Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
MATH 112
Section 4.3: GCDs and LCMs
Prof. Jonathan Duncan
Walla Walla College
Fall Quarter, 2006
Conclusion
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Outline
1
Greatest Common Factors
2
Least Common Multiples
3
Relationships Between GCFs and LCMs
4
Conclusion
Conclusion
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
A Motivating Example
In our last section in chapter 4, we will examine two different
numbers which can be built from the factors of a pair of given
numbers.
Example
A quilter is using a rectangular piece of cloth 300 inches by 90
inches in size. He wishes to make a large quilt out of perfect
squares. What are the dimensions of the largest possible square (in
whole inches) which will exactly use up the fabric?
We need an x by x square where x is a divisor of both 300 and 90.
How do we go about finding x?
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
The Greatest Common Factor
Greatest Common Factor
The greatest common factor of two numbers a and b, written as
GCF (a, b), is the largest number which is a factor of both a and b.
Properties of the GCF
The GCF of a and b has the following properties:
It is a factor of both a and b, so it is less than both.
Prime numbers a and b have a GCF of 1.
Even numbers a and b have an even GCF of 2 or more.
Odd numbers a and b have an odd GCF.
Modeling the GCF
The greatest common factor can be modeled using cuisinare rods.
Use rods to model GCF (8, 12).
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Finding the GCF by Listing Factors
There are several ways we can try to find Greatest Common
Factors. One of the most basic methods is to list factors.
Listing Factors
To find GCF (a, b) list all factors of a and all factors of b and find
the largest number common to both lists.
Example
Find GCF (84, 126) by listing factors.
Advantages and Disadvantages
Advantages: Visual and convincing
Disadvantages: time consuming, hard for larger numbers,
lots of “extra” information required.
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Finding the GCF by Intuition
Sometimes it is possible to just “know” the answer to a question
based on previous experience.
Intuition
To find GCF (a, b) think about the factors of a and b and see if the
largest common factor “comes” to you.
Example
Use intuition to find GCF (64, 12).
Advantages and Disadvantages
Advantages: it is fast!
Disadvantages: it is hard for large numbers, of questionable
accuracy, and does not work for everybody
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Finding the GCF by Repeated Division
If you don’t have good intuition, there is still a better method to
find the GCF than listing out all the factors.
Repeated Division
To find GCF (a, b) using repeated division, repeatedly divide a and
b by common prime factors until this can no longer be done.
Then, the GCF is the product of the common prime factors.
Example
Use repeated division to find GCF (135, 75).
Advantages and Disadvantages
Advantages: systematic and relatively quick
Disadvantages: involves division and recognizing common
prime factors.
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Finding the GCF by Prime Factorization
The last method we will examine for finding the GCF is by using
prime factorizations. It is in many ways similar to the repeated
division and listing of factors methods seen earlier.
Prime Factorization
To find GCF (a, b), write the prime factorization of a and b. The
GCF is the product of all prime factors common to a and b.
Example
Use prime factorizations to find GCF (84, 126)
Advantages and Disadvantages
Advantages: systematic and relatively quick
Disadvantages: requires complete factorization
Conclusion
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Another Important Number
Another important number which appears in many computations
has to do with common multiples instead of common factors.
Example
Your school is having a fund raiser at which your class is
responsible for selling vegie-burgers. When you go to the store to
buy supplies, you find that the patties come un packs of 12, but
the buns come in packs of 8. What is the smallest number of each
you could purchase so that you have the same number of patties
as buns?
We need a number m which is something times 8 and something
else times 12 and as small as possible.
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
The Least Common Multiple
The Least Common Multiple
The least common multiple of two numbers a and b, written
LCM(a, b), is the smallest number which is both a multiple of a
and a multiple of b.
Properties of the LCM
The LCM of a and b has the following properties:
It is a multiple of both a and b, so at least as big as both.
It is less than or equal to a × b.
Even numbers a and b have an even GCF.
Odd numbers a and b have an odd GCF.
Modeling the LCM
The least common multiple can be modeled using cuisinare rods.
Use rods to model LCM(4, 6).
Conclusion
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Finding the LCM by Listing Multiples
As with the greatest common factor, the simplest way to approach
this problem may be to just start listing multiples.
Listing Multiples
To find LCM(a, b) first list multiples of a up to a × b and multiples
of b up to a × b. The smallest multiple common to both lists is
the LCM.
Example
Find LCM(12, 30).
Advantages and Disadvantages
What are some of the advantages and disadvantages of this
method of finding the LCM?
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Finding the LCM by Prime Factorization
As with the GCF, perhaps the best method for finding the LCM is
through the use of prime factorization.
Prime Factorization
To find LCM(a, b) first write the prime factorization of a and b.
Then look for the shortest list of primes which contains all factors
in both the list for a and b including repeated primes. The product
of this list is the LCM.
Example
Find LCM(12, 30).
Advantages and Disadvantages
What are some of the advantages and disadvantages of this
method of finding the LCM?
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
The GCF and LCM
The prime factorization method for finding the GCF and LCM of
two numbers is somewhat similar. Let’s examine some of the
relationships between GCFs and LCMs, starting with an example.
Example
Find both the GCF and LCM of 504 and 98
There are several things we can do to solve this problem:
Use prime factorizations
Use a Venn Diagram of factors
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Conclusion
Relating the GCF and LCM
There is a relationship between the GCF and LCM of two numbers
due to their prime factorizations.
Example
Find the GCF and LCM of a and b.
a
4
6
8
9
10
b
6
10
12
15
15
GCF (a, b)
2
2
4
3
5
LCM(a, b)
12
30
24
45
30
The General Relationship
In general, a × b = GCF (a, b) × LCM(a, b), or stated another way,
a×b
LCM(a, b) = GCF
(a,b) .
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Some Closing Examples
Sometimes using the relationships between GCFs and LCMs can
help in finding either or both of these numbers.
Example
Find LCM(1485, 825).
Example
If GCF (45, x) = 9 and LCM(45, x) = 135 find x.
Conclusion
Greatest Common Factors
Least Common Multiples
Relationships Between GCFs and LCMs
Important Concepts
Things to Remember from Section 4.3
1
The definition of a greatest common factor
2
How to find greatest common factors
3
The definition of a least common multiple
4
How to find least common multiples
5
Relationships between GCFs and LCMs
Conclusion