218 CHAPTER 5 Number Theory and the Real Number System Example 4 illustrates this procedure. Example 4 Using Prime Factorization to Determine the GCD Determine the GCD of 36 and 90. Solution The branching method of determining the prime factors of 36 and 90 is illustrated in Fig. 5.4. 1. The prime factorization of 36 is 2 3 , 2 2 ⋅ and the prime factorization of 90 is 2 3 5. 2 ⋅ ⋅ 2. The prime factors with the smallest exponents that appear in each of the factorizations of 36 and 90 are 2 and 3 .2 Note that because 5 is not in the prime factorization of 36, it is not included when determining the GCD. 3. The product of the factors determined in Step 2 is 2 3 2 9 18. 2 ⋅ = ⋅ = The GCD of 36 and 90 is 18. It is the largest natural number that divides both 36 and 90. 7 Now try Exercise 41 Did You Know? Friendly Numbers The ancient Greeks often thought of numbers as having human qualities. For example, the numbers 220 and 284 were considered “friendly” or “amicable” numbers because each number was the sum of the other number’s proper factors . (A proper factor is any factor of a number other than the number itself.) If you sum all the proper factors of 284 (1 2 4 71 142), + + + + you get the number 220, and if you sum all the proper factors of 220 (12 4 5101120 22 + + + + + + + + 44 55 110), + + you get 284. Example 5 Determining the GCD Determine the GCD of 315 and 450. Solution 1. The prime factorization of 315 is 3 5 7, 2 ⋅ ⋅ and the prime factorization of 450 is 2 3 5 . 2 2 ⋅ ⋅ You should verify these answers using either the branching method or the division method. 2. The prime factors with the smallest exponents that appear in each of the factorizations of 315 and 450 are 32 and 5. 3. The product of the factors determined in Step 2 is 3 5 9 5 45. 2 ⋅ = ⋅ = The GCD of 315 and 450 is 45. It is the largest natural number that will evenly divide both 315 and 450. 7 Now try Exercise 43 Figure 5.4 36 90 6 2 3 2 3 2 5 3 3 6 10 9 Least Common Multiple To perform addition and subtraction of fractions (Section 5.3), we use the least common multiple (LCM). We will now discuss how to determine the LCM of a set of numbers. Definition: Least Common Multiple The least common multiple (LCM) of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set. What is the least common multiple of 12 and 18? One way to determine the LCM is to list the multiples of each number: 36 72 108 144 36 72 108 144 Multiples of12 12, 24, , 48, 60, , 84, 96, , 120, 132, , Multiples of18 18, , 54, , 90, , 126, , 162, { } { } … … Some common multiples of 12 and 18 are 36, 72, 108, and 144, in bold above. The least common multiple, 36, is the smallest number that is divisible by both 12 and 18. Usually, the most efficient method of determining the LCM is to use prime factorization. Least Common Multiple and Greatest Common Divisor Instructor Resources for Section 5.1 in MyLab Math • Objective-Level Videos 5.1 • Interactive Concept Video: Primes and Perfect Numbers • Interactive Concept Video: Difference between Least Common Multiple and Greatest Common Factor • Animation: Least Common Multiple and Greatest Common Divisor • PowerPoint Lecture Slides 5.1 • MyLab Exercises and Assignments 5.1
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