3.3: Prime Factorization of Natural Numbers
- Page ID
- 48845
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- be able to determine the factors of a whole number
- be able to distinguish between prime and composite numbers
- be familiar with the fundamental principle of arithmetic
- be able to find the prime factorization of a whole number
Factors
From observations made in the process of multiplication, we have seen that
\((\text{factor}) \cdot (\text{factor}) = \text{product}\)
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:
3 is a factor of 27, since \(27 \div 3 = 9\), or \(3 \cdot 9 = 27\).
7 is a factor of 56, since \(56 \div 7 = 8\), or \(7 \cdot 8 = 56\).
4 is not a factor of 10, since \(10 \div 4 = \text{2R2}\). (There is a remainder.)
Determining the Factors of a Whole Number
We can use the tests for divisibility from [link] to determine all the factors of a whole number.
Find all the factors of 24.
Solution
\(\begin{array} {lll} {\text{Try 1:}} & {24 \div 1 = 24} & {\text{1 and 24 are factors}} \\ {\text{Try 2:}} & {\text{24 is even, so 24 is divisible by 2.}} & {} \\ {} & {24 \div 2 = 12} & {\text{2 and 12 are factors}} \\ {\text{Try 3:}} & {2 + 4 = 6 \text{ and 6 is divisible by 3, so 24 is divisible by 3.}} & {} \\ {} & {24 \div 3 = 8} & {\text{3 and 8 are factors}} \\ {\text{Try 4:}} & {24 \div 4 = 6} & {\text{4 and 6 are factors}} \\ {\text{Try 5:}} & {24 \div 5 = \text{4R4}} & {\text{5 is not a factor}} \end{array}\)
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.
6
- Answer
-
1, 2, 3, 6
Practice Set A
12
- Answer
-
1, 2, 3, 4, 6, 12
Practice Set A
18
- Answer
-
1, 2, 3, 6, 9, 18
Practice Set A
5
- Answer
-
1, 5
Practice Set A
10
- Answer
-
1, 2, 5, 10
Practice Set A
33
- Answer
-
1, 3, 11, 33
Practice Set A
19
- Answer
-
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 twofactors (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.
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.
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.
Determine which whole numbers are prime and which are composite.
39. Since 3 divides into 39, the number 39 is composite: \(39 \div 3 = 13\)
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.
3
- Answer
-
prime
Practice Set B
16
- Answer
-
composite
Practice Set B
21
- Answer
-
composite
Practice Set B
35
- Answer
-
composite
Practice Set B
47
- Answer
-
prime
Practice Set B
29
- Answer
-
prime
Practice Set B
101
- Answer
-
prime
Practice Set B
51
- Answer
-
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.
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.
\(10 = 5 \cdot 2\). Both 2 and 5 are primes. Therefore, \(2 \cdot 5\) is the prime factorization of 10.
11. The number 11 is a prime number. Prime factorization applies only to composite numbers. Thus, 11 has no prime factorization.
\(60 = 2 \cdot 30\). The number 30 is not prime: \(30 = 2 \cdot 15\).
\(60 = 2 \cdot 2 \cdot 15\)
The number 15 is not prime: \(15 = 3 \times 5\)
\(60 = 2 \cdot 2 \cdot 3 \cdot 5\)
We'll use exponents.
\(60 = 2^2 \cdot 3 \cdot 5\)
The numbers 2, 3, and 5 are each prime. Therefore, \(2^2 \cdot 3 \cdot 5\) 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
- Divide the number repeatedly by the smallest prime number that will divide into it a whole number of times (without a remainder).
- 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.
- Continue this process until the quotient is smaller than the divisor.
- 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 [link] to help find the primes that divide the given number.
Find the prime factorization of 60.
