Section 2.1 Factors and Multiples

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1 Section 2.1 Factors and Multiples When you want to prepare a salad, you select certain ingredients (lettuce, tomatoes, broccoli, celery, olives, etc.) to give the salad a specific taste. You can think of the salad as the final result, or product, of combining the ingredients. In general, the order in which you combine these ingredients does not affect the taste and nutritional value of the salad. We saw in the previous chapter that whenever we multiply two or more whole numbers, we get another whole number as the final result. For example, = We can think of the numbers 17, 3 and 11 as the ingredients that multiplied together make up the salad, or final result of 561. In mathematics, the ingredients are called factors and we call the salad a product or multiple. Thus, we say that the whole numbers 17, 3 and 11 are the factors that multiplied together give the product 561. The number 561 is a multiple of 17, 3, and 11, which means that 561 is divisible by 17, 3, and 11 because $561 can be evenly divided among 17 people, among 3 people or among 11 people without resorting to cents. Example In the following equation, name the factors and the multiple. =, Answer: The numbers 13, 47, 5 and 241 are factors of the product 736,255, whereas the product 736,255 is a multiple of 13, 47, 5 and 241. This means that 736,255 is divisible by 13, 47, 5 and 241. You can think of 736,255 as being the salad and the numbers 13, 47, 5 and 241 the ingredients that multiplied together give 736,255. Copyright 2014 Luis Soto-Ortiz 104

2 Example In the following equation, name the factors and the product. =, Answer: The numbers 6, 4, 23 and 11 are factors of 6,072, whereas the number 6,072 is a multiple of 6, 4, 23 and 11. Note: The order in which you multiply the factors doesn t affect the final product because multiplication is commutative. For example: = = = 80 Whenever we can write a whole number as the product of a set of whole numbers (its factors), we say that the product is divisible by those numbers. That is, any whole number is divisible by its factors. Example Since 7 8 = 56, we know that 56 is divisible by 7 and that 56 is divisible by 8. Therefore, 56 is a multiple of 7 and 8, while 7 and 8 are factors of 56. The complete list of factors of 56 is 1, 2, 4, 14, 28 and 56. factors multiple factors multiple Since 7x8=56, then 56 is the product and 7 and 8 are factors of 56. Notice that whenever we divide a product by one of its factors, the quotient is also a factor and the remainder is always zero. Copyright 2014 Luis Soto-Ortiz 105

3 Example Since 2 28 = 56, we know that 56 is divisible by 2 and that 56 is divisible by 28. Therefore, 56 is a multiple of 2 and 28, while 2 and 28 are factors of 56. factors multiple factors multiple Since 2x28=56, then 56 is the product and 2 and 28 its factors. Notice that whenever we divide a product by one of its factors, the quotient is also a factor and the remainder is always zero. Note: 1 is a factor of any whole number because 1 =. This also means that any whole number is divisible by 1. For example, 1 is a factor of 17 since 1 17 = 17. Therefore, 17 is divisible by 1. 1 is a factor of 5,788 since 1 5,788 = 5,788. Therefore, 5,788 is divisible by 1. Let s check that the remainder is zero: remainder Factor Factor Multiple - : 0 Copyright 2014 Luis Soto-Ortiz 106

4 The following are special whole numbers that you should become familiar with: Number Definition Examples Even Any number that is divisible by 2. It has 2 0,2,4,6,8,10,12,14,16, as a factor. Odd Any number that is not divisible by 2. It 1,3,5,7,9,11,13,15,17, does not have 2 as a factor. Prime Any whole number that has exactly 2 different factors: 1 and the number itself. 2,3,5,7,11,13,17,19,23,... Composite Any whole number greater than 1 that is 4,6,8,9,10,12,14,15,16, not a prime number. Note: The number 1 is not a prime number because it has only one factor (itself), since 1 1 = 1. Moreover, 1 is not a composite number either. In fact, 1 is the only whole number that is neither prime nor composite. Table of Prime Numbers Less Than 1, Copyright 2014 Luis Soto-Ortiz 107

