1. FACTors
and
Multiples

2. Factors
Factors: are whole numbers (not including 0) that are multiplied together to give a product
- divides into another whole number evenly with no remainder (= NO DECIMALS!)
Sometimes represented by F(x) or F(X)
5 x 2 = 10
Factors Product
Product: is the answer to a multiplication
question

3. 56 ÷ 8 = 7 Factor Factor – a number that divides evenly into another.
Factors
Factor – a number that divides evenly into another.
56 ÷ 8 = 7
Factor

4. Factors Factoring – is breaking a number down into all its factors
Examples
Factors of 6: F(6) = (1x6), (2x3) = 1,2,3,6
Factors of 24: F(24) = (1x24), (2x12), (3x8), (6x4),
1, 2, 3, 4, 6, 8, 12, 24

5. Factors
For Example:
F(12) the factors of twelve are F(12) = 1, 2, 3, 4, 6, 12 because:
1x12 = x6= x4=12
All these numbers can multiply together to give a product of twelve.
F(7) = 7, 1
HINT *** 1 and the number itself are always factors. ***

6. What are the Factors?
6 x 7 = 42
6 & 7
7 x 9 = 63
7 & 9
8 x 6 = 48
8 & 6
4 & 9
4 x 9 = 36

7. What are the Factors?
42 ÷ 7 = 6 7
63 ÷ 9 = 7 9
48 ÷ 6 = 8 6 9
36 ÷ 9 = 4

8. Factor Rainbows
One way to visualize all the factors of a number is to use factor rainbow.
Factor Rainbow
For Example: List all the factors of 20
F(20) = 1x20, 2x10, 4x5
= 1, 2, 4, 5, 10, 20
HINT: It helps to list the factors from smallest to largest.

9. Examples Find the factors (F) of the following:
a). F(6) = (1x6), (2x3)
= 1, 2,3, 6
b). F(36) = (1x36), (2x18), (3x12), (4x9), (6x6)
= 1, 2, 3, 4, 6, 9, 18, 36
c). F(18) = (1x18), (2x9), (3x6)
= 1, 2,3, 6, 9, 18
d). F(3) = (1x3)
= 1, 3
e). F(48) = (1x48), (2x24), (4x12), (6x8)
= 1, 2, 4, 6, 8, 12, 24

10. IN Class Practice Find the factors of the following numbers a). 5
j). 21

11. Find the factors of the following numbers
a). 5 F(5) = 1, 5
b) F(30) = 1,2, 3, 5, 6, 10, 15, 30
c). 4 F(4) = 1, 2, 4
d). 12 F(12) = 1, 2, 3, 4, 6, 12
e) F(8) = 1, 2, 4, 8
f) F(15) = 1, 3, 5, 15
g) F(36) = 1, 2, 3, 4, 6, 9, 12, 18, 36
h) F(40) = 1, 2, 4, 5, 8, 10, 20, 40
i) F(18) = 1, 2, 3, 6, 9, 18
j) F(21) = 1, 3, 7, 21

12. Practice List all the factors of each number 32 19 35 17 38 46 27 2533 12 16 45

13. Practice List all the factors of each number. 13. 39 14. 26 15. 37
16. 28
17. 93
18. 97
19. 69
20. 18
21. 85
22. 42
23. 72
24. 15

14. Practice List all the factors of each number
32 F(32) = 1x32, 2x16, 4x8 = 1, 2, 4,8, 16, 32
19 F(19) = 1x = 1, 19
35 F(35) = 1x35, 5x = 1, 5, 7, 35
F(17) = 1x = 1, 17
F(38) 1x38, 2x = 1, 2, 19, 38
F(46) = 1x46, 2x = 1, 2, 23,46
F(27) = 1x27, 3x = 1, 3, 9, 27
F(25) = 1x25, 5x = 1, 5, 25
F(33) = 1x33 3x = 1, 3, 11, 33
F(12) = 1x12,2x6,3x4 = 1, 2, 3, 4, 6, 12
F(16) 1x16,2x8, 4x = 1, 2, 4, 8, 16
F(45) 1x45, 3x15, 5x9 = 1, 3, 5, 9, 15, 45

