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Chapter 2

Fractions

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Section 2.1

Factors, Prime Factorizations, andLeast Common Multiples

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Factors of a Natural Number

Rule for Finding Factors of a Natural Number

Divide the natural number by each of the numbers 1, 2, 3, and so on. If the natural number is divisible by one of these numbers, then both the divisor and the quotient are factors of the natural number. Continue until the factors begin to repeat.

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Rules for Divisibility

A NUMBER IS DIVISIBLE BY

IF

2 The number is even.

3 The sum of the digits is divisible by 3.

4 The number named by the last two digits is divisible by 4.

5 The last digit is either 0 or 5.

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Rules for Divisibility

A NUMBER IS DIVISIBLE BY

IF

6 The number is even and the sum of the digits is divisible by 3.

8 The number named by the last three digits is divisible by 8.

9 The sum of digits is divisible by 9.

10 The last digit is 0.

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Example

Find the factors of each number.

a. 64

b. 60

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Solution Strategya. Divide 64 by 1, 2, 3, and so on.

64 ÷ 1 = 64 1 and 64 are factors.

64 ÷ 2 = 32 2 and 32 are factors.

64 ÷ 3 Does not divide evenly.

64 ÷ 4 = 16 4 and 16 are factors.

64 ÷ 5 Does not divide evenly.

64 ÷ 6 Does not divide evenly.

64 ÷ 7 Does not divide evenly.

64 ÷ 8 = 8 8 and 8 are factors.

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Solution Strategya. (continued)

64 ÷ 10 Does not divide evenly.

64 ÷ 12 Does not divide evenly.

64 ÷ 13 Does not divide evenly.

64 ÷ 14 Does not divide evenly.

64 ÷ 15 Does not divide evenly.

64 ÷ 16 = 4 16 and 4 are factors.

Repeat factors-Stop!

64 ÷ 9 Does not divide evenly.

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Solution Strategyb. Divide 60 by 1, 2, 3, and so on.

60 ÷ 1 = 60 1 and 60 are factors.

60 ÷ 2 = 30 2 and 30 are factors.

60 ÷ 3 = 20 3 and 20 are factors.

60 ÷ 4 = 15 4 and 15 are factors.

60 ÷ 5 = 12 5 and 12 are factors.

60 ÷ 6 = 10 6 and 10 are factors.

60 ÷ 7 Does not divide evenly.

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Solution Strategyb. (continued)

60 ÷ 8 = 10 Does not divide evenly.

60 ÷ 9 Does not divide evenly.

60 ÷ 10 10 and 6 are factors. Repeat factors-Stop!

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Prime and Composite Numbers

Prime

A natural number greater than 1 that has only two factors (divisors), namely, 1 and itself.

Composite

A natural number greater than 1 that has more than two factors (divisors).

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Prime Numbers Less Than 100

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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Example

Determine whether each number is prime, composite, or neither.

a. 16 b. 13 c. 23

d. 72 e. 0 f. 19

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Solution Strategy

a. 16 composite Factors: 1, 2, 4, 8, and 16

b. 13primeFactors: 1 and

itself

c. 23 prime Factors: 1 and itself

d. 72compositeFactors: 1, 2, 3, 4,

6, 8, 9, 12, 18, 24, 36, and 72

e. 0 neitherBy definition

f. 19 prime Factors: 1 and itself

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Factoring a Number

To factor a number means to express the number as a product of factors.

As an example, consider the number 18. There are three two-number factorizations.

118 18 29 18 36 18

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Prime Factoring a Number

A prime factorization of a whole number is a factorization in which each factor is prime.

The prime factorization of 18 is

233 232 18

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Factor Trees

A factor tree is an illustration used to determine the prime factorization of a composite number.

18 233 232

or

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Example

Find the prime factorization of each number.

a. 60 b. 90 c. 288

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Solution Strategy

a. 60

The prime factorization of 60 is 2 • 2 • 3 • 5 = 22 • 3 • 5.

60

10

5

6

2 23

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Solution Strategy

b. 90

The prime factorization of 90 is 2 • 3 • 3 • 5 = 2 • 32 • 5.

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Solution Strategy

c. 288

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Solution Strategy

c. 288

The prime factorization of 288 is 2 • 2 • 2 • 2 • 2 • 3 • 3 = 25 • 32.

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Least Common Multiple

A multiple of a number is the product of the number and any natural number.

A common multiple is a multiple that is shared by a set of two or more natural numbers.

A least common multiple or LCM is the smallest multiple shared by a set of two or more numbers.

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Finding the Least Common Multiple

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Example

Find the LCM of 4, 6, and 10 using prime factorization.

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Solution Strategy

4 = 2 • 2

6 = 2 • 3

10 = 2 • 5

LCM = 2 • 2 • 3 • 5 = 60

Find the LCM of 4, 6, and 10 using prime factorization.

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Alternative Method of Finding the LCM

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Example

Find the LCM of 4, 6, and 10 using the alternative method.

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Solution Strategy

LCM = 2 • 2 • 3 • 5 = 60

Find the LCM of 4, 6, and 10 using the alternative method.

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Example Apply your knowledge

You have volunteered to be the “barbeque chef ” for a school party. From experience, you know that shrimp should be turned every 2 minutes and salmon should be turned every 3 minutes. How often will they be turned at the same time?

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Solution Strategy

LCM = 2 • 3 = 6

2 = 1 • 23 = 1 • 3

The shrimp and salmon will be turned at the same time every 6 minutes.