5 Example Make a table of all the factors of 24. Answer: , 2, 3, 4, 6, 8, 12 and 24 are factors of 24 because 1x24=24 2x12=24 3x8=24 4x6=24 The number at the top of the table shown in blue (24) is a multiple of all the numbers in green. Conversely, the numbers in green are factors of 24. In the table above, we see that 24 is a factor and multiple of itself. In fact, any whole number is a factor and multiple of itself because we can always write 1 =. Example Make a table of all the factors of 120. Answer: , 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120 are factors of 120 because 1x120=120 2x60=120 3x40=120 4x30=120 5x24=120 6x20=120 8x15=120 10x12=120 Copyright 2014 Luis Soto-Ortiz 108

6 Example Make a table of all the factors of 700. Answer: , 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350 and 700 are factors of 700 because 1x700=700 2x350=700 4x175=700 5x140=700 7x100=700 10x70=700 14x50=700 20x35=700 25x28=700 The number 700 at the top of the table (shown in blue) is a multiple of all the numbers in green. Similarly, the numbers in green are factors of 700. Please seek help from your instructor if you have difficulty understanding the difference between a multiple and a factor. Here is an instructional video on what a multiple of a number is: Copyright 2014 Luis Soto-Ortiz 109

7 Classwork Is 4 a factor of 216? Explain why or why not. 2. Is 17 a factor of 91? Explain why or why not. 3. Given that = 1105 state whether the following statements are true or false: A. 5 is a factor of B. 17 is a factor of C is a factor of 13. D is a multiple of 17. E is a multiple of 5. F is divisible by 13. G. 5 is a factor of Is 200 a multiple of 10? 5. Is 512 a multiple of 16? 6. Is 720 a multiple of 11? 7. Is 77 a multiple of 11? 8. Is 11 a multiple of 77? 9. Write a table of all the factors of Write a table of all the factors of True or false: a. 7 is a prime number. b. 90 is a prime number. Copyright 2014 Luis Soto-Ortiz 110

8 c. 63 is a composite number. d. 15 is a prime number. e. 41 is a composite number. f. 28 is a composite number. g. 9 is a prime number. h. 1 is a prime number. i. 1 is a composite number. j. 945 is a composite number. k. 121 is a prime number. l. 200 is a composite number. 12. Write a table of all the factors of True or false: a. 31 is a prime number. b. 70 is a composite number. c. 81 is a composite number. d. 9 is a prime number. e. 596 is a composite number. f. 45 is a prime number. g. 111 is a prime number. h. 441 is a prime number. i. 3,250 is a composite number. j. 777 is a composite number. Copyright 2014 Luis Soto-Ortiz 111

9 k. 169 is a prime number. l. 1,011 is a prime number. 14. T/F 90 is divisible by T/F 648 is divisible by T/F 7,326 is divisible by T/F 916 is divisible by T/F 5,783 is divisible by T/F 3,111 is divisible by T/F 12 is divisible by 8. Copyright 2014 Luis Soto-Ortiz 112

10 CW 2.1 Solutions: = 216 ( ) A. True B. True C. False D. True E. True F. True G. False 4. Yes 5. Yes 6. No 7. Yes 8. No a. True b. False c. True d. False e. False f. True g. False h. False i. False j. True k. False l. True a. True b. True c. True d. False e. True f. False g. False h. False i. True j. True k. False l. False 14. T 15. F 16. T 17. T 18. F 19. T 20. F Copyright 2014 Luis Soto-Ortiz 113

11 Homework Is 9 a factor of 325? Explain why or why not. 2. Is 12 a factor of 240? Explain why or why not. 3. Given that 8 19 = 152 state whether the following statements are true or false: A. 8 is a factor of 152. B. 19 is a factor of 152. C. 152 is a multiple of 8. D. 152 is divisible by 19. E. 152 is a multiple of 19. F. 152 is a factor of 8. G. 8 is a factor of Is 34,590 a multiple of 5? 5. Is 400 a multiple of 1,200? 6. Is 6 a factor of 30? 7. Is 144 divisible by 8? 8. Is 341 divisible by 3? 9. Write a table of all the factors of Write a table of all the factors of True or false: a. 33 is a prime number. b. 29 is a prime number. Copyright 2014 Luis Soto-Ortiz 114

