Free Pre-Algebra
Lesson 8  page 1
Lesson 8
Factor Pairs
Measuring more accurately requires breaking our inches into fractions of an inch, little parts smaller than a whole inch. You
can think ahead and see that once we’ve measured using fractions, we’ll need to be able to add and multiply the fractions in
order to find area and perimeter. Fractions are useful numbers
Why is this Funny?
in lots of situations.
Many, many people find working with fractions difficult, but bit
by bit we’ll overcome the reasons for this as we learn. The first
reason is that using fractions requires a thinking of numbers
multiplicatively, as products of their factors. Thinking
multiplicatively will be a great help as we learn fraction skills.
Remember that a product is the answer to a multiplication problem? The numbers that are multiplied are called factors. In
the equation 2 • 3 = 6, 2 and 3 are the factors, and 6 is the product. Some people remember the vocabulary with the
phrase “the factory makes the product.” After we use the terminology a bit in context it should feel more familiar.
A Multiplication Table
Memorization has gone out of fashion in schools, and many students are pretty rocky on their times tables. Some students
have learning disabilities that make memorization difficult, but if at all possible, it’s a good investment of time to refresh your
multiplication memories. When you have to stop and use your
calculator or think out simple multiplication and division problems
in the middle of long, multi-step math problems, it’s much easier
to lose your place and get confused. Your brain is overloaded,
because there is too much to figure out and pay attention to.
Memorizing and allowing the answers to become automatic frees
up your brain for the more complicated parts of the problem.
Try making a table like this yourself in a spreadsheet program.
You can make blank copies to fill in for practice, and you can use
one while you’re working on this homework.
Practice NOW!
Cover up the filled out table above and see how many you can
fill in yourself from memory.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 2
Idle Observations
As you’re filling in your multiplication table, you might notice yourself writing certain numbers several times. Here we see all
the 24s highlighted:
We can write the multiplication problems that produced 24 on the table:
3 • 8 = 24
and also 4 • 6 = 24
8 • 3 = 24
6 • 4 = 24
We say, “3 and 8 are factors of 24.” “24 is the product of 3 and 8.”
“4 and 6 are factors of 24.” “24 is the product of 4 and 6.”
The Area is the Product, the Sides are Factors
A concrete way to think about factors and products is
with rectangles.
A Question
Are these all the number pairs that multiply to give 24 for
the answer? If we were to extend the table, we’d find
another:
If you had 24 tiles to play with, could you find other
rectangles with area 24 square units?
2 • 12 = 24 and
12 • 2 = 24
If we extended it even further, we’d find yet another:
This is because 1 • 24 = 24 also.
Our goal in this lesson will be to list all the pairs of numbers that multiply to make some given number. Remember that
multiplication and division are related operations? We’ll use that relationship to find factor pairs.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 3
Divisibility
We say a number is divisible by another number if it divides evenly, that is, the answer is another whole number. For
example, 20 is divisible by 5, but 20 is not divisible by 7. Since 20 is divisible by 5, 5 is a factor of 20, and since 20 is not
divisible by 7, 7 is not a factor of 20.
Divisibility Pictures
Beyond the Multiplication Table
Refering to a large multiplication table is an inefficient way to find factor pairs. A better way is to use the
relationship between multiplication and division. If 8 • 3 = 24, then 24 ÷ 3 = 8. If we’re looking for factor
pairs that multiply to make 24, we should divide 24 by every possible factor.
The spreadsheet on the left divides 24 by each of the numbers 1 through 24. The highlighted rows are
whole number factors. The highlighted factor pairs are:
1 • 24
24 • 1
2 • 12
12 • 2
3•8
8•3
4•6
6•4
Notice that all of the pairs are represented in the first four entries. After that we get no new pairs, only the
same ones again, reversed. To be really efficient, it would be great to know when we can stop trying to
find new pairs, while still not missing any.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 4
Checking the Fewest Possible Factors
To the right is a copy of the same spreadsheet from the previous page. Look at the relationship between
the left-hand numbers, in the N column, and the right-hand numbers, in the 24 ÷ N column. Notice that for
the numbers highlighted in pink, the left-hand number is smaller than the right-hand number, but that
beginning at 5, the left-hand number is larger than the right-hand number.
