Copyright © Cengage Learning. All rights reserved. CHAPTER 4 Number Theory Copyright © Cengage Learning. All rights reserved.

Greatest Common Factor and Least Common Multiple SECTION 4.3 Greatest Common Factor and Least Common Multiple Copyright © Cengage Learning. All rights reserved.

What Do You Think? What does it mean to say that the GCF and the LCM let us decompose composite numbers? How are the GCF and the LCM related to each other?

Investigation A – Cutting Squares Using Number Theory Concepts Let’s say that a teacher has a rectangular sheet of cardboard 420 centimeters long and 378 centimeters wide and that he wants to cut that sheet into many squares, all of the same size. What are the dimensions of the largest possible square (whose length is a whole number) that will create no waste? Discussion: Strategy 1: Use Guess–Check–Revise Looking at the numbers, we can see that they are both divisible by 2, so we could make 2 by 2 squares.

Investigation A – Discussion cont’d Using the divisibility rule for 3, we can see that they are also divisible by 3, so we can make 3 by 3 squares. We can proceed in this manner, checking for divisibility by 4, 5, 6, and so on, until we find that we can go no further. Strategy 2: Use Prime Factorization 378 = 2  3  3  3  7 420 = 2  2  3  5  7 Because we are looking for a number that divides both 378 and 420, an equivalent statement is that we are looking for a number that is a factor of both.

Investigation A – Discussion cont’d In fact, we are looking for the largest such common factor. By looking at the prime factorization of both numbers, we can see that they have in common a 2, a 3, and a 7. If we multiply the three numbers, the product of these three numbers (42) will also be a factor of both 378 and 420. In fact, 42 is the greatest number that divides both 378 and 420. If we express this conclusion using the term factor instead of divides, we can say that 42 is the greatest common factor of 378 and 420.

The Greatest Common Factor

The Greatest Common Factor Whenever we examine two natural numbers, we can create a set of numbers called their common factors. The greatest of these common factors is called the greatest common factor (GCF). We use the notation GCF(a, b) to express the GCF of two natural numbers a and b. Note: In a similar fashion, we can speak of the GCF of three or more numbers.

The Greatest Common Factor Exploring the greatest common factor concept Before we examine techniques for finding the GCF of two or more numbers, let us take some time to explore the concept. What can we say about the size of the GCF of any two numbers a and b? For example, will it always be smaller than the two numbers? Can it be in between the two numbers? The GCF of two numbers is generally smaller than either of them, because it has to be a factor of each. However, there are some cases in which the GCF of two numbers is not smaller than either of them.

The Greatest Common Factor Although the GCF can never be larger than either of the numbers, it can be equal to the smaller of the two numbers. For example, the GCF of 4 and 8 is 4. In a manner similar to a detective story, we can make predictions about the GCF if we know just a little bit about the two numbers. For example, if you know that two numbers are prime, what can you say about their GCF?

The Greatest Common Factor Prime numbers have only two factors, and one of them is 1. Because the other factor will be the number itself, we can conclude that if the two numbers are prime, their GCF is 1. What can you say about the GCF of two even numbers? Their GCF will be at least 2.

Investigation B – Methods for Finding the GCF Let us now investigate how we might determine the GCF of two numbers. Rather than give an efficient procedure right away, we will build the foundation of this procedure, much like taking care while constructing the foundation of a house. Using only the definition of GCF, how would you determine GCF(45, 60)?

Investigation B – Discussion Strategy 1: Use Factorization We could, as we just saw, determine all the factors of each number and then find the largest of the common factors: Factors of 45 = {1, 3, 5, 9, 15, 45} Factors of 60 = {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60} Common factors = {1, 3, 5, 15} We see from this list that 15 is the GCF of 45 and 60.

Investigation B – Discussion cont’d Strategy 2: Use Intuition Or Number Sense A student who is highly intuitive and has good number sense might just know that 15 divides both these numbers. The fact that 15 divides both numbers simply means that 15 is a common factor. How might you reason that, in fact, 15 is the GCF? Let us represent the results of dividing each number by 5 (which we know is not the GCF) and by 15. 45 = 5  9 45 = 15  3 60 = 5  12 60 = 15  4

Investigation B – Discussion cont’d When we divide 45 and 60 by 5, we are left with 9 and 12. When we divide 45 and 60 by 15, we are left with 3 and 4. One difference between 9 and 12 and 3 and 4 is that 3 and 4 have no common factors. When two numbers have no factors in common other than 1, they are said to be relatively prime. Because 3 and 4 are relatively prime, 15 is the GCF of 45 and 60.

Investigation B – Discussion cont’d Strategy 3: Repeatedly Divide By Prime Numbers What about people who did not have the intuition and number sense to see this? Is their only recourse the long way we saw above? Another procedure involves an adaptation of the long-division algorithm. The next problem illustrates a systematic application in that we begin with the smallest prime divisor and then move up. That is, we first divide both numbers by 3.