15. Practice List all the factors of each number.
F(39) = 1x39, 3x = 1, 3, 13, 39
F(26) = 1x26, 2x = 1, 2, 13, 26
F(37) = 1x = 1, 37
F(28) = 1x28, 2x14, 4x7 = 1, 2, 4, 7,14, 28
F(93) = 1x93, 3x = 1, 3, 31, 93
F(97) = 1x = 1, 97
F(69) = 1x69, 3x = 1, 3, 23, 69
F(18) = 1x18, 2x9, 3x6 = 1, 2,3, 6, 9, 18
F(85) = 1x85 5x = 1, 5, 17, 35
F(42) = 1x42, 2x21, 6x7 = 1, 2, 6, 7, 21, 42
F(72) = 1x72, 2x36, 3x24, 6x12, 8x9
= 1,2,3,4,6,8,9,12,24,36,72
F(15) = 1x15, 3x5 = 1, 3, 5, 15

16. The Greatest Common Factor
The Greatest Common Factor (GCF) is the largest factor that divides all the numbers
To find the Greatest Common Factor (GCF) of two numbers:
List all the factors of each number
If there are no common factors, the GCF is 1
For Example
F(18) = 1, 2, 3, 6, 9, 18 F(25) = 1, 5, 25
F(24) = 1, 2, 3, 4, 6, 8, 12, 24 F(17) = 1, 17
The GCF is 6 The GCF is 1

17. Class Practice Find the Greatest Common Factor of each set.
12 & F(12) = 1, 2, 3, 4, 6, 12 The GCF = 3
F(15) = 1, 3, 5, 15
4 & 9
3. 16 & 24
4. 30 & 18
12, 30 & 9

18. Class Practice Find the Greatest Common Factor of each set.
12 & F(12) = 1, 2, 3, 4, 6, 12 The GCF = 3
F(15) = 1, 3, 5, 15
4 & F(4) = 1, 2, 4 The GCF = 1
F(9) = 1, 3, 9
3. 16 & F(16) = 1, 2, 4, 8, The GCF = 8
F (24) = 1, 2, 3, 4, 6, 8, 12, 24
4. 30 & F(30) 1,2,3,5,6,10,15, The GCF = 6
F(18) 1,2,3, 6,9,18
5. 12, 30 & 9 F(9) = 1, 3, The GCF = 3
F(30) 1,2, 3,5,6,10,15,30
F(12) = 1, 2, 3, 4, 6, 12

19. For Example: 4 x 6 = 24 means add four groups of 6
Multiplication
Multiplication: is the repeated addition of groups of numbers
For Example: 4 x 6 = means add four groups of 6
  +  
= 24
6 x 4 = 24
 +  + 
= 9
3x3 = 9

20. Multiplication Multiplication can also be expressed using arrays
An array is an arrangement of columns and rows
Can also be called a matrix.
Show how multiplication can be shown as repeated addition
Shows how division can be shown as fair shares.
Columns
Rows ●●●● Rows x Columns
●●●● = 5 x 4
●●●● = 20
●●●● Therefore = 20 dots in the array altogether
●●●●

21. How could you show your work?
Rectangles Question
Find all the possible whole number lengths and widths of rectangles with each area (A=Length x Width) given below.
a) 12 cm2
b) 20 cm2
c) 17 cm2
d) 24 cm2
How could you show your work?

22. Rectangles Question - Answers
12 cm2
OPTION A: USE ARRAYS
●● ●●●● ●●●●●●●●●●●●●
●● ●●●● x 12 = 12 cm2
●● ●●●●
●● x4 =12 cm2 ●●
6x2 = 12 cm2
OPTION B: USE SHAPES
1 x 12 = 12 cm2
6x2 = 12 cm x4 = 12 cm2
12 cm
4cm
1cm
6cm
3cm
2cm

23. Rectangles Question - Answers
B) 20 cm2
2 x 10 = 20 cm x 4 = 20 cm x 20 = 20 cm2
20 cm
1cm
5cm
10 cm
4cm
2cm

24. Rectangles Question - Answers
C) 17 cm2
1 x 17 = 17 cm2
17 cm
1cm

25. Rectangles Question - Answers
D) 24 cm2
2 x 12 = 24 cm2 6 x 4 = 24 cm2 3 x 8 = 24 cm x 24 = 24 cm2
3cm
24 cm
1cm
4cm
12 cm
8cm
6cm
2cm

26. How could you show your work?
Rectangles Question
Find all the possible whole number lengths and widths of rectangles with each area (A=Length x Width) given below.
a) 12 cm2 = 1cm x 12cm, 2x6, 3x4
b) 20 cm2 = 1cm x 20cm, 4x5, 2x10
c) 17 cm2 = 1cm x 17cm
d) 24 cm2 = 1cm x 24cm, 2x12, 3x8, 4x6
How could you show your work?

