the least common multiple

1. The Least Common Multiple (LCM)
Frank Ma © 2011

2. The Least Common Multiple (LCM)
We say m is a multiple of x if m can be divided by x.

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Searching Method
Methods of Finding LCM

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. b. Construct the LCM of x2y3z, x3yz4, x3zw
Methods of Finding LCM

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Ex. A. Find the LCM of the given expressions.
Methods of Finding LCM
xy3, xy1. x2y, xy32. xy3, xyz3.
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.