12 c. 2 is a composite number. d. 777 is a prime number. e. 11 is a prime number. f. 35 is a prime number. g. 450 is a composite number. h. 3 is a prime number. i. 21 is a composite number. j is a composite number. k. 144 is a prime number. l. 27 is a prime number. 12. Write a table of all the factors of True or false: a. 27 is a prime number. b. 63 is a composite number. c. 47 is a composite number. d. 33 is a prime number. e. 286 is a composite number. f. 121 is a prime number. g. 360 is a prime number. h. 8,674 is a prime number. i. 835 is a composite number. Copyright 2014 Luis Soto-Ortiz 115

13 j. 23 is a composite number. k. 2 is a prime number. l. 999 is a prime number. 14. T/F 63 is divisible by T/F 532 is divisible by T/F 2,763 is divisible by T/F 322 is divisible by T/F 32 is divisible by T/F 430 is divisible by T/F 72 is divisible by 5. Copyright 2014 Luis Soto-Ortiz 116

14 HW 2.1 Solutions: = 240 ( ). 3. A. True B. True C. True D. True E. True F. False G. False 4. Yes 5. No 6. Yes 7. Yes 8. No a. False b. True c. False d. False e. True f. False g. True h. True i. True j. True k. False l. False Answer: a. False b. True c. False d. False e. True f. False g. False h. False i. True j. False k. True l. False 14. T 15. F 16. T 17. F 18. T 19. T 20. F Copyright 2014 Luis Soto-Ortiz 117

15 Section 2.2 Rules of Divisibility In the previous section, we learned what it means for a number to be divisible by another number. In particular, if we have =, this means that and are both factors of. This also means if we divide by or by, the remainder will be zero. Therefore, = means that is divisible by and that is divisible by. For example, since 15 2 = 30, both 15 and 2 are factors of 30. This means that 30 is divisible by 15 and that 30 is divisible by 2. This means that we get a zero remainder when we divide 30 by 2 and a zero reminder when we divide 30 by 15. Checking that the remainder is zero is a way to test divisibility. factors multiple factors remainder multiple F F M - : 0 Unfortunately, the long division process might be time consuming in some instances, depending on the numbers that are being divided. Therefore, it is advantageous to memorize the following divisibility rules of whole numbers and apply them as appropriate. There are many divisibility rules, but only the most basic and easy to remember are presented in this table. Divisible by Condition Examples 1 All whole numbers are divisible by 1. 0,1,2,3,4,5,6,7,8,9,10,11,12,13, 2 If the number is even. 0,2,4,6,8,10,12,14,16,18,20,22, 3 If the sum of the digits is divisible by 3. 0,3,6,9,12,15,18,21,24,27,30,33, 4 If the 2 rightmost digits are divisible by 4. 0,4,8,12,16,20,24,28,32,36,40,44, 5 If the number ends with a 5 or 0. 0,5,10,15,20,25,30,35,40,45,50,... 6 If the number is divisible by 2 and by 3. 0,6,12,18,24,30,36,42,48,54,60, 9 If the sum of the digits is divisible by 9. 0,9,18,27,36,45,54,63,72,81,90, 10 If the number ends with 0 0,10,20,30,40,50,60,70,80,90,100, Copyright 2014 Luis Soto-Ortiz 118

16 Example Determine whether the number 345,726 is divisible by 1, 2, 3, 5 or 6. Answer: 345,726 is divisible by 1 because all whole numbers are divisible by ,726 is divisible by 2 because 345,726 is an even number. 345,726 is divisible by 3 because the sum of the digits = 27 and 27 is divisible by ,726 is not divisible by 5 because the rightmost digit is not 5 or ,726 is divisible by 6 because 345,726 is divisible by 2 and by 3. Example Determine whether the number 68,970 is divisible by 1, 2, 3, 4, 9 or 10. Answer: 68,970 is divisible by 1 because all whole numbers are divisible by 1. 68,970 is divisible by 2 because 68,970 is an even number. 68,970 is divisible by 3 because the sum of the digits = 30 and 30 is divisible by 3. 68,970 is not divisible by 4 because the number formed by the 2 rightmost digits is 70, but 70 is not divisible by 4. 68,970 is not divisible by 9 because the sum of the digits = 30 and 30 is not divisible by 9. 68,970 is divisible by 10 because the rightmost digit is a zero. Copyright 2014 Luis Soto-Ortiz 119

