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### 1 5 the least common multiple

1. 1. The Least Common Multiple (LCM) Frank Ma © 2011
2. 2. The Least Common Multiple (LCM) We say m is a multiple of x if m can be divided by x.
3. 3. For example, 12 is a multiple of 2 The Least Common Multiple (LCM) We say m is a multiple of x if m can be divided by x.
4. 4. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) We say m is a multiple of x if m can be divided by x.
5. 5. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, We say m is a multiple of x if m can be divided by x.
6. 6. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
7. 7. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
8. 8. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. We say m is a multiple of x if m can be divided by x.
9. 9. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
10. 10. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
11. 11. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
12. 12. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
13. 13. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, and that 12xy is the LCM of 4x and 6y. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
14. 14. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, and that 12xy is the LCM of 4x and 6y. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, We give two methods for finding the LCM. I. The searching method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
15. 15. The least common multiple (LCM) of two or more numbers is the smallest common multiple of all the given numbers. For example, 12 is a multiple of 2. 12 is also a multiple of 3, 4, 6 or 12 itself. For example, 12 is the LCM of 4 and 6, xy is the LCM of x and y, and that 12xy is the LCM of 4x and 6y. The Least Common Multiple (LCM) A common multiple of two or more quantities is a multiple of all the given quantities. For example, 24 is a common multiple of 4 and 6, We give two methods for finding the LCM. I. The searching method II. The construction method xy2 is a multiple of x, xy2 is also a multiple of y or xy. xy2 is a common multiple of x and y. We say m is a multiple of x if m can be divided by x.
16. 16. Searching Method Methods of Finding LCM
17. 17. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Methods of Finding LCM
18. 18. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. Methods of Finding LCM
19. 19. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: Methods of Finding LCM
20. 20. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, Methods of Finding LCM
21. 21. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … Methods of Finding LCM
22. 22. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM
23. 23. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method:
24. 24. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method: Given two or more quantities, I. factor each quantity completely,
25. 25. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
26. 26. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
27. 27. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
28. 28. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
29. 29. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
30. 30. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
31. 31. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
32. 32. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
33. 33. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
34. 34. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
35. 35. Searching Method Given a list of quantities, list the multiples of the largest quantity and test them one by one until we find the LCM. Example A. Find the LCM of 18, 24, 16. List the multiples of 24: 24, 48, 72, 96, 120, 144, … so the LCM of {18, 24, 16} is 144. Methods of Finding LCM Example B. a. Construct the LCM of 18, 24, 20. Factor each number completely 18 = 2*32 24 = 233 20 = 225 Take the highest of the each factor, so the LCM is 23325 = 360. Construction Method: Given two or more quantities, I. factor each quantity completely, II. multiply the highest power of each factor that appears in the factorization to get the LCM.
36. 36. b. Construct the LCM of x2y3z, x3yz4, x3zw Methods of Finding LCM
37. 37. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. Methods of Finding LCM
38. 38. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = Methods of Finding LCM
39. 39. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3 Methods of Finding LCM
40. 40. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3 Methods of Finding LCM
41. 41. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4 Methods of Finding LCM
42. 42. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM
43. 43. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2
44. 44. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity.
45. 45. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1)
46. 46. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1)
47. 47. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM =
48. 48. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)
49. 49. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)
50. 50. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
51. 51. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
52. 52. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
53. 53. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
54. 54. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
55. 55. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
56. 56. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
57. 57. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2 c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
58. 58. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2(x – 2)2 c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
59. 59. b. Construct the LCM of x2y3z, x3yz4, x3zw All quantities are factored, Take the highest power of each factor. The LCM = x3y3z4w. Methods of Finding LCM d. Construct the LCM of x4 – 4x2, x2 – 4x +4 Factor each quantity. x4 – 4x2 = x2(x2 – 4) = x2(x + 2)(x – 2) x2 – 4x + 4 = (x – 2)2 Take the highest power of each factor to get the LCM = x2(x – 2)2(x + 2) c. Construct the LCM of x2 – 3x + 2, x2 – x – 2 Factor each quantity. x2 – 3x + 2 = (x – 2)(x – 1) x2 + x – 2 = (x + 2)(x – 1) Take the highest power of each factor to get the LCM = (x – 2)(x – 1)(x +2)
60. 60. The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. Methods of Finding LCM
61. 61. The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Methods of Finding LCM
62. 62. Methods of Finding LCM The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?
63. 63. Science Math. English Arts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Methods of Finding LCM The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?
64. 64. Science Math. English Arts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Methods of Finding LCM The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?
65. 65. Science Math. English Arts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need?
66. 66. Science Math. English Arts A 2 yrs 3 yrs 3 yrs B 3 yrs 3 yrs 2 yrs C 2 yrs 2 yrs 4 yrs 1 yrs Science-3 yr Math-3yrs English-4 yrs Arts-1yr Methods of Finding LCM The Construction Method takes just enough of each factor to make the LCM which "covers" all the quantities on the list. The following is an example that illustrates the principle behind this method . Example C. A student wants to take enough courses so she meets the requirement to apply to colleges A, B, and C. The following is a table of requirements for each college, how many years of each subject does she need? So we take the highest power of each factor to make the LCM.
67. 67. Ex. A. Find the LCM of the given expressions. Methods of Finding LCM xy3, xy1. x2y, xy32. xy3, xy3. 4xy3, 6xy4. x2y, xy3, xy25. 9xy3, 12x3y, 15x2y36. (x + 2), (x – 1)7. (x + 2), (2x – 4)8. (4x + 12), (10 – 5x)9. (4x + 6), (2x + 3)10. (9x + 18), (2x + 4)11. (–x + 2), (4x – 8) 12. x2(x + 2), (x + 2)13. (x – 1)(x + 2), (x + 2)(x – 2)14. (x – 3)(x – 3), (3 – x)(x + 2)17. x2(x – 2), (x – 2)x18. 19. x2(x + 2), (x + 2)(–x – 2)20.(x – 1)(1 – x), (–1 – x)(x + 1) (3x + 6)(x – 2), (x + 2)(x + 1)15. (x – 3)(2x + 1), (2x + 1)(x – 2)16. x2 – 4x + 4, x2 – 421. x2 – x – 6, x2 – x – 222. x2 + 2x – 3, 1 – x225. –x2 – 2x + 3, x2 – x26. x2 – x – 6, x2 – 923. x2 + 5x – 6, x2 – x – 624. 2x2 + 5x – 3, x – 4x327. x2 – 5x + 4, x3 – 16x28.