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Mathematics LibreTexts

1.4: Prime Factorization

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    22462
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    In the statement 3 · 4 = 12, the number 12 is called the product, while 3 and 4 are called factors.

    Example 1

    Find all whole number factors of 18.

    Solution

    We need to find all whole number pairs whose product equals 18. The following pairs come to mind.

    1 · 18 = 18 and 2 · 9 = 18 and 3 · 6 = 18.

    Hence, the factors of 18 are (in order) 1, 2, 3, 6, 9, and 18.

    Exercise

    Find all whole number factors of 21.

    Answer

    1, 3, 7, and 21

    Divisibility

    In Example 1, we saw 3 · 6 = 18, making 3 and 6 factors of 18. Because division is the inverse of multiplication, that is, division by a number undoes the multiplication of that number, this immediately provides

    18 ÷ 6 = 3 and 18 ÷ 3=6.

    That is, 18 is divisible by 3 and 18 is divisible by 6. When we say that 18 is divisible by 3, we mean that when 18 is divided by 3, there is a zero remainder.

    Divisible

    Let a and b be whole numbers. Then a is divisible by b if and only if the remainder is zero when a is divided by b. In this case, we say that “b is a divisor of a.”

    Example 2

    Find all whole number divisors of 18.

    Solution

    In Example 1, we saw that 3 · 6 = 18. Therefore, 18 is divisible by both 3 and 6 (18 ÷ 3 = 6 and 18 ÷ 6 = 3). Hence, when 18 is divided by 3 or 6, the remainder is zero. Therefore, 3 and 6 are divisors of 18. Noting the other products in Example 1, the complete list of divisors of 18 is 1, 2, 3, 6, 9, and 18.

    Exercise

    Find all whole number divisors of 21.

    Answer

    1, 3, 7, and 21.

    Example 1 and Example 2 show that when working with whole numbers, the words factor and divisor are interchangeable.

    Factors and Divisors

    If c = a · b, then a and b are called factors of c. Both a and b are also called divisors of c.

    Divisibility Tests

    There are a number of very useful divisibility tests.

    Divisible by 2. If a whole number ends in 0, 2, 4, 6, or 8, then the number is called an even number and is divisible by 2. Examples of even numbers are 238 and 1,246 (238 ÷ 2 = 119 and 1, 246 ÷ 2 = 623). A number that is not even is called an odd number. Examples of odd numbers are 113 and 2,339.

    Divisible by 3. If the sum of the digits of a whole number is divisible by 3, then the number itself is divisible by 3. An example is 141. The sum of the digits is 1 + 4 + 1 = 6, which is divisible by 3. Therefore, 141 is also divisible by 3 (141 ÷ 3 = 47).

    Divisible by 4. If the number represented by the last two digits of a whole number is divisible by 4, then the number itself is divisible by 4. An example is 11,524. The last two digits represent 24, which is divisible by 4 (24 ÷ 4 = 6). Therefore, 11,524 is divisible by 4 (11, 524 ÷ 4=2, 881).

    Divisible by 5. If a whole number ends in a zero or a 5, then the number is divisible by 5. Examples are 715 and 120 (715÷5 = 143 and 120÷5 = 24).

    Divisible by 6. If a whole number is divisible by 2 and by 3, then it is divisible by 6. An example is 738. First, 738 is even and divisible by 2. Second, 7+3+8=18, which is divisible by 3. Hence, 738 is divisible by 3. Because 738 is divisible by both 2 and 3, it is divisible by 6 (738 ÷ 6 = 123).

    Divisible by 8. If the number represented by the last three digits of a whole number is divisible by 8, then the number itself is divisible by 8. An example is 73,024. The last three digits represent the number 24, which is divisible by 8 (24÷8 = 3). Thus, 73,024 is also divisible by 8 (73, 024÷8 = 9, 128).

    Divisible by 9. If the sum of the digits of a whole number is divisible by 9, then the number itself is divisible by 9. An example is 117. The sum of the digits is 1 + 1 + 7 = 9, which is divisible by 9. Hence, 117 is divisible by 9 (117 ÷ 9 = 13).

