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Factors
Numbers that are multiplied together to give you another number are
factors of that number.
Example
6 × 5 = 30
6 and 5 are factors of 30.
Remember that the product of two numbers is called a multiple.
6 (factor) × 5 (factor)
=
30 (multiple)
30 is a multiple of both 5 and 6.
Factors are useful when determining how many
different ways these boxes can be stacked and
arranged to be quickly counted in a warehouse.
Numbers can be broken down to their factors. For example:
Represent 6 by showing the factors of 6:
6×1
3×2
1, 2, 3 and 6 are all factors of 6.
Represent 24 by showing the factors of 24:
24 × 1
12 × 2
6×4
8×3
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1, 2, 3, 4, 6, 8 and 12 are all factors of 24.
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Finding the Factors of a Number
You can use the steps below to find the factors of any whole number.
Step 1
1 and the number are always factors.
Step 2
Divide the number by 2 and see if you get a number without a decimal.
Step 3
•
If you get a whole number, then 2 and the whole
number you get are both factors of the number.
•
If you get a decimal number, then 2 is not a factor.
Keep dividing by consecutive numbers (3, 4, 5, 6, etc.) until you
have found all the factors.
You will know when all the factors have been found when you
divide by a number that you already have as a factor, or when you
get the same number as the number you are dividing by. Each
factor should only be listed once.
Examples
A) Identify the factors of 64
64 ÷ 1 = 64
64 ÷ 2 = 32
64 ÷ 3 = decimal number
64 ÷ 4 = 16
64 ÷ 5 = decimal number
64 ÷ 6 = decimal number
64 ÷ 7 = decimal number
64 ÷ 8 = 8
1 and 64 are pairs of factors of 64
2 and 32 are pairs of factors of 64
3 is not a factor of 64
4 and 16 are pairs of factors of 64
5 is not a factor of 64
6 is not a factor of 64
7 is not a factor of 64
8 is a factor of 64
The set of factors of a number are listed in order from LEAST to
GREATEST and separated by commas.
The set of factors of 64 is:
64: {1, 2, 4, 8, 16, 32, 64}
Notice that 8 × 8 = 64, but 8 is only listed once in the list of factors.
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B) Is 5 a factor of 125?
Looking at Hints for Dividing, we see the following.
Rule: 5 is a factor if the number ends in a 0 or 5.
Conclusion: 5 is a factor of 125.
To find another factor, we can divide 125 by 5.
125 ÷ 5 = 25
Conclusion: 5 and 25 are both factors of 125
Greatest Common Factors
Common factors are used in many ways. For example, we use common factors when
deciding on the largest size of tile to cover a floor without having any left over.
Example
What is the largest size of tile that will cover a
6 feet × 4 feet floor?
Factors of 6 are 1, 2, 3 and 6.
Factors of 4 are 1, 2 and 4.
The largest size of tile to cover the floor would be
2 feet by 2 feet.
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The Greatest Common Factor (GCF) of a pair of numbers is the largest factor
found in the factors sets of both the numbers.
Follow these steps to find the greatest common factor of two numbers.
1. List ALL the factors for both numbers in order from least to greatest.
2. Determine the largest factor that is common to both sets.
Example
Find the greatest common factor of 20 and 45.
The factors of 20 are: 1,
2,
4,
The factors of 45 are:
1,
3,
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Mathematics
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5,
5,
10,
9,
20.
The greatest common
factor of 20 and 45 is 5.
15, 45.
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Practice: Finding Factors
1. A factor is missing in each of the questions below.
Write in the missing factor.
a. 2 ×
b.
=
× 12
c. 9 ×
=
24
d. 7 ×
=
48
108
= 56
e.
× 7 = 77
f.
× 4 = 64
2. Use a calculator or another method to list the set of factors for the following
numbers.
a.
20
b.
45
c.
36
d.
80
E.g., 20: {1, 2, 4, 5, 10, 20}
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e.
24
f.
51
g.
90
h.
33
i.
48
j.
56
3. Name 3 sets of numbers (all less than 50) that have the same greatest common
factor.
4. Name the factors of each number. Find the greatest common factor of each pair of
numbers
Factors
a)
E.g.
Number
1, 2, 3, 4, 6, 9, 12, 18, 36
36
1, 2, 4, 7, 14, 28
28
Greatest Common
Factor
4
b)
21
49
c)
16
40
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5.
The art class is creating a wall mural. They want to cut a large, rectangular piece
of wood into equal pieces. Some art students will then take a piece to paint for
the mural. The wood is 14 feet by 6 feet. What is the largest size of piece the
students can cut to make all pieces equal?
6.
The period 2 CALM classes are meeting in the common room to hear a career
speaker. The speaker would like tables with equal groups of students. If class
B2 has 32 students and class D2 has 28 students, what number of students
should sit at each table for the least amount of tables?
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