Investigation B – Discussion cont’d At this point, we move up to 5 because 15 and 20 are both divisible by 5. The resulting quotients, 3 and 4, have no factors in common. The GCF of 45 and 60 is the product of their common factors: 3  5 = 15.

Investigation B – Discussion cont’d Strategy 4: Use Prime Factorization This strategy uses the Fundamental Theorem of Arithmetic. We first determine the prime factorization of each number and then look for common factors. If we look at the prime factorizations of 45 and 60 and circle the factors that the two numbers have in common, we have the following: We can further refine this procedure by using exponents: 45 = 32  51 60 = 22  31  51

Investigation B – Discussion cont’d The GCF is determined by examining those factors that both numbers have in common and then taking the smallest exponent in each case. The common factors of 45 and 60 are 3 and 5. The smallest exponent of 3 is 1, and the smallest exponent of 5 is 1. Thus 3  5 is the GCF.

Least Common Multiple

Least Common Multiple Let us begin our investigation of the least common multiple (LCM) by having you think about the following question before reading on: What do you think “least common multiple” means? Write a definition in informal language. Let us examine this concept word by word, using a specific example. Suppose we wanted to find the least common multiple of 8 and 12. At the most basic level, we can start listing multiples of 8 and 12 until we find the first multiple they have in common.

Least Common Multiple Multiples of 8 = {8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, . . .} Multiples of 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, . . .} What multiples do the two numbers have in common? These two numbers have many common multiples: {24, 48, 72, 96, . . .}. Because the least of the common multiples is 24, we say that LCM(8, 12) = 24.

Least Common Multiple Whenever we examine two natural numbers, we can create a set of numbers called their common multiples. The least of these common multiples is called the least common multiple (LCM). We use the notation LCM (a, b) to express the LCM of two natural numbers, a and b. Exploring the least common multiple concept Looking at the process we used to find the LCM, the active reader might think at this point, “Yes, this will work, but what if the numbers are larger?

Least Common Multiple What can we say about the size of the LCM of any two numbers a and b? The LCM of two numbers is generally greater than either of them, because it has to be a multiple of each. However, there are some cases in which the LCM of two numbers is not greater than either of them.

Least Common Multiple When one number is a multiple of the other, the LCM will be equal to the larger number; for example, LCM(5, 10) = 10. Now let me give you two more numbers. Without doing any computation, what can you tell me about the LCM of 18 and 40? Here are several possible responses: • It is greater than 40. • It is less than or equal 18  40 to or 720. • It is a multiple of both numbers. • It ends in a zero, because all multiples of 40 end in 0.

Strategies for Finding the LCM

Strategies for Finding the LCM Now let us examine how we might find the actual LCM of 18 and 40. Again, there are many ways to determine the LCM of two numbers. We will examine several ways. One way of determining the LCM of 18 and 40 is to construct the LCM by beginning with one of the numbers and applying our understanding of LCM.

Strategies for Finding the LCM This process is illustrated in the following discussion: Reasoning The work The LCM must contain all 18 = 2  3  3, the factors of 18. LCM(18, 40) must contain: 2  3  3 Now, in order also to be a 40 = 2  2  2  5 multiple of 40, the LCM will have to contain all the factors in the prime factorization of 40.

Strategies for Finding the LCM Reasoning The work Looking now at the factors of 18, LCM(18, 40) what factors of 40 are we missing? must contain: We need to put two more 2s and 2  2  2  3  3  5 one 5 into our prime factorization That is, of LCM(18, 40). LCM(18, 40) = 360.

Strategies for Finding the LCM Reasoning The work Another way of illustrating this process is to note the prime factorizations of 18 and 40 and realize that the least common multiple must contain all the factors in either number with no redundancies. That is, the LCM needs to contain three 2s, two 3s, and one 5.

Using Prime Factorization and Exponents to Find the LCM

Using Prime Factorization and Exponents to Find the LCM There is a more formal way to find the LCM, and this is connected to one of the ways in which we found the GCF. This method comes from representing the prime factorization of each number in exponential form: 18 = 2  3  3 = 21  32 40 = 2  2  2  5 = 23  51 When finding the GCF, we took the smaller exponent of all common factors.

Using Prime Factorization and Exponents to Find the LCM In order for a number to be the LCM, it must contain all the factors in either number. For example, because the prime factorization of 18 contains a 2, the LCM must contain a 2. However, because the prime factorization of 40 contains three 2s, the LCM must contain three 2s. Thus, when we examine the prime factorization of each number, whenever there is a common factor, we must take the greater exponent. Using this method, we find that LCM(18, 40) = 23  32  51.

Using Prime Factorization and Exponents to Find the LCM The only factor that 18 and 40 have in common is 2, and the greatest exponent above 2 is 3 (meaning that 2  2  2 is a factor of 40). Therefore, the prime factorization of the LCM must contain 23. The noncommon factors are 3 and 5, so 32 and 51 are also placed in the prime factorization of the LCM.

Investigation C – Relationships Between the GCF and the LCM It is important to note that although the GCF and LCM are different concepts, they are also closely related. One of the goals of the next few investigations is for you to deepen your understanding of each of these concepts and also to understand more deeply how they are related.

