ANSWERS 1. 5 7 2. 2 21, 3 14, 6 7 3. 2 2 2 3 3 4. 45 15 5 5. 2 3 3 13 2 90 3 45 3 15 90 2 3 3 5 6. 30: 1, 2, 3, 5, 6, 10, 15, 30 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 6 is the greatest common factor. 7. 16: 1, 2, 4, 8, 16 24: 1, 2, 3, 4, 6, 8, 12, 24 32: 1, 2, 4, 8, 16, 32 The GCF is 8. 8. 30 2 3 5 36 2 2 3 3 The GCF is 2 3 6. 9. 15 10. GCF is 1; 30 and 49 are relatively prime

Name 2.2 Exercises Section Date Find the prime factorization of each number. 1. 18 2. 22 ANSWERS 1. 3. 30 4. 35 2. 3. 4. 5. 51 6. 42 5. 6. 7. 7. 63 8. 94 8. 9. 10. 9. 70 10. 90 11. 12. 11. 66 12. 100 13. 14. 15. 13. 130 14. 88 16. 17. 18. 15. 315 16. 400 19. 20. 17. 225 18. 132 19. 189 20. 330 143

ANSWERS 21. 22. 23. In later mathematics courses, you often will want to find factors of a number with a given sum or difference. The following exercises use this technique. 21. Find two factors of 24 with a sum of 10. 24. 25. 26. 22. Find two factors of 15 with a difference of 2. 27. 28. 29. 23. Find two factors of 30 with a difference of 1. 30. 31. 24. Find two factors of 28 with a sum of 11. 32. 33. 34. 35. Find the greatest common factor (GCF) for each of the following groups of numbers. 25. 4 and 6 26. 6 and 9 36. 27. 10 and 15 28. 12 and 14 29. 21 and 24 30. 22 and 33 31. 20 and 21 32. 28 and 42 33. 18 and 24 34. 35 and 36 35. 18 and 54 36. 12 and 48 144

ANSWERS 37. 12, 36, and 60 38. 15, 45, and 90 37. 38. 39. 39. 105, 140, and 175 40. 17, 19, and 31 40. 41. 41. 25, 75, and 150 42. 36, 72, and 144 42. 43. 43. A natural number is said to be perfect if it is equal to the sum of its counting number divisors, except itself. (a) Show that 28 is a perfect number. (b) Identify another perfect number less than 28. 44. 45. 46. 44. Find the smallest natural number that is divisible by all of the following: 2, 3, 4, 6, 8, 9. 45. Tom and Dick both work the night shift at the steel mill. Tom has every sixth night off, and Dick has every eighth night off. If they both have August 1 off, when will they both be off together again? 46. Mercury, Venus, and Earth revolve around the sun once every 3, 7, and 12 months, respectively. If the three planets are now in the same straight line, what is the smallest number of months that must pass before they line up again? 145

Answers 1. 2 3 3 3. 2 3 5 5. 3 17 7. 3 3 7 9. 2 5 7 11. 2 3 11 13. 2B130 15. 3 3 5 7 5B65 13 130 2 5 13 17. 3 3 5 5 19. 3B189 21. 4, 6 23. 5, 6 25. 2 27. 5 3B63 3B21 7 189 3 3 3 7 29. 3 31. 1 33. 6 35. 18 37. 12 39. 35 41. 25 43. 45. August 25 146