Transcript Prime Factorization, Greatest Common Fac

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 5 2 3 3
5 6
2
10
18
180
3
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 the
Prime Factors using
the Factor Tree Method
Factor Tree Method
96
• The CORRECT answer:
– must be only PRIME
numbers
– must multiply to give
the specified quantity
8 × 12
4 × 2
2 × 2
2 × 6
2 × 3
2×2×2×2×2×3 = 96
Factor Tree Method cont.
There is more than one way to solve the same problem
96
8
4 × 2
2 × 2
96
× 12
4 × 24
2 × 6
2 × 3
96= 2×2×2×2×2×3
2 × 2
6 × 4
2 × 3
96= 2×2×2×2×2×3
2 × 2
Determining the
Prime Factors using
The Stacked Method
The Stacked Method
1)
2)
3)
4)
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 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
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
Determining the
Least Common Multiple
Of Two Composite Numbers
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
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.
The GCF of 36 and 90
Method 1
1) List the factors of each number.
36: 1
2
3
4
36 18
24
9
2
3
5
90 45
30
90: 1
6
6
9
18 15 10
2) Circle the common factors.
3) The greatest of these will be your Greatest Common Factor:
18
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
the Greatest Common Factor:
2 ● 3 ● 3 = 18
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.
The LCM of 12 and 15
Method 1
1) List the first few multiples of each number.
12: 12
24
15: 15 30
36 48 60 72 84 90 108 120
45
60
75
90
105
120 135
2) Circle the common multiples.
3) The least of these will be your Least Common Multiple:
60
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
Least Common Multiple :
3 ● 2 ● 2 ● 5 = 60
Note that the common factor, 3, was only used once.
Method 3: Find both GCF and LCM at Once.
The GCF and LCM of 72 and 90
1) Make the following table.
9
72
2
8
4
90
10
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:
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.