    Prime Numbers

    We begin with the definition of a prime number.

    Prime Number

    A whole number (other than 1) is a prime number if its only factors (divisors) are 1 and itself. Equivalently, a number is prime if and only if it has exactly two factors (divisors).

    Example 3

    Which of the whole numbers 12, 13, 21, and 37 are prime numbers?

    Solution

    • The factors (divisors) of 12 are 1, 2, 3, 4, 6, and 12. Hence, 12 is not a prime number.
    • The factors (divisors) of 13 are 1 and 13. Because its only divisors are 1 and itself, 13 is a prime number.
    • The factors (divisors) of 21 are 1, 3, 7, and 21. Hence, 21 is not a prime number.
    • The factors (divisors) of 37 are 1 and 37. Because its only divisors are 1 and itself, 37 is a prime number.

    Exercise

    Which of the whole numbers 15, 23, 51, and 59 are prime numbers?

    Answer

    23 and 59

    Example 4

    List all the prime numbers less than 20.

    Solution

    The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19.

    You try it!

    List all the prime numbers less than 100.

    Composite Numbers

    If a whole number is not a prime number, then it is called a composite number.

    Example 5

    Is the whole number 1,179 prime or composite?

    Solution

    Note that 1 + 1 + 7 + 9 = 18, which is divisible by both 3 and 9. Hence, 3 and 9 are both divisors of 1,179. Therefore, 1,179 is a composite number.

    Exercise

    Is the whole number 2,571 prime or composite?

    Answer

    Composite

    Factor Trees

    We will now learn how to express a composite number as a unique product of prime numbers. The most popular device for accomplishing this goal is the factor tree.

    Example 6

    Express 24 as a product of prime factors.

    Solution

    We use a factor tree to break 24 down into a product of primes.

    Screen Shot 2019-08-07 at 8.00.41 PM.png

    At each level of the tree, break the current number into a product of two factors. The process is complete when all of the “circled leaves” at the bottom of the tree are prime numbers. Arranging the factors in the “circled leaves” in order,

    24 = 2 · 2 · 2 · 3.

    The final answer does not depend on product choices made at each level of the tree. Here is another approach.

    Screen Shot 2019-08-07 at 8.00.48 PM.png

    The final answer is found by including all of the factors from the “circled leaves” at the end of each branch of the tree, which yields the same result, namely 24 = 2 · 2 · 2 · 3.

    Alternate Approach

    Some favor repeatedly dividing by 2 until the result is no longer divisible by 2. Then try repeatedly dividing by the next prime until the result is no longer divisible by that prime. The process terminates when the last resulting quotient is equal to the number 1.

    Screen Shot 2019-08-07 at 8.00.56 PM.png

    The first column reveals the prime factorization; i.e., 24 = 2 · 2 · 2 · 3.

    Exercise

    Express 36 as a product of prime factors.

    Answer

    2 · 2 · 3 · 3.

    The fact that the alternate approach in Example 6 yielded the same result is significant.

    Unique Factorization Theorem

    Every whole number can be uniquely factored as a product of primes.

    This result guarantees that if the prime factors are ordered from smallest to largest, everyone will get the same result when breaking a number into a product of prime factors.

    Exponents

    We begin with the definition of an exponential expression.

    Exponents

    The expression am is defined to mean

    \( a^{m}=\underbrace{a \cdot a \cdot \ldots \cdot a}_{m \text { times }}\)

    The number a is called the base of the exponential expression and the number m is called the exponent. The exponent m tells us to repeat the base a as a factor m times.

    Example 7

    Evaluate 25, 23, and 52.

    Solution

    • In the case of 25, we have

    25 = 2 · 2 · 2 · 2 · 2

    = 32.

    • In the case of 33, we have

    33 = 3 · 3 · 3

    = 27.

    • In the case of 52, we have

    52 = 5 · 5

    = 25.

    Exercise

    Evaluate: 35.

    Answer

    243.

    Example 8

    Express the solution to Example 6 in compact form using exponents.

    Solution

    In Example 6, we determined the prime factorization of 24.