17 Example Determine whether 476,306 is divisible by 9 by applying an appropriate divisibility rule. Check your answer by performing the long division. Answer: Since the sum of the digits is = 26 and 26 is not divisible by 9, this means that 476,306 is not divisible by 9 either. To check the answer, we perform the long division and note that the remainder is not zero, as expected F F M - : 0 Since the reminder is not zero, 9 is not a factor of 476,306. We also conclude that 476,306 is not a multiple of 9. Example Determine whether 128,975 is divisible by 5 by applying an appropriate divisibility rule. Check your answer by performing the long division. Answer: Since the number 128,975 has a digit 5 in the ones place, the number 128,975 is divisible by 5. To check the answer, we perform the long division and note that the remainder is zero, as expected. Copyright 2014 Luis Soto-Ortiz 120

18 F F M - : 0 A zero remainder means that the number 128,975 is divisible by 5. Hence, 5 is a factor of 128,975 and 128,975 is a multiple of 5. Instructional videos on the application of the Rules of Divisibility can be found in the following websites: Classwork 2.2 The following questions ask you to determine whether a number is a factor of the given number. You may use any method to determine this, including the rules of divisibility that were presented in this section. 1. Is 2 a factor of 7,986? 2. Is 8 a factor of 6039? 3. Is 5 a factor of 34,780? 4. Is 8 a factor of 7,432? 5. Is 10 a factor of 7,901? Copyright 2014 Luis Soto-Ortiz 121

19 6. Is 7 a factor of 7,910? 7. Is 9 a factor of 666? 8. Is 538 divisible by 2? 9. Is 7,872 divisible by 3? 10. Is 345 divisible by 5? 11. Is 9 a factor of 3,673,909? 12. Is 4 a factor of 845,912? 13. Is 2 a factor of 67,932,663? 14. Is 3 a factor of 852,504? 15. Is 9 a factor of 852,504? 16. Is 10 a factor of 89,015? 17. Is 9 a factor of 10,203? 18. Is 6,340 divisible by 5? 19. Is 48 divisible by 3? 20. Is 48 divisible by 2? 21. Is 8,360 divisible by 4? 22. Is 34,785 divisible by 4? 23. Is 678,021 divisible by 5? 24. Is 678,021 divisible by 3? 25. Is 30 divisible by 10? 26. Is 827 divisible by 2? 27. Is 7,212 divisible by 3? Copyright 2014 Luis Soto-Ortiz 122

21 CW 2.2 Solutions: 1. Yes 2. No 3. Yes 4. Yes 5. No 6. Yes 7. Yes 8. Yes 9. Yes 10. Yes 11. No 12. Yes 13. No 14. Yes 15. No 16. No 17. No 18. Yes 19. Yes 20. Yes 21. Yes 22. No 23. No 24. Yes 25. Yes 26. No 27. Yes 28. Yes 29. Yes 30. No Copyright 2014 Luis Soto-Ortiz 124

22 Homework 2.2 The following questions ask you to determine whether a number is a factor of the given number. You may use any method to determine this, including the rules of divisibility that were presented in this section. 1. Is 9 a factor of 504? 2. Is 6 a factor of 530? 3. Is 2 a factor of 687,421? 4. Is 5 a factor of 120? 5. Is 7 a factor of 821? 6. Is 10 a factor of 16,785? 7. Is 9 a factor of 440? 8. Is 470 divisible by 2? 9. Is 16,002 divisible by 3? 10. Is 120 divisible by 3? 11. Is 4 a factor of 32,719? 12. Is 2 a factor of 97,456,031? 13. Is 6 a factor of 34,692? 14. Is 3 a factor of 600? 15. Is 9 a factor of 5,555? 16. Is 2 a factor of 90? 17. Is 3 a factor of 90? 18. Is 80 divisible by 3? Copyright 2014 Luis Soto-Ortiz 125

23 19. Is 145 divisible by 5? 20. Is 620 divisible by 4? 21. Is 9,879 divisible by 9? 22. Is 10 divisible by 2? 23. Is 774,645 divisible by 5? 24. Is 666,666 divisible by 9? 25. Is 5,145 divisible by 10? 26. Is 654 divisible by 2? 27. Is 16,428 divisible by 4? 28. Is 1,736 divisible by 3? 29. Is 417,370 divisible by 5? 30. Is 720 divisible by 4? Copyright 2014 Luis Soto-Ortiz 126

