The document discusses the least common multiple (LCM) of numbers. The LCM of two or more numbers is the smallest positive number that is a multiple of each number. Two methods are presented for finding the LCM: 1) listing the multiples of each number and finding their common multiples, and 2) factorizing each number and taking the highest power of each prime factor. Examples are provided to illustrate both methods. The least common denominator (LCD) of fractions is also introduced as being the LCM of the denominators.

1. LCM and LCD

2. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers.

3. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6.

4. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, …

5. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,…

6. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12,

7. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12.

8. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM.

9. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number,

10. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers.

11. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers. Example B. Find the LCM of 8, 9, and 12.

12. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers. Example B. Find the LCM of 8, 9, and 12. The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 …

13. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers. Example B. Find the LCM of 8, 9, and 12. The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 … The first number that is also a multiple of 8 and 9 is 72.

14. LCM and LCD Definition of LCM The least common multiple (LCM) of two or more numbers is the least number that is the multiple of all of these numbers. Example A. Find the LCM of 4 and 6. The multiples of 4 are 4, 8, 12, 16, 20, 24, … The multiples of 6 are 6, 12, 18, 24, 30,… The smallest of the common multiples is 12, then LCM{4, 6 } = 12. We may improve the above listing-method for finding the LCM. Given two or more numbers, we start with the largest number, list its multiples in order until we find the one that can divide all the other numbers. Example B. Find the LCM of 8, 9, and 12. The largest number is 12 and the multiples of 12 are 12, 24, 36, 48, 60, 72, 84 … The first number that is also a multiple of 8 and 9 is 72. Hence LCM{8, 9, 12} = 72.

15. LCM and LCD But when the LCM is large, the listing method is cumbersome.

16. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead.

17. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM:

18. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely

19. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations.

20. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product.

21. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}.

22. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely,

23. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23

24. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5

25. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32

26. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor:

27. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor:

28. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 *32 From the factorization select the highest degree of each prime factor

29. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 *5 18 = 2 *32 From the factorization select the highest degree of each prime factor: 23, 32, 5,

30. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

31. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.

32. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360. The LCM of the denominators of a list of fractions is called the least common denominator (LCD).

33. LCM and LCD But when the LCM is large, the listing method is cumbersome. It's easier to find the LCM by constructing it instead. To construct the LCM: a. Factor each number completely b. For each prime factor, take the highest power appearing in the factorizations. The LCM is their product. Example C. Construct the LCM of {8, 15, 18}. Factor each number completely, 8 = 23 15 = 3 * 5 18 = 2 * 32 From the factorization select the highest degree of each prime factor: 23, 32, 5, then LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360. The LCM of the denominators of a list of fractions is called the least common denominator (LCD). Following is an application of the LCM.

34. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6.

35. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. In picture: Mary Joe Chuck

36. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? In picture: Mary Joe Chuck

37. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck

38. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching.

39. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, …

40. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM.

41. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices

42. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 Joe gets 12* 3

43. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 Joe gets 12* = 4 slices 3

44. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* Joe gets 12* = 4 slices 4 3

45. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3

46. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3 1 Chuck gets 12* 6

47. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3 1 Chuck gets 12* = 2 slices 6

48. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3 1 Chuck gets 12* = 2 slices 6 In total, that is 4 + 2 + 3 = 9 slices,

49. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3 1 Chuck gets 12* = 2 slices 6 9 In total, that is 4 + 2 + 3 = 9 slices, or of the pizza. 12

50. LCM and LCD Example D. From one pizza, Joe wants 1/3, Mary wants 1/4 and Chuck wants 1/6. How many equal slices should we cut the pizza into and how many slices should each person take? What is the fractional amount of the pizza they want in total? In picture: Mary Joe Chuck We find the LCM of 1/3, 1/4, 1/6 by searching. The multiples of 6 are 6, 12, 18, 24, … Since 12 is also the multiple of 3 and 4, then 12 is the LCM. Hence we should cut it into 12 slices and 1 1 Mary gets 12* = 3 slices Joe gets 12* = 4 slices 4 3 1 Chuck gets 12* = 2 slices 6 9 3 In total, that is 4 + 2 + 3 = 9 slices, or of the pizza. = 12 4

51. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take?

52. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12

53. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator.

54. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem:

55. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b

56. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. 16

57. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. The new denominator is 48, 16

58. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. The new denominator is 48, then the new numerator is 48* 16 9 16

59. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. The new denominator is 48, then the new numerator is 48* 16 9 3 16

60. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. The new denominator is 48, then the new numerator is 48* = 27. 16 9 3 16

61. LCM and LCD Your Turn: From one pizza, Joe wants 3/8 of it, Mary wants 1/6 of it and Chuck wants 5/12 of it. How many equal slices should we cut the pizza and how many slices should each person take? 1 4 In the above example, we found that is the same . 3 12 The following theorem tells us how to convert the denominator of a fraction to a fraction with a different denominator. Multiplier Theorem: To convert the fraction into a fraction with denominator d, the new numerator is * d. a b a b 9 Example D: Convert to a fraction with denominator 48. The new denominator is 48, then the new numerator is 48* = 27. 16 9 3 9 27 Hence = 16 48 . 16

62. LCM and LCD Exercise A. Find the LCM. 1. a.{6, 8} b. {6, 9} c. {3, 4} d. {4, 10} 2. a.{5, 6, 8} b. {4, 6, 9} c. {3, 4, 5} d. {4, 6, 10} a.{6, 8, 9} b. {6, 9, 10} c. {4, 9, 10} d. {6, 8, 10} 4. a.{4, 8, 15} b. {8, 9, 12} c. {6, 9, 15} 5. a.{6, 8, 15} b. {8, 9, 15} c. {6, 9, 16} 6. a.{8, 12, 15} b. { 9, 12, 15} c. { 9, 12, 16} 7. a.{8, 12, 18} b. {8, 12, 20} c. { 12, 15, 16} 8. a.{8, 12, 15, 18} b. {8, 12, 16, 20} 9. a.{8, 15, 18, 20} b. {9, 16, 20, 24}

63. LCM and LCD B. Convert the fractions to fractions with the given denominators. 10. Convert to denominator 12. 11. Convert to denominator 24. 12. Convert to denominator 36. 13. Convert to denominator 60. 2 3 5 7 3 , 4 , 6 , 4 1 3 5 3 6 , 4 , 6 , 8 7 5 8 11 12 , 4 , 9 , 6 9 7 13 11 10 , 12 , 5 , 15