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Exponents, Roots, Factorization of Whole Numbers: Prime Factorization of Natural Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses prime factorization of natural numbers. By the end of the module students should be able to determine the factors of a whole number, distinguish between prime and composite numbers, be familiar with the fundamental principle of arithmetic and find the prime factorization of a whole number.

Section Overview

  • Factors
  • Determining the Factors of a Whole Number
  • Prime and Composite Numbers
  • The Fundamental Principle of Arithmetic
  • The Prime Factorization of a Natural Number

Factors

From observations made in the process of multiplication, we have seen that

( factor ) ⋅ ( factor ) = product ( factor ) ⋅ ( factor ) = product size 12{ \( "factor" \) cdot " " \( "factor" \) =" product"} {}

Factors, Product

The two numbers being multiplied are the factors and the result of the multiplication is the product. Now, using our knowledge of division, we can see that a first number is a factor of a second number if the first number divides into the second number a whole number of times (without a remainder).

One Number as a Factor of Another

A first number is a factor of a second number if the first number divides into the second number a whole number of times (without a remainder).

We show this in the following examples:

Example 1

3 is a factor of 27, since 27÷3=927÷3=9 size 12{"27"÷3=9} {}, or 3⋅9=273⋅9=27 size 12{3 cdot 9="27"} {}.

Example 2

7 is a factor of 56, since 56÷7=856÷7=8 size 12{"56"÷7=8} {}, or 7⋅8=567⋅8=56 size 12{7 cdot 8="56"} {}.

Example 3

4 is not a factor of 10, since 10÷4=2R210÷4=2R2 size 12{"10"÷4=2R2} {}. (There is a remainder.)

Determining the Factors of a Whole Number

We can use the tests for divisibility from (Reference) to determine all the factors of a whole number.

Sample Set A

Example 4

Find all the factors of 24.

Try 1: 24÷1=241 and 24 are factors Try 2:24 is even, so 24 is divisible by 2. 24÷2=122 and 12 are factors Try 3: 2+4=6and 6 is divisible by 3, so 24 is divisible by 3. 24÷3=83 and 8 are factors Try 4: 24÷4=64 and 6 are factors Try 5: 24÷5=4R45 is not a factor.Try 1: 24÷1=24 size 12{"24"÷1="24"} {}1 and 24 are factors Try 2:24 is even, so 24 is divisible by 2. 24÷2=12 size 12{"24"÷2="12"} {}2 and 12 are factors Try 3: 2+4=6 size 12{2+4=6} {}and 6 is divisible by 3, so 24 is divisible by 3. 24÷3=8 size 12{"24"÷3=8} {}3 and 8 are factors Try 4: 24÷4=6 size 12{"24"÷4=6} {}4 and 6 are factors Try 5: 24÷5=4R4 size 12{"24"÷5=4R4} {}5 is not a factor.

The next number to try is 6, but we already have that 6 is a factor. Once we come upon a factor that we already have discovered, we can stop.

All the whole number factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Practice Set A

Find all the factors of each of the following numbers.

Exercise 1

6

Solution

1, 2, 3, 6

Exercise 2

12

Solution

1, 2, 3, 4, 6, 12

Exercise 3

18

Solution

1, 2, 3, 6, 9, 18

Exercise 4

5

Solution

1, 5

Exercise 5

10

Solution

1, 2, 5, 10

Exercise 6

33

Solution

1, 3, 11, 33

Exercise 7

19

Solution

1, 19

Prime and Composite Numbers

Notice that the only factors of 7 are 1 and 7 itself, and that the only factors of 3 are 1 and 3 itself. However, the number 8 has the factors 1, 2, 4, and 8, and the number 10 has the factors 1, 2, 5, and 10. Thus, we can see that a whole number can have only two factors (itself and 1) and another whole number can have several factors.

We can use this observation to make a useful classification for whole numbers: prime numbers and composite numbers.

Prime Number

A whole number (greater than one) whose only factors are itself and 1 is called a prime number.

The Number 1 is Not a Prime Number

The first seven prime numbers are 2, 3, 5, 7, 11, 13, and 17. Notice that the whole number 1 is not considered to be a prime number, and the whole number 2 is the first prime and the only even prime number.

