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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.

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

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

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Practice set b

Determine which of the following whole numbers are prime and which are 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 2 size 12{"10"=5 cdot 2} {} . Both 2 and 5 are primes. Therefore, 2 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 30 size 12{"60"=2 cdot "30"} {} . The number 30 is not prime: 30 = 2 15 size 12{"30"=2 cdot "15"} {} .

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

The number 15 is not prime: 15 = 3 5 size 12{"15"=3 cdot 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 size 12{"60"=2 rSup { size 8{2} } cdot 3 cdot 5} {}

The numbers 2, 3, and 5 are each prime. Therefore, 2 2 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 [link] to help find the primes that divide the given number.

Sample set c

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:

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

We use exponents when possible.

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

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

Use exponents.

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

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Find the prime factorization of 31.

31 is not divisible by 2 Its last digit is not even 31 ÷ 2 = 15 R1 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 = 10 R1 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 = 6 R1 The quotient, 6, is larger than the divisor, 5. Continue. 31 is not divisible by 7. Divide by 7. 31 ÷ 7 = 4 R1 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

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Practice set c

Find the prime factorization of each whole number.

22

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

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40

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

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48

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

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63

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

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945

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

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1,617

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

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Exercises

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

14 = 7 size 12{"14"=7 cdot } {}

2

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20 = 4 size 12{"20"=4 cdot } {}

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36 = 9 size 12{"36"=9 cdot } {}

4

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42 = 21 size 12{"42"="21"} {}

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44 = 4 size 12{"44"=4 cdot } {}

11

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38 = 2 size 12{"38"=2 cdot } {}

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

3 2 size 12{3 cdot 2} {}

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

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

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

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

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

56

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

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220

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

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

55

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

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209

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

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

26

2 13 size 12{2 cdot "13"} {}

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54

2 3 3 size 12{2 cdot 3 rSup { size 8{3} } } {}

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56

2 3 7 size 12{2 rSup { size 8{3} } cdot 7} {}

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480

2 5 3 5 size 12{2 rSup { size 8{5} } cdot 3 cdot 5} {}

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2,025

3 4 5 2 size 12{3 rSup { size 8{4} } cdot 5 rSup { size 8{2} } } {}

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Exercises for review

( [link] ) Round 26,584 to the nearest ten.

26,580

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( [link] ) How much bigger is 106 than 79?

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

true

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( [link] ) Find the quotient. 10 , 584 ÷ 126 size 12{"10","584" div "126"} {} .

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( [link] ) Find the value of 121 81 + 6 2 ÷ 3 size 12{ sqrt {"121"} - sqrt {"81"} +6 rSup { size 8{2} } div 3} {} .

14

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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