27. Multiples
Multiples is the product of a number and any other whole number.
Zero (0) is a multiple of every number
Sometimes represented by M(x) or M(x)
For Example
Multiples of 4: M(4) = (4x0), (4x1), (4x2), (4x3), (4x4), (4x5),
= 0, 4, 8, 12, 16, 20, 24 … …
Multiples of 6: M(6) = (6x0), (6x1), (6x2), (6x3), (6x4), (6x5), …
= 0, 6, 12, 18, 24, 30, …
A common multiple of 4 and 6 is 24

28. IN CLASS PRACTICE List the first 7 multiples of each number M (5) =

29. IN CLASS PRACTICE List the first 7 multiples of each number

30. The Lowest Common Multiple (LCM)
The Lowest Common Multiple (LCM) of two numbers is the
smallest number (not zero) that is a multiple of all numbers.
To find the LCM of two numbers:
List the multiples of each number.
Sometimes, the LCM will be the numbers multiplied together.
For Example:
M(2) = 2, 4, 6, 8, 10,12, … M(3) = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, …
M(3) = 3, 6, 9, 12, 15, … M(8) = 8, 16, 24, 32 40, 48 …
The LCM = 6 The LCM = 24

31. CLASS PRACTICE Find the Lowest Common Multiple (LCM) for each pair.
6 and 8
13 and 2
15 and 9

32. CLASS PRACTICE Find the Lowest Common Multiple (LCM) for each pair.
6 and 8 M(6) = 6, 12, 18, 24, 30, 36, 42, 48, 54
M(8) = 8, 16, 24, 32
LCM = 24
b) 13 and 2 M(13) = 13, 26, 39
M(2) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26
LCM = 26
15 and 9 M(15) = 15, 30, 45
M(9) = 9, 18, 27, 36, 45, 54
LCM = 45

33. GCF and LCM Problem Solving
How can you tell if a word problem requires you to use Greatest Common Factor or Least Common Multiple to solve?

34. If it is a GCF Problem What is the question asking us?
Do we have to split things into smaller sections?
Are we trying to figure out how many people we can invite?
Are we trying to arrange something into rows or groups?

35. GCF Problem - Example
Samantha has two pieces of cloth. One piece is 72 cm wide and the other piece is 90 cm wide. She wants to cut both pieces into strips of equal width that are as wide as possible.
How wide should she cut the strips?

36. Key Information The pieces of cloth are 72 and 90 cm wide.
How wide should she cut the strips so that they are the largest possible equal lengths?

37. Key Information
This problem can be solved using Greatest Common Factor because we are cutting or “dividing” the strips of cloth into smaller pieces (factor) of 72 and 90.
Find the GCF of 72 and 90

38. Pictures
Fabric #1: 90 cm
Fabric #2 72 cm

39. Calculations F(90) = 1x90, 2x45, 5x18, 6x15, 9x10
= 1, 2, 5, 6, 9, 10, 15, 18, 45, 90
F(72) = 1x72, 2x36, 3x24, 4x18, 6x12,
= 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
GCF = 18
Samantha should cut each piece to be 18 cm wide

40. Concluding Sentence
Samantha should cut each piece to be 18 cm wide

41. If it is a LCM Problem What is the question asking us?
Do we have an event that is or will be repeating over and over?
Will we have to purchase or get multiple items in order to have enough?
Are we trying to figure out when something will happen again at the same time?

42. LCM Problem - Example
Ben exercises every 12 days and Isabel every 8 days. Ben and Isabel both exercised today.
How many days will it be until they exercise together again?

43. Key Information
Ben exercises every 12 days and Isabel every 8 days and they both exercised today.
How many days is it until they will both exercise on the same day again.

44. Key Information
This problem can be solved using Least Common Multiple. We are trying to figure out when will be the next time they are exercising together.
Find the LCM of 12 and 8.

45. Calculations M(8) = 0, 8, 16, 24, 32,40, 48 M(12) = 0, 12, 24, 36, 48
LCM = 24

46. Concluding Sentence
Ben and Isabel would exercise on the same day every 24 days.

47. QUIZ!!!!!!
On a sheet of notebook paper, tell whether the following word problems could be solved using GCF or LCM…
Then solve each problem.

48. QUIZ Answers…
1.) GCF
2.) GCF
3.) LCM
4.) LCM
5.) LCM
6.) GCF

49. Question #1
Mrs. Evans has 120 crayons and 30 pieces of paper to give to her students. What is the largest # of students she can have in her class so that each student gets equal # of crayons and equal # of paper.