Composite Number

A whole number composed of factors other than itself and 1 is called a compos­ite number. Composite numbers are not prime numbers.

Some composite numbers are 4, 6, 8, 9, 10, 12, and 15.

Sample Set B

Determine which whole numbers are prime and which are composite.

Example 5

39. Since 3 divides into 39, the number 39 is composite: 39÷3=1339÷3=13

Example 6

47. A few division trials will assure us that 47 is only divisible by 1 and 47. Therefore, 47 is prime.

Practice Set B

Determine which of the following whole numbers are prime and which are composite.

Exercise 8

3

Solution

prime

Exercise 9

16

Solution

composite

Exercise 10

21

Solution

composite

Exercise 11

35

Solution

composite

Exercise 12

47

Solution

prime

Exercise 13

29

Solution

prime

Exercise 14

101

Solution

prime

Exercise 15

51

Solution

composite

The Fundamental Principle of Arithmetic

Prime numbers are very useful in the study of mathematics. We will see how they are used in subsequent sections. We now state the Fundamental Principle of Arithmetic.

Fundamental Principle of Arithmetic

Except for the order of the factors, every natural number other than 1 can be factored in one and only one way as a product of prime numbers.

Prime Factorization

When a number is factored so that all its factors are prime numbers. the factori­zation is called the prime factorization of the number.

The technique of prime factorization is illustrated in the following three examples.

  1. 10=5⋅210=5⋅2 size 12{"10"=5 cdot 2} {}. Both 2 and 5 are primes. Therefore, 2⋅52⋅5 size 12{2 cdot 5} {} is the prime factorization of 10.
  2. 11. The number 11 is a prime number. Prime factorization applies only to composite numbers. Thus, 11 has no prime factorization.
  3. 60=2⋅3060=2⋅30 size 12{"60"=2 cdot "30"} {}. The number 30 is not prime: 30=2⋅1530=2⋅15 size 12{"30"=2 cdot "15"} {}.

60 = 2 ⋅ 2 ⋅ 15 60 = 2 ⋅ 2 ⋅ 15 size 12{"60"=2 cdot 2 cdot "15"} {}

The number 15 is not prime: 15=3⋅515=3⋅5 size 12{"15"=3 cdot 5} {}

60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 size 12{"60"=2 cdot 2 cdot 3 cdot 5} {}

We'll use exponents.

60 = 2 2 ⋅ 3 ⋅ 5 60 = 2 2 ⋅ 3 ⋅ 5 size 12{"60"=2 rSup { size 8{2} } cdot 3 cdot 5} {}

The numbers 2, 3, and 5 are each prime. Therefore, 22⋅3⋅522⋅3⋅5 size 12{2 rSup { size 8{2} } cdot 3 cdot 5} {} is the prime factorization of 60.

The Prime Factorization of a Natural Number

The following method provides a way of finding the prime factorization of a natural number.

The Method of Finding the Prime Factorization of a Natural Number

  1. Divide the number repeatedly by the smallest prime number that will divide into it a whole number of times (without a remainder).
  2. When the prime number used in step 1 no longer divides into the given number without a remainder, repeat the division process with the next largest prime that divides the given number.
  3. Continue this process until the quotient is smaller than the divisor.
  4. The prime factorization of the given number is the product of all these prime divisors. If the number has no prime divisors, it is a prime number.

We may be able to use some of the tests for divisibility we studied in (Reference) to help find the primes that divide the given number.

Sample Set C

Example 7

Find the prime factorization of 60.

Since the last digit of 60 is 0, which is even, 60 is divisible by 2. We will repeatedly divide by 2 until we no longer can. We shall divide as follows:

graphics2.png 30 is divisible by 2 again. 15 is not divisible by 2, but it is divisible by 3, the next prime. 5 is not divisble by 3, but it is divisible by 5, the next prime. 30 is divisible by 2 again. 15 is not divisible by 2, but it is divisible by 3, the next prime. 5 is not divisble by 3, but it is divisible by 5, the next prime.

The quotient 1 is finally smaller than the divisor 5, and the prime factorization of 60 is the product of these prime divisors.

