Prime Factorization, Greatest Common Factor, & Least Common Multiple

Prime Factorization,Greatest Common Factor, &

Least Common Multiple

EDTE 203

Introduction

• Determining Prime Factorization• Determining the Greatest Common Factor

(GCF)• Determining the Least Common Multiple

(LCM)

Introduction• The facts you will learn will give you a variety of

information about prime factorization, GCF, and LCM.• This lesson will show you different ways to calculate the prime factors of composite numbers.• This lesson will show you how to use the prime factors to calculate the GCF and LCM of two composite numbers.• You will learn how prime factorization equates to everyday life.

Essential Question

The Essentials We Hope To Discover

The Essential Question• How do prime factorization, greatest common

factor, and least common multiple help you to understand the world?

2 3 32

510

2 6

180

18

3

5

Background Information

The Basic facts you need you to know about prime factorization, GCF, and

LCM

History of Prime Factorization, Greatest Common Factor, & Least Common Multiple

• Originated around 300 B.C. through the “Theorem of (unique) prime factorization”

• Started with Euclid’s “Property of Natural Numbers” (e.g., 24= 2 2 2 3)∙ ∙ ∙

History of Prime Factorization, Greatest Common Factor, & Least Common Multiple cont.• The Theorem of Prime Factorization was further proven through the work of Gauss and Ernst Eduard Kummer• Prime Factorization is the foundation for finding the

Greatest Common Factor and the Least Common Multiple

Solving for Prime Factorization, GCF, and LCM.• There are 2 Ways determine the prime factors

– Factor Tree Method– Stacked Method

• Determining the GCF and LCM– GCF– LCM

Determining thePrime Factors using

the Factor Tree Method

Factor Tree Method96

8 × 12

4 × 2 2 × 6

2 × 2 2 × 3

2×2×2×2×2×3 = 96

96

8 × 12

4 × 2 2 × 6

2 × 2 2 × 3

2×2×2×2×2×3 = 96

• The CORRECT answer:– must be only PRIME

numbers– must multiply to give

the specified quantity

Factor Tree Method cont.

There is more than one way to solve the same problem96

8 × 12

4 × 2 2 × 6

2 × 2 2 × 3 96= 2×2×2×2×2×3

96

8 × 12

4 × 2 2 × 6

2 × 2 2 × 3 96= 2×2×2×2×2×3

96

4 × 24

2 × 2 6 × 4

2 × 3 2 × 2 96= 2×2×2×2×2×3

96

4 × 24

2 × 2 6 × 4

2 × 3 2 × 2 96= 2×2×2×2×2×3

Determining the Prime Factors using

The Stacked Method

The Stacked Method1) Begin by dividing the specified

quantity by any PRIME number that divides equally, (hint; if it is even try dividing by 2)

2) Reduce the quotient, dividing again by a PRIME number

3) Continue reducing the quotient until both the divisor and the quotient are prime numbers.

4) Re-write the prime numbers as a multiplication problem. (if the final quotient is 1 it doesn’t need included in the answer)

• The CORRECT answer:– must be only prime numbers– must multiply to give the

specified quantity

Determining the Greatest Common Factor

Of Two Composite Numbers

Solving for the Greatest Common Factor

363 × 12

3 × 4 2 × 2

2 × 2 × 3 × 3 = 36

546 × 9

3 × 2 3 × 3

2 × 3 × 3 × 3 = 54

1) Find the prime factorization of the given quantities2) Determine what factors they have in common.

Determining the Least Common Multiple

Of Two Composite Numbers

Solving for the Least Common Multiple

363 × 12

3 × 4 2 × 2

2 × 2 × 3 × 3 = 36

546 × 9

2 × 3 3 × 3

2 × 3 × 3 × 3 = 54

Finding the Greatest Common Factor

of Two Numbers

must be common to both numbers. WeWe are looking for a factor. The factor

need to pick the greatest of suchcommon factors.

Method 1

The GCF of 36 and 90

1) List the factors of each number.

36: 1 2 3 4 6

36 18 24 9

2) Circle the common factors.

90: 1 2 3 5 6 9

90 45 30 18 15 10

3) The greatest of these will be your Greatest Common Factor:

18

Method 2

The GCF of 36 and 90

1) Prime factor each number.

36 = 2 ● 2 ● 3 ● 3

2) Circle each pair of common prime factors.

90 = 2 ● 3 ● 3 ● 5

3) The product of these common prime factors will be

2 ● 3 ● 3 = 18the Greatest Common Factor:

Finding the Least Common Multiple

of Two Numbers

must be common to both numbers. WeWe are looking for a multiple. The multiple

need to pick the least of suchcommon multiples.

Method 1

The LCM of 12 and 15

1) List the first few multiples of each number.

12: 12 24 36 48 60 72 84 90 108 120

2) Circle the common multiples.

15: 15 30 45 60 75 90 105 120 135

3) The least of these will be your Least Common Multiple:

60

Method 2

The LCM of 12 and 15.

1) Prime factor each number.

12 = 2 ● 2 ● 3

2) Circle each pair of common prime factors.

15 = 5 ● 3

4) Multiply together one factor from each circle to get the

3 ● 2 ● 2 ● 5 = 60Least Common Multiple :

3) Circle each remaining prime factor.

Note that the common factor, 3, was only used once.

Method 3: Find both GCF and LCM at Once.

1) Make the following table.

72 90

The GCF and LCM of 72 and 90

2) Divide each number by a common factor.

3) Divide the new numbers by a common factor. Repeat this process until there is no longer a common factor.

9

8 1024 5

The product of the factors on the left is the GCF:

9 ● 2 = 18

The product of the factors on the left AND bottom is the LCM: 9 ● 2 ● 4 ● 5 = 360

Journal & Summary

• Nine people plan to share equally 24 stamps from one set and 36 stamps from another set. Explain why 9 people cannot share the stamps equally.

• What's is the LCM for two numbers that have no common factors greater than 1? Explain your reasoning.