Solution
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:
\(\begin{array} {l} {\text{30 is divisible by 2 again}} \\ {\text{15 is not divisible by 2, but it is divisible by 3, the next prime}} \\ {\text{5 is not divisible by 3, but it is divisible by 5, the next prime.}} \end{array}\)
The quotient 1 is finally smaller than the divisor 5, and the prime factorization of 60 is the product of these prime divisors.
\(60 = 2 \cdot 2 \cdot 3 \cdot 5\)
We use exponents when possible.
\(60 = 2^2 \cdot 3 \cdot 5\)
Find the prime factorization of 441.
Solution
441 is not divisible by 2 since its last digit is not divisible by 2.
441 is divisible by 3 since \(4 + 4 + 1 = 9\) and 9 is divisible by 3.
\(\begin{array} {l} {\text{147 is divisible by } 3(1 + 4 + 7 = 12).} \\ {\text{49 is not divisible by 3, nor is it divisible by 5. It is divisible by 7.}} \end{array}\)
The quotient 1 is finally smaller than the divisor 7, and the prime factorization of 441 is the product of these prime divisors.
\(441 = 3 \cdot 3 \cdot 7 \cdot 7\)
Use exponents.
\(441 = 3^2 \cdot 7^2\)
Find the prime factorization of 31.
Solution
\(\begin{array} {ll} {\text{31 is not divisible by 2}} & {\text{Its last digit is not even}} \\ {} & {31 \div 2 = \text{15R1}} \\ {} & {\text{The quotient, 15, is larger than the divisor, 3. Continue.}} \\ {\text{31 is not divisible by 3}} & {\text{The digits } 3 + 1 = 4, \text{ and 4 is not divisible by 3.}} \\ {} & {31 \div 3 = \text{10R1}} \\ {} & {\text{The quotient, 10, is larger than the divisor, 3. Continue.}} \\ {\text{31 is not divisible by 5}} & {\text{The last digit of 31 is not 0 or 5.}} \\ {} & {31 \div 5 = \text{6R1}} \\ {} & {\text{The quotient, 6, is larger than the divisor, 5. Continue.}} \\ {\text{31 is not divisible by 7.}} & {\text{Divide by 7.}} \\ {} & {31 \div 7 = \text{4R1}} \\ {} & {\text{The quotient, 4 is smaller than the divisor, 7.}} \\ {} & {\text{We can stop the process and conclude that 31 is a prime number.}} \end{array}\)
The number 31 is a prime number
Practice Set C
Find the prime factorization of each whole number.
22
- Answer
-
\(22 = 2 \cdot 11\)
Practice Set C
40
- Answer
-
\(40 = 2^3 \cdot 5\)
Practice Set C
48
- Answer
-
\(48 = 2^4 \cdot 3\)
Practice Set C
63
- Answer
-
\(63 = 3^2 \cdot 7\)
Practice Set C
945
- Answer
-
\(945 = 3^3 \cdot 5 \cdot 7\)
Practice Set C
1,617
- Answer
-
\(1617 = 3 \cdot 7^2 \cdot 11\)
Practice Set C
17
- Answer
-
17 is prime
Practice Set C
61
- Answer
-
61 is prime
Exercises
For the following problems, determine the missing factor(s).
Exercise \(\PageIndex{1}\)
\(14 = 7 \cdot \)
- Answer
-
2
Exercise \(\PageIndex{2}\)
\(20 = 4 \cdot \)
Exercise \(\PageIndex{3}\)
\(36 = 9 \cdot \)
- Answer
-
4
Exercise \(\PageIndex{4}\)
\(42 = 21 \cdot \)
Exercise \(\PageIndex{5}\)
\(44 = 4 \cdot \)
- Answer
-
11
Exercise \(\PageIndex{6}\)
\(38 = 2 \cdot \)
Exercise \(\PageIndex{7}\)
\(18 = 3 \cdot \).
- Answer
-
\(3 \cdot 2\)
Exercise \(\PageIndex{8}\)
\(28 = 2 \cdot \).