60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 60 = 2 ⋅ 2 ⋅ 3 ⋅ 5 size 12{"60"=2 cdot 2 cdot 3 cdot 5} {}

We use exponents when possible.

60 = 2 2 ⋅ 3 ⋅ 5 60 = 2 2 ⋅ 3 ⋅ 5 size 12{"60"=2 rSup { size 8{2} } cdot 3 cdot 5} {}

Example 8

Find the prime factorization of 441.

441 is not divisible by 2 since its last digit is not divisible by 2.

441 is divisible by 3 since 4+4+1=94+4+1=9 size 12{4+4+1=9} {} and 9 is divisible by 3.

441 divided by 3 is 147. 147 divided by 3 is 49. 49 divided by 7 is 7. 7 divided by 7 is 1. 147 is divisible by 3(1+4+7=12). 49 is not divisible by 3, nor is it divisible by 5. It is divisible by 7. 147 is divisible by 3(1+4+7=12) size 12{3 \( 1+4+7="12" \) } {}. 49 is not divisible by 3, nor is it divisible by 5. It is divisible by 7.

The quotient 1 is finally smaller than the divisor 7, and the prime factorization of 441 is the product of these prime divisors.

441 = 3 ⋅ 3 ⋅ 7 ⋅ 7 441 = 3 ⋅ 3 ⋅ 7 ⋅ 7 size 12{"441"=3 cdot 3 cdot 7 cdot 7} {}

Use exponents.

441 = 3 2 ⋅ 7 2 441 = 3 2 ⋅ 7 2 size 12{"441"=3 rSup { size 8{2} } cdot 7 rSup { size 8{2} } } {}

Example 9

Find the prime factorization of 31.

31 is not divisible by 2 Its last digit is not even 31÷2=15R1 The quotient, 15, is larger than the divisor, 3. Continue. 31 is not divisible by 3 The digits 3 + 1 = 4, and 4 is not divisible by 3. 31÷3=10R1 The quotient, 10, is larger than the divisor, 3. Continue. 31 is not divisible by 5 The last digit of 31 is not 0 or 5. 31÷5=6R1 The quotient, 6, is larger than the divisor, 5. Continue. 31 is not divisible by 7. Divide by 7. 31÷7=4R1 The quotient, 4, is smaller than the divisor, 7. We can stop the process and conclude that 31 is a prime number. 31 is not divisible by 2 Its last digit is not even 31÷2=15R1 The quotient, 15, is larger than the divisor, 3. Continue. 31 is not divisible by 3 The digits 3 + 1 = 4, and 4 is not divisible by 3. 31÷3=10R1 The quotient, 10, is larger than the divisor, 3. Continue. 31 is not divisible by 5 The last digit of 31 is not 0 or 5. 31÷5=6R1 The quotient, 6, is larger than the divisor, 5. Continue. 31 is not divisible by 7. Divide by 7. 31÷7=4R1 The quotient, 4, is smaller than the divisor, 7. We can stop the process and conclude that 31 is a prime number.

The number 31 is a prime number

Practice Set C

Find the prime factorization of each whole number.

Exercise 16

22

Solution

22=2⋅1122=2⋅11 size 12{"22"=2 cdot "11"} {}

Exercise 17

40

Solution

40=23⋅540=23⋅5 size 12{"40"=2 rSup { size 8{3} } cdot 5} {}

Exercise 18

48

Solution

48=24⋅348=24⋅3 size 12{"48"=2 rSup { size 8{4} } cdot 3} {}

Exercise 19

63

Solution

63=32⋅763=32⋅7 size 12{"63"=3 rSup { size 8{2} } cdot 7} {}

Exercise 20

945

Solution

945=33⋅5⋅7945=33⋅5⋅7 size 12{"945"=3 rSup { size 8{3} } cdot 5 cdot 7} {}

Exercise 21

1,617

Solution

1617=3⋅72⋅111617=3⋅72⋅11 size 12{"1617"=3 cdot 7 rSup { size 8{2} } cdot "11"} {}

Exercise 22

17

Solution

17 is prime

Exercise 23

61

Solution

61 is prime

Exercises

For the following problems, determine the missing factor(s).

