1. The Least Common Multiple (LCM)

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. (LCM) The Searching Method
Methods of Finding LCM
The Minimum Coverage and The Construction Method

17. (LCM) The Searching Method
Example B. Find the LCM of 18, 24, 16.
Methods of Finding LCM
The Minimum Coverage and The Construction Method

18. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
Methods of Finding LCM
The Minimum Coverage and The Construction Method

19. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24:
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The largest one on the list.

20. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The largest one on the list.

21. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. Find the LCM of 18, 24, 16.
List the multiples of 24: 24, 48, 72, 96, 120, 144, …
Methods of Finding LCM
The Minimum Coverage and The Construction Method
The 1st one
that can also be
divided by 16&18.The largest one on the list.

22. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. 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
The Minimum Coverage and The Construction Method
The 1st one
that can also be
divided by 16&18.

23. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. 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
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
The 1st one
that can also be
divided by 16&18.

24. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. 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
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
The 1st one
that can also be
divided by 16&18.
Following are two concrete examples that demonstrate
the principle of constructing the “the minimum coverage”.

25. (LCM) The Searching Method
To find the LCM of a list of numbers, test the multiples of
the largest number, one by one, until we find the LCM.
Example B. 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
The Minimum Coverage and The Construction Method
The construction method is based on the principle of building
"the minimal (smallest, least) coverage which is necessary
to fulfill assorted requirements or conditions."
Following are two concrete examples that demonstrate
the principle of constructing the “the minimum coverage”.
We will construct the LCD utilizing this principle later.
The 1st one
that can also be
divided by 16&18.

26. The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list.
Methods of Finding LCM

27. The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Methods of Finding LCM

28. Methods of Finding LCM
The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?

29. 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 minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?

30. 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
Methods of Finding LCM
The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?

31. 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
Methods of Finding LCM
The minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?

32. 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 minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?

33. 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 minimal-coverage-construction method takes just
enough of each specification to build the minimum which
"covers" all the requirements on the list. The following is an
example that illustrates the principle behind this method .
Example B. 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?
form the least amount of required classes needed to be taken.

34. LCM and LCD
Here is another example of fulfilling "the minimal" requirements.

35. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
LCM and LCD
Here is another example of fulfilling "the minimal" requirements.

36. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
Here is another example of fulfilling "the minimal" requirements.

37. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
Here is another example of fulfilling "the minimal" requirements.

38. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
Here is another example of fulfilling "the minimal" requirements.

39. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
Here is another example of fulfilling "the minimal" requirements.

40. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
Here is another example of fulfilling "the minimal" requirements.

41. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots.
Here is another example of fulfilling "the minimal" requirements.

42. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots. Apu took 1 apple and 1 banana,
Here is another example of fulfilling "the minimal" requirements.

43. Example C. There are three identical
boxes with the same content.
Apu took some items from the 1st
box and what’s left is shown here.
Bolo took some items from the 2nd
box and what’s left is shown here.
Cato took some items from the 3rd
box and what’s left is shown here.
What is the least amount of items
possible in box originaIly?
What’s the least amount of items
that each person took?
LCM and LCD
What’s left after
Apu took some
items.
What’s left after
Bolo took some
items.
What’s left after
Cato took some
items.
The least amount of items in the box consist of 2 apples,
5 bananas and 4 carrots. Apu took 1 apple and 1 banana,
Bolo took 2 carrots, and Cato took 2 banana and 1 carrot.
Here is another example of fulfilling "the minimal" requirements.

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

45. But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD

46. Example D. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

47. To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

48. To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

49. To construct the LCM:
a. Factor each number completely
Example D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

50. 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 D. Construct the LCM of {8, 15, 18}.
Factor each number completely,
8 = 23
15 = 3 * 5
18 = 2 * 32
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

51. 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 D. 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:
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

52. 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 D. 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,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must be divisible by 8 = 23
so it must have the factor 23

53. 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 D. 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
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must be divisible by 18 = 2*32
so it must have the factor 32
(we have enough 2’s already)

54. 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 D. 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,
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method
The LCM must has the factor 5
to be divisible by 15 = 3*5
(we have enough 3’s already)

55. 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 D. 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, so the LCM{8, 15, 18} = 23*32*5 = 8*9*5 = 360.
But when the LCM is large, the listing method is cumbersome.
It's easier to find the LCM by constructing it instead because
the LCM is a type of minimal coverage of the list of numbers.
LCM and LCD
The LCM–Construction Method

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

57. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM =
Methods of Finding LCM

58. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3
Methods of Finding LCM

59. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3
Methods of Finding LCM

60. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4
Methods of Finding LCM

61. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM

62. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking the highest power of each
factor. The LCM = x3y3z4w.
Methods of Finding LCM
c. Construct the LCM of x2 – 3x + 2, x2 – x – 2

63. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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.

64. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)

65. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)

66. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM =

67. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)

68. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)

69. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

70. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

71. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

72. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

73. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

74. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

75. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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
Taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

76. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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
Taking 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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

77. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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
Taking the highest power of each factor to get the
LCM = x2(x + 2)(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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

78. b. Construct the LCM of x2y3z, x3yz4, x3zw
All quantities are factored, taking 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
Taking the highest power of each factor to get the
LCM = x2(x + 2)(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)
Taking the highest power of each factor to get the
LCM = (x – 2)(x – 1)(x +2)

79. Clearing Denominator with LCD

80. The LCD is used for clearing denominators to simplify fractional
calculations.
Clearing Denominator with LCD

81. Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.

82. Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

83. Example E.
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

84. Example E.
Their LCD is (x – 2)(x + 4).
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

85. Example E.
Their LCD is (x – 2)(x + 4).
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2 (x – 2)(x + 4)( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

86. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2 (x – 2)(x + 4)
x + 1
( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

87. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2 (x – 2)(x + 4)
x + 1
( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

88. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2 (x – 2)(x + 4)
x + 1
( )
= (x + 1)(x + 4)
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

89. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2 (x – 2)(x + 4)
x + 1
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

90. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2
x + 4
2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

91. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2
x + 4
2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
x + 4
2x – 1
(x – 2)(x + 4)( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

92. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2
x + 4
2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
= (2x – 1)(x – 2)x + 4
2x – 1
(x – 2)(x + 4)( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

93. Example E.
Their LCD is (x – 2)(x + 4).
x– 2
( ) (x – 2)(x + 4)
a. Find the LCD of
x + 1
x – 2 , x + 4
2x – 1
b. Simplify
x + 1
x – 2
x + 4
2x – 1
(x – 2)(x + 4)
c. Simplify (x – 2)(x + 4)
x + 1
( )
( )
= (x + 1)(x + 4)
= x2 + 5x + 4
= (2x – 1)(x – 2)
= 2x2 – 5x + 2
x + 4
2x – 1
(x – 2)(x + 4)( )
Clearing Denominator with LCD
The LCD is used for clearing denominators to simplify fractional
calculations. Likewise the LCD is utilized in the same manner
and purposes for simplifying rational expression calculations.
Here is a typical sequence of steps of rational operations.

94. x– 2
–
x + 4
( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
Clearing Denominator with LCD

95. x– 2
–
x + 4
)(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD

96. x– 2
–
x + 4
( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD

97. x– 2
–
x + 4
( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD

98. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
Clearing Denominator with LCD

99. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
Clearing Denominator with LCD

100. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2

101. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2

102. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2
x– 2
–
x + 4
x + 1 2x – 1
e. Combine

103. = (x + 1)(x + 4) – (2x – 1)(x – 2)
x– 2
–
x + 4
( )(x – 2)(x + 4)
Clearing Denominator with LCD
d. Simplify
x + 1 2x – 1
(x + 4) (x – 2)
x – 2
–
x + 4
( ) (x – 2)(x + 4)x + 1 2x – 1
= x2 + 5x + 4 – [2x2 – 5x + 2]
= x2 + 5x + 4 – 2x2 + 5x – 2
= – x2 + 10x + 2
x– 2
–
x + 4
x + 1 2x – 1
This is the next topic.
e. Combine

104. 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.

105. Multiplication and Division of Rational Expressions
33. x – 2
x2 – 9
( –
x + 1
x2 – 2x – 3
)
34. x + 3
x2 – 4
( – 2x + 1
x2 + x – 2
)
35. x – 1
x2 – x – 6
( –
x + 1
x2 – 2x – 3 )
36.
x + 2
x2 – 4x + 3
( – 2x + 1
x2 + 2x – 3)
29. 4 – x
x – 3
( –
x – 1
2x + 3
) * LCD
30. 3 – x
x + 2
( –
2x + 3
x – 3
)
Ex B. Find the LCD of the terms then expand and simplify
the following expressions. (These are from the last section.)
31. 3 – 4x
x + 1
( – 1 – 2x
x + 3
)
32. 5x – 7
x + 5
( –
4 – 5x
x – 3
)
* LCD
* LCD
* LCD
* LCD
* LCD
* LCD
* LCD