Investigation C – Discussion If you take any two numbers, the product of those numbers is always identical to the product of their GCF and LCM; that is, for any natural numbers a and b, GCF(a, b)  LCM(a, b) = a  b Using Venn diagrams Venn diagrams can help us to understand why this relationship GCF(a, b)  LCM(a, b) = a  b is true. However, using these diagrams here requires a slightly unorthodox use of the concepts of set, intersection, and union.

Investigation C – Discussion cont’d Let us examine this connection among GCF, LCM, and sets. Consider the prime factorization of 18 and 40 as sets; that is, Prime factorization of 18 = {2, 3, 3} Prime factorization of 40 = {2, 2, 2, 5}

Investigation C – Discussion cont’d If we place these elements in a Venn diagram, we have Figure 4.11. How does this Venn diagram connect to the GCF of 18 and 40? How does it connect to the LCM of 18 and 40? Figure 4.11

Investigation C – Discussion cont’d From this unorthodox representation, the GCF of 18 and 40 can be seen as the intersection of the two sets; that is, GCF(18, 40) = 2. The LCM of 18 and 40 can be seen as the union of the two sets; that is, LCM(18, 40) = 3  3  2  2  2  5 = 360. In this case, we see that the factors fall within three distinct regions of the Venn diagram. The first region contains those numbers that are factors of 18 but not of 40(3  3), the second region contains common factors (2), and the third region contains those numbers that are factors of 40 but not of 18(2  2  5).

Investigation C – Discussion cont’d Now examine the equation connecting the two concepts in this light: 18  40 = GCF(18, 40)  LCM(18, 40). We can see that 18 and 40 both consist of numbers from two regions, the GCF consists of numbers from the common region, and the LCM consists of numbers from all three regions.

Yet Another Way to Find the LCM

Yet Another Way to Find the LCM This relationship between the GCF and the LCM has a very practical application. It yields another way to find the LCM of two numbers. Consider, for example, finding LCM(40, 72). The equation in Investigation C is an algebraic equation, and thus we can use the rules of algebra to solve the equation for LCM(a, b). Doing so, we have

Yet Another Way to Find the LCM As you may have already discovered, finding the GCF of two numbers is generally much easier than finding the LCM. Thus, to find LCM(40, 72), we have only to find GCF(40, 72) and then solve the equation. Because GCF(40, 72) = 8, we have If you first divide the 72 by 8, you then have the relatively simple multiplication problem 40  9, which is 360.

Investigation D – Going Deeper into the GCF and the LCM The following problem provides an opportunity to develop your problem-solving toolbox while applying your understanding of GCF, LCM, and their relationship. If the GCF of 45 and x is 9, and the LCM of 45 and x is 135, find x. Discussion: There are several different ways to solve this problem.

Investigation D – Discussion cont’d Strategy 1: Think Of The Characteristics Of GCF and LCM First, let’s look at the other two numbers. 45 = 3  3  5 135 = 3  3  3  5

Investigation D – Discussion cont’d If 9 is the GCF of 45 and x, that means x is a multiple of 9. If the LCM of 45 and x is 135, that means that the prime factorization of x has to have at least three 3s. Using this line of reasoning, we find that our first candidate for the answer is a multiple of 9 that has three 3s: 3  3  3 = 27 It also turns out, not surprisingly, that 27 is the correct answer.

Investigation D – Discussion cont’d Strategy 2: Build x From The Ground Up We could also proceed by building this number x. Since we know that the GCF of 45 and x is 9, we know that x must be at least 9, that is 3  3. Since we know that the LCM of 45 and x is 135, we know that: 45 = 3  3  5 x = 3  3  ? 135 = 3  3  3  5

Investigation D – Discussion cont’d Because 135 has 3  3  3, x must be at least 3  3  3 = 27. The only other possibility now for x would be 3  3  3  5, but the GCF of 45 and 135 could be 45, so x cannot be 135.

Relationships of Operations on the Set of Natural Numbers

Relationships of Operations on the Set of Natural Numbers One of the important “big ideas” of arithmetic is the part—whole aspect of numbers—which is connected to how numbers are decomposed and composed. In the lower elementary grades, the primary focus is on additive decompositions. For example, 10 can be decomposed into 9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5,and so on.

Relationships of Operations on the Set of Natural Numbers This decomposition is a critical part of knowing how to add and subtract when regrouping and is a critical part of good mental arithmetic and estimating. In the upper elementary grades, understanding multiplicative decompositions is developed. For example, 12 can be decomposed multiplicatively in several ways—12  1, 6  2, and 4  3, as well as its prime decomposition, 2  2  3.

Relationships of Operations on the Set of Natural Numbers The figure shows that when we examine the way two numbers are related, we can decompose them additively or multiplicatively. Figure 4.12

Relationships of Operations on the Set of Natural Numbers In the former case, relating two numbers often leads to statements that use the words more than or less than. In the latter case, relating two numbers often leads to statements involving concepts and terms such as divisibility, ratio, and percent.