MATH 112 Section 4.3: Gcds and Lcms

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Greatest Common Factors Least Common Multiples Relationships Between GCFs and LCMs Conclusion

MATH 112 Section 4.3: GCDs and LCMs

Prof. Jonathan Duncan

Walla Walla University

Autumn Quarter, 2007 Greatest Common Factors Least Common Multiples Relationships Between GCFs and LCMs Conclusion

Outline

1 Greatest Common Factors

2 Least Common Multiples

3 Relationships Between GCFs and LCMs

4 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 Conclusion

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 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 Conclusion

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). 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).

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 b GCF (a, b) LCM(a, b) 4 6 2 12 6 10 2 30 8 12 4 24 9 15 3 45 10 15 5 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 Conclusion

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. Greatest Common Factors Least Common Multiples Relationships Between GCFs and LCMs Conclusion

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