1. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 2.3 Least Common Multiple

2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Understand the meaning of the term least common multiple. o Use prime factorizations to find the LCM of a set of numbers. o Recognize the application of the LCM concept in a word problem.

3. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Least Common Multiple (LCM) The least common multiple (LCM) of two (or more) whole numbers is the smallest number that is a multiple of each of these numbers. Least Common Multiple (LCM)

4. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. To Find the LCM of a Set of Counting Numbers 1.Find the prime factorization of each number. 2.Identify the prime factors that appear in any one of the prime factorizations. 3.Form the product of these primes using each prime the most number of times it appears in any one of the prime factorizations. Finding the Least Common Multiple (LCM)

5. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM of 20 and 45. Solution Step 1:Prime factorizations: Step 2:2, 3 and 5 are the only prime factors. Example 1: Finding the Least Common Multiple (LCM)

6. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Step 3:Most of each factor in any one factorization: two 2s(in 20) two 3s(in 45) one 5(one in 20 and one in 45) 180 is the smallest number divisible by both 20 and 45. (Note also that the LCM, 180, contains all the factors of the numbers 20 and 45.) Example 1: Finding the Least Common Multiple (LCM) (cont.)

7. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM of 12, 18, and 54. Solution Step 1:Prime factorizations: Step 2:2 and 3 are the only prime factors. Example 2: Finding the Least Common Multiple (LCM)

8. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Step 3:Most of each factor in any one factorization: four 2s(in 48) two 3s(in 18) Example 2: Finding the Least Common Multiple (LCM) (cont.)

9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM of 36, 24 and 48. Solution Step 1:Prime factorizations: Step 2:_____ and _____ are the only prime factors. 32 Completion Example 3: Finding the Least Common Multiple (LCM)

10. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Step 3:Most of each factor in any one factorization: _______(in 48) _______(in 36) So, LCM = _______ = _______. _______ is the smallest number divisible by the numbers 36, 24, and 48. two 3s four 2s 144 2 4 · 3 2 Completion Example 3: Finding the Least Common Multiple (LCM) (cont.) 144

11. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Finding the Least Common Multiple (LCM) Find the LCM of 27, 30, and 42. Solution

12. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM of 8 and 25. Solution In this case, where the two numbers have no common prime factors, the LCM is the product of the two numbers. Example 5: Finding the Least Common Multiple (LCM)

13. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM for 27, 30, and 42, and then state how many times each number divides into the LCM. Solution Recall from Example 4, Example 6: Prime Factorizations and the Least Common Multiple (LCM)

14. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Prime Factorizations and the Least Common Multiple (LCM) (cont.) So, 27 divides into 1890 70 times; 30 divides into 1890 63 times; 42 divides into 1890 45 times.

15. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Find the LCM for 12, 18, and 66, and then state how many times each number divides into the LCM. Solution Example 7: Prime Factorizations and the Least Common Multiple (LCM)

16. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Prime Factorizations and the Least Common Multiple (LCM) (cont.) So, 12 divides into 396 33 times; 18 divides into 396 22 times; 66 divides into 396 6 times.

17. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Application with Least Common Multiple (LCM) Suppose three weather satellites—A, B, and C—are orbiting the Earth at different times. Satellite A takes 24 hours, B takes 18 hours, and C takes 12 hours. If they are directly above each other now, as shown in part a. of the figure on the next slide, in how many hours will they again be directly above each other in the position shown in part a.? How many orbits will each satellite have made in that time?

18. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Application with Least Common Multiple (LCM) (cont.)

19. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Application with Least Common Multiple (LCM) (cont.) Solution Study the diagram on the previous slide. When the three satellites are again in the position shown in part a., each will have made some number of complete orbits. Since A takes 24 hours to make one complete orbit, the solution must be a multiple of 24. Similarly, the solution must be a multiple of 18 and a multiple of 12 to allow for complete orbits of satellites B and C. The solution is the LCM of 24, 18, and 12.

20. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Application with Least Common Multiple (LCM) (cont.) Thus, the satellites will align again at the position shown in 72 hours (or 3 days). Note that: Satellite A will have made 3 orbits: 24 ⋅ 3 = 72; Satellite B will have made 4 orbits: 18 ⋅ 4 = 72; Satellite C will have made 6 orbits: 12 ⋅ 6 = 72.

21. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems 1.Find the LCM for each of the following sets of numbers. a. 28, 70 b. 30, 40, 50 c. 18, 36, 66 2.Use prime factorizations to find the LCM of 168 and 140 and to tell how many times each number divides into the LCM.

22. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1.a. 140b. 600c. 396 2.840; 168 divides into 840 5 times, 140 divides into 840 6 times.