    24 = 2 · 2 · 2 · 3

    Because 2 · 2 · 2=23, we can write this more compactly.

    24 = 23 · 3

    Exercise

    Prime factor 54.

    Answer

    2 · 3 · 3 · 3

    Example 9

    Evaluate the expression 23 · 32 · 52.

    Solution

    First raise each factor to the given exponent, then perform the multiplication in order (left to right).

    23 · 32 · 52 = 8 · 9 · 25

    = 72 · 25

    = 1800

    Exercise

    Evaluate: 33 · 52.

    Answer

    675

    Application

    A square is a rectangle with four equal sides.

    Area of a Square

    Let s represent the length of each side of a square.

    Screen Shot 2019-08-07 at 8.08.27 PM.png

    Because a square is also a rectangle, we can find the area of the square by multiplying its length and width. However, in this case, the length and width both equal s, so A = (s)(s) = s2. Hence, the formula for the area of a square is

    A = s2.

    Example 10

    The edge of a square is 13 centimeters. Find the area of a square.

    Solution

    Substitute s = 13 cm into the area formula.

    A = s2

    = (13 cm)2

    = (13 cm)(13 cm)

    = 169 cm2

    Hence, the area of the square is 169 cm2; i.e., 169 square centimeters.

    Exercise

    The edge of a square is 15 meters. Find the area of a square.

    Answer

    225 square meters

    Exercises

    In Exercises 1-12, find all divisors of the given number.

    1. 30

    2. 19

    3. 83

    4. 51

    5. 91

    6. 49

    7. 75

    8. 67

    9. 64

    10. 87

    11. 14

    12. 89


    In Exercises 13-20, which of the following numbers is not divisible by 2?

    13. 117, 120, 342, 230

    14. 310, 157, 462, 160

    15. 30, 22, 16, 13

    16. 382, 570, 193, 196

    17. 105, 206, 108, 306

    18. 60, 26, 23, 42

    19. 84, 34, 31, 58

    20. 66, 122, 180, 63


    In Exercises 21-28, which of the following numbers is not divisible by 3?

    21. 561, 364, 846, 564

    22. 711, 850, 633, 717

    23. 186, 804, 315, 550

    24. 783, 909, 504, 895

    25. 789, 820, 414, 663

    26. 325, 501, 945, 381

    27. 600, 150, 330, 493

    28. 396, 181, 351, 606


    In Exercises 29-36, which of the following numbers is not divisible by 4?

    29. 3797, 7648, 9944, 4048

    30. 1012, 9928, 7177, 1592

    31. 9336, 9701, 4184, 2460

    32. 2716, 1685, 2260, 9788

    33. 9816, 7517, 8332, 7408

    34. 1788, 8157, 7368, 4900

    35. 1916, 1244, 7312, 7033

    36. 7740, 5844, 2545, 9368


    In Exercises 37-44, which of the following numbers is not divisible by 5?

    37. 8920, 4120, 5285, 9896

    38. 3525, 7040, 2185, 2442

    39. 8758, 3005, 8915, 3695

    40. 3340, 1540, 2485, 2543

    41. 2363, 5235, 4145, 4240

    42. 9030, 8000, 5445, 1238

    43. 1269, 5550, 4065, 5165

    44. 7871, 9595, 3745, 4480


    In Exercises 45-52, which of the following numbers is not divisible by 6?

    45. 328, 372, 990, 528

    46. 720, 288, 148, 966

    47. 744, 174, 924, 538

    48. 858, 964, 930, 330

    49. 586, 234, 636, 474

    50. 618, 372, 262, 558

    51. 702, 168, 678, 658

    52. 780, 336, 742, 312


    In Exercises 53-60, which of the following numbers is not divisible by 8?

    53. 1792, 8216, 2640, 5418

    54. 2168, 2826, 1104, 2816

    55. 8506, 3208, 9016, 2208

    56. 2626, 5016, 1392, 1736

    57. 4712, 3192, 2594, 7640

    58. 9050, 9808, 8408, 7280

    59. 9808, 1232, 7850, 7912

    60. 3312, 1736, 9338, 3912


    In Exercises 61-68, which of the following numbers is not divisible by 9?