24 HW 2.2 Solutions: 1. Yes 2. No 3. No 4. Yes 5. No 6. No 7. No 8. Yes 9. Yes 10. Yes 11. No 12. No 13. Yes 14. Yes 15. No 16. Yes 17. Yes 18. No 19. Yes 20. Yes 21. No 22. Yes 23. Yes 24. Yes 25. No 26. Yes 27. Yes 28. No 29. Yes 30. Yes Copyright 2014 Luis Soto-Ortiz 127

25 Section 2.3 Prime Factorization Recall that a prime number has exactly two different factors: 1 and the number itself. For your convenience, here again is a list of all the prime numbers that are less than 1000: Table of Prime Numbers Less Than 1, We have learned that whole numbers have factors, and thus can be written in factorized form. For example, some factorizations of the number 360 are These are four different factorizations of 360 because when we multiply the whole numbers the product is = = = = Copyright 2014 Luis Soto-Ortiz 128

26 A factorization of a number shows factors that multiplied together give the original number. Note that although 4, 6 and 5 are factors of 360, the expression is not a factorization of 360 because Recall that the symbol means not equal to. Example Write 5 different factorizations of 2,000. Answer: 2,000 = ,000 = ,000 = ,000 = ,000 = In some applications, it is useful to factorize a whole number using only prime factors. The prime factorization of a number entails breaking or splitting a number into factors that are prime numbers, and that gives back the original number when we multiply the prime factors. For example, the prime factorization of 2,000 is because all these factors of 2,000 are prime numbers, and when we multiply we get back 2,000. The prime factorization of 90 is because = 90 and the numbers 2, 3 and 5 are prime. Note: 1 is not a prime number because it has only one factor: itself 1 1 = 1. Copyright 2014 Luis Soto-Ortiz 129

27 A method to find the prime factorization of any whole number involves constructing a tree of factors. Each factor appearing in the tree must be either a prime number or a composite number. Hence, 1 should not appear in a tree of factors because 1 is neither prime nor composite. The approach to construct a tree of factors is to split, or factor, the original number into a product of prime and/or composite factors, and then continue splitting these factors until we are left with prime factors at the end of the branches. For example, to find the prime factorization of 12, we begin by factoring 12 in any way we choose, as long as the factors are prime or composite. At the end of the branches, we will be left with only prime numbers that multiplied together give the original number we started with (12). The numbers in red are the prime factors of 12, and so the prime factorization of 12 in expanded form is. If you are familiar with exponents, you can write the prime factorization in exponential form as. You will learn more about exponential notation in Section 2.5. Example Write the prime factorization of 45 in expanded form. Answer: 45 = Using exponents, the prime factorization of 45 in exponential form is given by 45 = 5 3. Copyright 2014 Luis Soto-Ortiz 130

28 Example Write the prime factorization of 120 in expanded form. Answer: 120 = Using exponents, the prime factorization of 120 is 120 = Example Write the prime factorization of 350 in expanded form. Answer: 350 = In exponential form, the answer is 350 = Copyright 2014 Luis Soto-Ortiz 131

29 Example Write the prime factorization of 504 in expanded form. Answer: 504 = In exponential form, the answer is 504 = Instructional video on finding the prime factorization of a whole number: The following website has an interactive tool to help you construct a tree of factors to find the prime factorization of any whole number: Classwork 2.3 Write the prime factorization of each number Copyright 2014 Luis Soto-Ortiz 132

35 HW 2.3 Solutions: = = = = ,300 = = = ,620 = = = = = = = = = = ,600 = = = = = 7 a prime number is its own prime factorization 23. 2,187 = ,750 = = ,455 = = ,936 = ,100 = = 5 17 Copyright 2014 Luis Soto-Ortiz 138

Section 1.6 Dividing Whole Numbers

Section 1.6 Dividing Whole Numbers We begin this section by looking at an example that involves division of whole numbers. Dale works at a Farmer s Market. There are 245 apples that he needs to put in