Once the number on the right is smaller than the number on the left, there can be no new pairs. If the
number on the left divides evenly, the other factor is smaller, so it’s one we’ve already seen. With this
observation we find that we have to check many fewer possible factors.
Steps for Finding All the Factor Pairs of a Given Number
If you’re using a calculator or a spreadsheet, you can list numbers until you see that the numbers on the
right are larger than those on the left, then stop your columns there. Your table or spreadsheet would look
like this:
and you know that that’s enough to give you all the pairs, because 5 is multiplied by
a number smaller than 5 to make 24.
But there is a way to know that you need only check 1 through 4 before you even start. 52 is 25, which is more than 24.
Since 5 times itself is greater than 24, 5 has to be multiplied by a number less than itself to equal 24.
If the square of a number is greater than the number we are factoring, that number does not need to be checked as a factor.
The N column consists only of numbers with squares less than the given number.
Example: Find all the factor pairs that have 15 as a product.
Step 1: Figure out how many numbers you need to check by finding the first square greater than 15. Since 42 is 16, and 16
is greater than 15, you need only check the numbers less than 4, that is, 1 through 3.
Step 2: Make a table, with the numbers 1 – 3 on the left. You can write the column heads as was shown in
the spreadsheet, or you can just write 15 on top.
Step 3: Divide 15 by each of the numbers on the left. (15 ÷ 1, 15 ÷ 2, etc.) If you’re
dividing with a calculator and the quotient has a decimal part, the number is not a factor.
If you’re dividing by hand (need a refresher?) and there is a remainder, the number is not a factor. Either
way, just cross out the number, since it is not a factor. If the division results in a whole number answer,
write that in the right-hand column. You’ve found a factor pair.
Step 4: Read the factor pairs from the table. The table shows:
The factor pairs that have a product of 15 are: 1 and 15, 3 and 5.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 5
Example: Find all the factor pairs that have 32 as a product.
Step 1: Since 62 is 36, and 36 is greater than 32, we need only check the numbers less than 6.
Steps 2 and 3: See the table to the right.
Step 4: The factor pairs that have a product of 32 are:
1 and 32, 2 and 16, 4 and 8.
Example: Find all the factor pairs that have 79 as a product.
Step 1: Since 92 is 81, and 81 is greater than 79, we need only check the numbers less than 9.
Steps 2 and 3: See the table to the right.
Step 4: The factor pairs that have a product of 79 are:
1 and 79.
(Aren’t you glad you didn’t need to check all the way up to 79?)
Divisibility Shortcuts
Since you’re doing a lot of dividing to find the factor pairs, here are some shortcuts to save you time.
Look at the Last Digit
If the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2. (2 is a factor.)
If the last digit is 5 or 0, the number is divisible by 5. (5 is a factor.)
378 is divisible by 2
370 is divisible by 5
Add the Digits
If the sum of the digits is divisible by 3, the number is divisible by 3. (3 is a factor.)
If the sum of the digits is divisible by 9, the number is divisible by 9. (9 is a factor.)
42 is divisible by 3 (4+2=6)
72 is divisible by 9 (7+2=9)
Quick Eliminations
If the number is not divisible by 2, it’s not divisible by ANY even number.
Same for 3. If it’s not divisible by 3, it’s not divisible by any multiple of 3.
And so on. If it’s not divisible by N, it’s not divisibly by any multiple of N.
*What’s a Multiple of 3?
*
37 is not divisible by 2, so I don’t
need to check 4, 6, 8, etc.
The multiples of a number are found in the row for that number on the multiplication table. They go on to infinity.