Exercise 24

14=7⋅14=7⋅ size 12{"14"=7 cdot } {}

Solution

2

Exercise 25

20=4⋅20=4⋅ size 12{"20"=4 cdot } {}

Exercise 26

36=9⋅36=9⋅ size 12{"36"=9 cdot } {}

Solution

4

Exercise 27

42=21⋅42=21⋅ size 12{"42"="21"} {}

Exercise 28

44=4⋅44=4⋅ size 12{"44"=4 cdot } {}

Solution

11

Exercise 29

38=2⋅38=2⋅ size 12{"38"=2 cdot } {}

Exercise 30

18=3⋅18=3⋅ size 12{"18"=3 cdot } {}
⋅⋅

Solution

3⋅23⋅2 size 12{3 cdot 2} {}

Exercise 31

28=2⋅28=2⋅ size 12{"28"=2 cdot } {}
⋅⋅

Exercise 32

300=2⋅5⋅300=2⋅5⋅ size 12{"300"=2 cdot 5 cdot } {}
⋅⋅

Solution

2⋅3⋅52⋅3⋅5 size 12{2 cdot 3 cdot 5} {}

Exercise 33

840=2⋅840=2⋅ size 12{"840"=2 cdot } {}
⋅⋅
⋅⋅

For the following problems, find all the factors of each of the numbers.

Exercise 34

16

Solution

1, 2, 4, 8, 16

Exercise 35

22

Exercise 36

56

Solution

1, 2, 4, 7, 8, 14, 28, 56

Exercise 37

105

Exercise 38

220

Solution

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220

Exercise 39

15

Exercise 40

32

Solution

1, 2, 4, 8, 16, 32

Exercise 41

80

Exercise 42

142

Solution

1, 2, 71, 142

Exercise 43

218

For the following problems, determine which of the whole numbers are prime and which are composite.

Exercise 44

23

Solution

prime

Exercise 45

25

Exercise 46

27

Solution

composite

Exercise 47

2

Exercise 48

3

Solution

prime

Exercise 49

5

Exercise 50

7

Solution

prime

Exercise 51

9

Exercise 52

11

Solution

prime

Exercise 53

34

Exercise 54

55

Solution

composite ( 5⋅115⋅11 size 12{5 cdot "11"} {})

Exercise 55

63

Exercise 56

1,044

Solution

composite

Exercise 57

924

Exercise 58

339

Solution

composite

Exercise 59

103

Exercise 60

209

Solution

composite ( 11⋅1911⋅19 size 12{"11" cdot "19"} {})

Exercise 61

667

Exercise 62

4,575

Solution

composite

Exercise 63

119

For the following problems, find the prime factorization of each of the whole numbers.

Exercise 64

26

Solution

2⋅132⋅13 size 12{2 cdot "13"} {}

Exercise 65

38

Exercise 66

54

Solution

2⋅332⋅33 size 12{2 cdot 3 rSup { size 8{3} } } {}

Exercise 67

62

Exercise 68

56

Solution

23⋅723⋅7 size 12{2 rSup { size 8{3} } cdot 7} {}

Exercise 69

176

Exercise 70

480

Solution

25⋅3⋅525⋅3⋅5 size 12{2 rSup { size 8{5} } cdot 3 cdot 5} {}

Exercise 71

819

Exercise 72

2,025

Solution

34⋅5234⋅52 size 12{3 rSup { size 8{4} } cdot 5 rSup { size 8{2} } } {}

Exercise 73

148,225

Exercises For Review

Exercise 74

((Reference)) Round 26,584 to the nearest ten.

Solution

26,580

Exercise 75

((Reference)) How much bigger is 106 than 79?

Exercise 76

((Reference)) True or false? Zero divided by any nonzero whole number is zero.

Solution

true

Exercise 77

((Reference)) Find the quotient. 10,584÷12610,584÷126 size 12{"10","584" div "126"} {}.

Exercise 78

((Reference)) Find the value of 121−81+62÷3121−81+62÷3 size 12{ sqrt {"121"} - sqrt {"81"} +6 rSup { size 8{2} } div 3} {}.

Solution

14

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