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MATH 112 Section 4.3: Gcds and Lcms

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MATH 112 Section 4.3: Gcds and Lcms 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). 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 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.
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