Multiples of 3:
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 6
Example: Find all the factor pairs whose product is 103.
These frames show the process step
by step using the divisiblity shortcuts.
Step 1: How far?
103 is greater than 102 = 100,
and less than 112 = 121,
so we check up to 10.
Step 2: Make the table.
Step 3: Check the factors.
1 is easy – it always works, and the
other factor is the number itself.
Step 3 continued:
103 is odd – the last digit is 3 – so it is
not divisible by 2 or any multiple of 2.
Cross out 2, 4, 6, 8, and 10.
Step 3 continued:
103 does not have 5 or 0 as the last
digit, so cross out 5.
Step 3 continued:
The sum of the digits of 103 is
1 + 0 + 3 = 4,
which is not divisible by 3 or 9.
Cross out 3 and 9.
Step 3 continued:
The last number to check is 7.
103 ÷ 7 = 14.714285…
(or 14 remainder 5),
so 7 is not a factor. Cross out 7.
Step 4: Read the factor pairs from the
table:
Find all the factor pairs whose
product is 103.
The only factor pair is 1 • 103.
Notice that we haven’t actually divided
anything yet. We’ve just used divisiblilty
shortcuts.

© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 7
Lesson 8: Factor Pairs
Worksheet
Name
Find all the factor pairs of the given numbers.
1. These numbers are greater than 52=25, but less than 62=36, so we need only check the numbers 1 through 5.
2. These numbers are greater than (or equal to) 62=36, but less than 72 = 49, so we need only check the
numbers 1 through 6.
3. Find all the factor pairs that have the given number as product
a. 48
b. 55
c. 70
© 2010 Cheryl Wilcox
d. 81
e. 105
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Lesson 8  page 8
Lesson 8: Factor Pairs
Homework 8A
Name
1. Evaluate.
2. Find the volume of each box.
a. 72 – 32
a. The length is 30 inches, the width is 5 less than the
length, and the height is twice the length.
L = 30, W = ______, H = _______
b. (7 – 3)2
c. 5 • 7 – 3
V=
b. The length is three times the width, the width is 20 cm,
and the height is 2 more than the width.
L =______, W = 20, H = _______
V=
d. 5(7 – 3)
c. The length is 3 more than the height, the width is 3 less
than the height, and the height is 13 feet.
L =______, W = _______, H =13
e. 52 – 3(14 – 8)
V=
3. Measure the lines to the nearest inch and to the nearest
sixteenth of an inch.
4. Write and use a formula.
a.
nearest inch __________ nearest sixteenth ___________
b.
nearest inch __________ nearest sixteenth ___________
© 2010 Cheryl Wilcox
a. The sea turtle swam 16 hours at 42 miles per hour. How
far did it travel?
b. A gamer performed 305 actions per minute for 214
minutes. How many actions were performed?
c. The average daily temperature at the location of the Mars
Rover is 35ºC. Find the temperature in degrees Fahrenheit.
The formula is F = 9C/5 + 32.
Free Pre-Algebra
Lesson 8  page 9
5. Write the related addition/subtraction equations with 15, 8
and 23.
6. a. Find the sum of 21 and 7.
b. Find the product of 21 and 7.
Write the related multiplication/division equations with 15, 8
and 120.
c. Find the quotient of 21 and 7.
d. Find the difference of 21 and 7.
7. Find all the factor pairs that have the given numbers as product.
a. 51
b. 52
c. 53
d. 54
e. 55
f. 56
g. 57
h. 58
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 10
Lesson 8: Factor Pairs
Homework 8A Answers
1. Evaluate.
2. Find the volume of each box.
a. 72 – 32
a. The length is 30 inches, the width is 5 less than the
length, and the height is twice the length.
49 – 9 = 40
L = 30, W = 25, H = 60
V = (30)(25)(60) = 45000
b. (7 – 3)2
42 = 16
c. 5 • 7 – 3
35 – 3 = 32
45,000 in3
b. The length is three times the width, the width is 20 cm,
and the height is 2 more than the width.