50. Question 1: Mrs. Evans and the Kindergarten Class (GCF)
Key Information
120 crayons
30 papers
- largest class size possible?
Pictures
120 crayons papers

51. Question 1: Mrs. Evans and the Kindergarten Class (GCF)
Calculations F(120) = 1x120, 2x60, 3x40, 4x30, 5x24, 6x20, 10x12 =1,2,3,4,5,6,10, 12,20,24,30, 40, 60, 120 F(30) = 1x30, 2x15, 3x10, 5x6 1, 2, 5, 6, 15, 30 The Greatest Common Factor = 30

52. Question 1: Mrs. Evans and the Kindergarten Class (GCF)
Concluding Sentence  The largest number of students Mrs. Evans can have in her class is 30. Each student will get 1 paper and 4 crayons.

53. Question #2
Rosa is making a game board that is 16 inches by 24 inches. She wants to use square tiles. What is the largest tile she can use?

54. Question 2: Rosa’s Game (GCF)
Key Information
16 x 24 inches board
Square tiles
- largest tile possible?
Pictures
24 inches
16 inches

55. Question 2: Rosa’s Game (GCF)
Calculations F(16) = 1x16, 2x8, 4x4 = 1, 2, 4, 8, 16 F(24) = 1x24, 2x12, 3x8, 4x6 = 1, 2, 3, 4, 6, 8, 12, 24 The Greatest Common Factor = 8

56. Question 2: Rosa’s Game (GCF)
Concluding Sentence  The largest tile that Rosa can use for her game is 8 inches.

57. Question #3
Z100 gave away a Z $100 bill for every 100th caller. Every 30th caller received free concert tickets. How many callers must get through before one of them receives both a coupon and a concert ticket?

58. Question 3: Radio Contest (LCM)
Key Information
$100 bill for 100th caller
Free Concert Ticket for 30th caller
Who gets both free ticket and $100?
Pictures
30th Caller
100th Caller

59. Question 3: Radio Contest (LCM)
Calculations M(100) = 100, 200, 300, 400 M(30) = 30, 60, 90, 120, 150, 180, 210, 240, 270, 300 The Lowest Common Multiple (LCM) = 300

60. Question 3: Radio Contest (LCM)
Concluding Sentence  The 300th caller would be the first to receive concert tickets and $100.

61. Question #4
Two bikers are riding a circular path. The first rider completes a round in 12 minutes. The second rider completes a round in 18 minutes. If they both started at the same place and time and go in the same direction, after how many minutes will they meet again at the starting point?

62. Question 4: The Bikers (LCM)
Key Information
Biker # 1 – 12 minutes/round
Biker #2 – 18 minutes/round
- When will they be at start again?
Pictures
Biker #2
18 minutes/round
Biker #1
12 minutes/round

63. Question 4: The Bikers (LCM)
Calculations M(12) = 12, 24, 36, 48 M(18) = 18, 36, 54, 72 The Lowest Common Multiple (LCM) is 36.

64. Question 4: The Bikers (LCM)
Concluding Sentence  The next time the two bikers will meet at start will be in 36 minutes.

65. Question #5
Sean has 8-inch pieces of toy train track and Ruth has 18-inch pieces of train track. How many of each piece would each child need to build tracks that are equal in length?

66. Question 5: Toy Trains (LCM)
Key Information
Sean: 8 inch track
Ruth: 18 inch track
# of parts to build tracks equal length?
Pictures
Sean: 8 inches
Ruth: 18 inches

67. Question 5: Toy Trains (LCM)
Calculations M(8) = 8, 16, 24, 32, 40, 48, 56, 64, 72 M(18) = 18, 36, 54, 72, 90, 108, The Lowest Common Multiple (LCM) is 72.

68. Question 5: Toy Trains (LCM)
Concluding Sentence  Ruth and Sean would need 72 pieces to make tracks of equal length.

69. Question #6
I am planting 50 apple trees and 30 peach trees. I want the same number and type of trees per row. What is the maximum number of trees I can plant per row?

70. Question 6: The Orchard (GCF)
Key Information
50 apple trees
30 peach trees
How many of same trees in each row?
Pictures
50 Trees
30 Trees

71. Question 6: The Orchard (GCF)
Calculations F(50) = 1x50, 2x25, 5x10 =1, 2, 5, 10, 25, 50 F(30) = 1x30, 2x15, 3x10, 5x6 = 1, 2, 3, 5, 6, 10, 15, 30 The Greatest Common Factor = 10

72. Question 6: The Orchard (GCF)
Concluding Sentence  The largest number of trees per row is 10.