Prime Factorization, Greatest Common Factor,

1
Prime Factorization,Greatest Common Factor,
Least Common Multiple

• EDTE 203

2
Introduction

• Determining Prime Factorization
• Determining the Greatest Common Factor (GCF)
• Determining the Least Common Multiple (LCM)

3
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.

4
Essential Question

• The Essentials We Hope To Discover

5
The Essential Question

• How do prime factorization, greatest common
  factor, and least common multiple help you to
  understand the world?

6
Background Information

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

7
History of Prime Factorization, Greatest Common
Factor, Least Common Multiple

• Originated around 300 B.C. through the Theorem
  of (unique) prime factorization
• Started with Euclids
• Property of Natural
• Numbers
• (e.g., 24 2223)

8
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

9
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

10
Determining thePrime Factors usingthe Factor
Tree Method
11
Factor Tree Method

• 96
• 12
• 4 2 2 6
• 2 2 2 3
• 222223 96

• The CORRECT answer
• must be only PRIME numbers
• must multiply to give the specified quantity

12
Factor Tree Method cont.

• There is more than one way to solve the same
  problem

• 96
• 12
• 4 2 2 6
• 2 2 2 3
• 96 222223

• 96
• 4 24
• 2 2 6 4
• 2 3 2 2
• 96 222223

13
Determining the Prime Factors usingThe Stacked
Method
14
The Stacked Method

• Begin by dividing the specified quantity by any
  PRIME number that divides equally, (hint if it
  is even try dividing by 2)
• Reduce the quotient, dividing again by a PRIME
  number
• Continue reducing the quotient until both the
  divisor and the quotient are prime numbers.
• Re-write the prime numbers as a multiplication
  problem. (if the final quotient is 1 it doesnt
  need included in the answer)
• The CORRECT answer
• must be only prime numbers
• must multiply to give the specified quantity

15
Determining the Greatest Common FactorOf Two
Composite Numbers
16
Solving for the Greatest Common Factor

1. Find the prime factorization of the given
   quantities
2. Determine what factors they have in common.

• 36
• 3 12
• 3 4
• 2 2
• 2 2 3 3 36

• 54
• 6 9
• 3 2 3 3
• 2 3 3 3 54

17
Determining the Least Common MultipleOf Two
Composite Numbers
18
Solving for the Least Common Multiple

• 36
• 3 12
• 3 4
• 2 2
• 2 2 3 3 36

• 54
• 6 9
• 2 3 3 3
• 2 3 3 3 54

19
Finding the Greatest Common Factor of Two Numbers
We are looking for a factor. The factor
must be common to both numbers. We
need to pick the greatest of such
common factors.
20
The GCF of 36 and 90
Method 1
1) List the factors of each number.
36 1 2 3 4 6 36 18
24 9
90 1 2 3 5 6 9 90
45 30 18 15 10
2) Circle the common factors.
3) The greatest of these will be your Greatest
Common Factor
18
21
The GCF of 36 and 90
Method 2
1) Prime factor each number.
36 2 ? 2 ? 3 ? 3
90 2 ? 3 ? 3 ? 5
2) Circle each pair of common prime factors.
3) The product of these common prime factors
will be
2 ? 3 ? 3 18
the Greatest Common Factor
22
Finding the Least Common Multiple of Two Numbers
We are looking for a multiple. The multiple
must be common to both numbers. We
need to pick the least of such
common multiples.
23
The LCM of 12 and 15
Method 1
1) List the first few multiples of each number.
12 12 24 36 48 60 72 84 90
108 120
15 15 30 45 60 75 90 105
120 135
2) Circle the common multiples.
3) The least of these will be your Least Common
Multiple
60
24
The LCM of 12 and 15.
Method 2
1) Prime factor each number.
12 2 ? 2 ? 3
15 5 ? 3
2) Circle each pair of common prime factors.
3) Circle each remaining prime factor.
4) Multiply together one factor from each circle
to get the
3 ? 2 ? 2 ? 5 60
Least Common Multiple
Note that the common factor, 3, was only used
once.
25
Method 3 Find both GCF and LCM at Once.
The GCF and LCM of 72 and 90
1) Make the following table.
72 90

9
8
10
2
4
5
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.
The product of the factors on the left is the
GCF
The product of the factors on the left AND bottom
is the LCM
9 ? 2 18
9 ? 2 ? 4 ? 5 360
26
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.