    61. 477, 297, 216, 991

    62. 153, 981, 909, 919

    63. 153, 234, 937, 675

    64. 343, 756, 927, 891

    65. 216, 783, 594, 928

    66. 504, 279, 307, 432

    67. 423, 801, 676, 936

    68. 396, 684, 567, 388


    In Exercises 69-80, identify the given number as prime, composite, or neither.

    69. 19

    70. 95

    71. 41

    72. 88

    73. 27

    74. 61

    75. 91

    76. 72

    77. 21

    78. 65

    79. 23

    80. 36


    In Exercises 81-98, find the prime factorization of the natural number.

    81. 224

    82. 320

    83. 108

    84. 96

    85. 243

    86. 324

    87. 160

    88. 252

    89. 32

    90. 128

    91. 360

    92. 72

    93. 144

    94. 64

    95. 48

    96. 200

    97. 216

    98. 392


    In Exercises 99-110, compute the exact value of the given exponential expression.

    99. 52 · 41

    100. 23 · 41

    101. 01

    102. 13

    103. 33 · 02

    104. 33 · 22

    105. 41

    106. 52

    107. 43

    108. 42

    109. 33 · 12

    110. 52 · 23


    In Exercises 111-114, find the area of the square with the given side.

    111. 28 inches

    112. 31 inches

    113. 22 inches

    114. 13 inches


    Create factor trees for each number in Exercises 115-122. Write the prime factorization for each number in compact form, using exponents.

    115. 12

    116. 18

    117. 105

    118. 70

    119. 56

    120. 56

    121. 72

    122. 270


    123. Sieve of Eratosthenes. This exercise introduces the Sieve of Eratosthenes, an ancient algorithm for finding the primes less than a certain number n, first created by the Greek mathematician Eratosthenes. Consider the grid of integers from 2 through 100.

    Screen Shot 2019-08-07 at 8.53.48 PM.png

    To find the primes less than 100, proceed as follows.

    i) Strike out all multiples of 2 (4, 6, 8, etc.)

    ii) The list’s next number that has not been struck out is a prime number.

    iii) Strike out from the list all multiples of the number you identified in step (ii).

    iv) Repeat steps (ii) and (iii) until you can no longer strike any more multiples.

    v) All unstruck numbers in the list are primes.

    Answers

    1. 1, 2, 3, 5, 6, 10, 15, 30

    3. 1, 83

    5. 1, 7, 13, 91

    7. 1, 3, 5, 15, 25, 75

    9. 1, 2, 4, 8, 16, 32, 64

    11. 1, 2, 7, 14

    13. 117

    15. 13

    17. 105

    19. 31

    21. 364

    23. 550

    25. 820

    27. 493

    29. 3797

    31. 9701

    33. 7517

    35. 7033

    37. 9896

    39. 8758

    41. 2363

    43. 1269

    45. 328

    47. 538

    49. 586

    51. 658

    53. 5418

    55. 8506

    57. 2594

    59. 7850

    61. 991

    63. 937

    65. 928

    67. 676

    69. prime

    71. prime

    73. composite

    75. composite

    77. composite

    79. prime

    81. 2 · 2 · 2 · 2 · 2 · 7

    83. 2 · 2 · 3 · 3 · 3

    85. 3 · 3 · 3 · 3 · 3

    87. 2 · 2 · 2 · 2 · 2 · 5

    89. 2 · 2 · 2 · 2 · 2

    91. 2 · 2 · 2 · 3 · 3 · 5

    93. 2 · 2 · 2 · 2 · 3 · 3

    95. 2 · 2 · 2 · 2 · 3

    97. 2 · 2 · 2 · 3 · 3 · 3

    99. 100

    101. 0

    103. 0

    105. 4

    107. 64

    109. 27

    111. 784 in2

    113. 484 in2

    115. 12 = 22 · 3

    117. 105 = 3 · 5 · 7

    119. 56 = 23 · 7

    121. 72 = 23 · 32

    123. Unstruck numbers are primes: 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

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