Exercise \(\PageIndex{9}\)
\(300 = 2 \cdot 5 \cdot \).
- Answer
-
\(2 \cdot 3 \cdot 5\)
Exercise \(\PageIndex{10}\)
\(840 = 2 \cdot \).
For the following problems, find all the factors of each of the numbers.
Exercise \(\PageIndex{11}\)
16
- Answer
-
1, 2, 4, 8, 16
Exercise \(\PageIndex{12}\)
22
Exercise \(\PageIndex{13}\)
56
- Answer
-
1, 2, 4, 7, 8, 14, 28, 56
Exercise \(\PageIndex{14}\)
105
Exercise \(\PageIndex{15}\)
220
- Answer
-
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220
Exercise \(\PageIndex{16}\)
15
Exercise \(\PageIndex{17}\)
32
- Answer
-
1, 2, 4, 8, 16, 32
Exercise \(\PageIndex{18}\)
80
Exercise \(\PageIndex{19}\)
142
- Answer
-
1, 2, 71, 142
Exercise \(\PageIndex{20}\)
218
For the following problems, determine which of the whole numbers are prime and which are composite.
Exercise \(\PageIndex{21}\)
23
- Answer
-
prime
Exercise \(\PageIndex{22}\)
25
Exercise \(\PageIndex{23}\)
27
- Answer
-
composite
Exercise \(\PageIndex{24}\)
2
Exercise \(\PageIndex{25}\)
3
- Answer
-
prime
Exercise \(\PageIndex{26}\)
5
Exercise \(\PageIndex{27}\)
7
- Answer
-
prime
Exercise \(\PageIndex{28}\)
9
Exercise \(\PageIndex{29}\)
11
- Answer
-
prime
Exercise \(\PageIndex{30}\)
34
Exercise \(\PageIndex{31}\)
55
- Answer
-
composite (\(5 \cdot 11\))
Exercise \(\PageIndex{32}\)
63
Exercise \(\PageIndex{33}\)
1,044
- Answer
-
composite
Exercise \(\PageIndex{34}\)
924
Exercise \(\PageIndex{35}\)
339
- Answer
-
composite
Exercise \(\PageIndex{36}\)
103
Exercise \(\PageIndex{37}\)
209
- Answer
-
composite \((11 \cdot 19)\)
Exercise \(\PageIndex{38}\)
667
Exercise \(\PageIndex{39}\)
4,575
- Answer
-
composite
Exercise \(\PageIndex{40}\)
119
For the following problems, find the prime factorization of each of the whole numbers.
Exercise \(\PageIndex{41}\)
26
- Answer
-
\(2 \cdot 13\)
Exercise \(\PageIndex{42}\)
38
Exercise \(\PageIndex{43}\)
54
- Answer
-
\(2 \cdot 3^3\)
Exercise \(\PageIndex{44}\)
62
Exercise \(\PageIndex{45}\)
56
- Answer
-
\(2^3 \cdot 7\)
Exercise \(\PageIndex{46}\)
176
Exercise \(\PageIndex{47}\)
480
- Answer
-
\(2^5 \cdot 3 \cdot 5\)
Exercise \(\PageIndex{48}\)
819
Exercise \(\PageIndex{49}\)
2,025
- Answer
-
\(3^4 \cdot 5^2\)
Exercise \(\PageIndex{50}\)
148,225
Exercises For Review
Exercise \(\PageIndex{51}\)
Round 26,584 to the nearest ten.
- Answer
-
26,580
Exercise \(\PageIndex{52}\)
How much bigger is 106 than 79?
Exercise \(\PageIndex{53}\)
True or false? Zero divided by any nonzero whole number is zero.
- Answer
-
true
Exercise \(\PageIndex{54}\)
Find the quotient. \(10,584 \div 126.\)
Exercise \(\PageIndex{55}\)
Find the value of \(\sqrt{121} - \sqrt{81} + 6^2 \div 3\).
- Answer
-
14