L = 60, W = 20, H = 22
V = (60)(20)(22) = 26400
26,400 cm3
d. 5(7 – 3)
5(4) = 20
c. The length is 3 more than the height, the width is 3 less
than the height, and the height is 13 feet.
L =16, W = 10, H =13
e. 52 – 3(14 – 8)
52 – 3(6) = 52 – 18 = 34
3. Measure the lines to the nearest inch and to the nearest
sixteenth of an inch.
V = (16)(10)(13) = 2080
2,080 ft3
4. Write and use a formula.
a. The sea turtle swam 16 hours at 42 miles per hour. How
far did it travel?
d = rt = (42)(16) = 672
672 miles
a.
nearest inch 2 in. nearest sixteenth 1 and 11/16 in.
b. A gamer performed 305 actions per minute for 214
minutes. How many actions were performed?
a = rt = (305)(214) = 65270
65,270 actions
c. The average daily temperature at the location of the Mars
Rover is 35ºC. Find the temperature in degrees Fahrenheit.
The formula is F = 9C/5 + 32.
b.
nearest inch 3 in. nearest sixteenth 2 and 5/8 in.
© 2010 Cheryl Wilcox
F = 9(35)/5 + 32 = 95
95ºF
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Lesson 8  page 11
5. Write the related addition/subtraction equations with 15, 8
and 23.
15 + 8 = 23
23 – 8 = 15
8 + 15 = 23
23 – 15 = 8
Write the related multiplication/division equations with 15, 8
and 120.
15 • 8 = 120
120 ÷ 8 = 15
8 • 15 = 120
120 ÷ 15 = 8
6. a. Find the sum of 21 and 7.
21 + 7 = 28
b. Find the product of 21 and 7.
21 • 7 = 147
c. Find the quotient of 21 and 7.
21 ÷ 7 = 3
d. Find the difference of 21 and 7.
21 – 7 = 14
7. Find all the factor pairs that have the given numbers as product.
a. 51
b. 52
c. 53
d. 54
e. 55
f. 56
g. 57
h. 58
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 12
Lesson 8: Factor Pairs
Homework 8B
Name
1. Evaluate.
2. Find the volume of each box.
a. 52 + 72
a. The length is 12 inches, the width is 4 less than the
length, and the height is 3 more than the length.
L = 12, W = ______, H = _______
b. (5 + 7)2
c. 3 • 5 + 7
V=
b. The length is four times the width, the width is 2 cm, and
the height is 8 more than the width.
L =______, W = 2, H = _______
V=
d. 3(5 + 7)
c. The length is 10 more than the height, the width is 10 less
than the height, and the height is 13 feet.
L =______, W = _______, H =13
e. 17 + 3(5 + 7)
V=
3. Measure the lines to the nearest inch and to the nearest
sixteenth of an inch.
4. Write and use a formula to answer the questions.
a.
nearest inch __________ nearest sixteenth ___________
a. The endurance rider rode for 12 hours at an average
speed of 8 miles per hour. How far did she travel?
b. A computer performed operations at the rate of 102 billion
operations per second for 30 seconds. How many
operations were performed?
c. Cast iron melts at approximately 1,370 °C. Find the
temperature in degrees Fahrenheit. The formula is
F = 9C/5 + 32.
b.
nearest inch __________ nearest sixteenth ___________
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 8  page 13
5. Write the related addition/subtraction equations with 12,
10 and 22.
6. a. Find the difference of 12 and 3.
b. Find the product of 12 and 3.
Write the related multiplication/division equations with 12, 10
and 120.
c. Find the sum of 12 and 3.
d. Find the quotient of 12 and 3.
7. Find all the factor pairs that have the given numbers as product.
a. 59
b. 60
c. 61
d. 62
e. 63
f. 64
g. 65
h. 66
© 2010 Cheryl Wilcox