Download Factors and Divisibility

Name
7.1
?
Algebraic
Reasoning—5.4.A
MATHEMATICAL PROCESSES
5.1.C, 5.1.D, 5.1.F
Factors and Divisibility
Essential Question
How can you tell whether one number is a factor of
another number?
Unlock
Unlock the
the Problem
Problem
Students in Carlo’s art class painted 32 square tiles
for a mosaic. They will arrange the tiles to make a
rectangle. Can the rectangle have 32 tiles arranged
into 3 equal rows, without gaps or overlaps?
A composite number is a whole number greater than
1 that has more than two factors. You can use models
and division to find factors of composite numbers.
One Way Draw a model.
Think: Try to arrange the tiles into 3 equal rows to make a rectangle.
A rectangle __ have 32 tiles arranged into 3 equal rows.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Artville/Getty Images
Another Way Use division.
If 3 is a factor of 32, then the unknown factor in 3 × ■ = 32 is a
whole number.
3)‾
3 2
▲ Mosaics are decorative patterns made
with pieces of glass or other materials.
Math Idea
A factor of a number divides
the number evenly. This means
the quotient is a whole number
and the remainder is 0.
Think: Divide to see
whether the unknown
factor is a whole
number.
Math Talk
Mathematical Processes
The unknown factor in 3 × ■ = 32 __ a whole number.
So, a rectangle __ have 32 tiles arranged in 3 rows.
Explain how the model relates
to the quotient and remainder
for 32 ÷ 3.
Module 7
291
Divisibility Rules A number is divisible by another
Divisibility Rules
number if the quotient is a counting number and the
remainder is 0.
Some numbers have a divisibility rule. You can use a
divisibility rule to tell whether one number is a factor
of another.
Is 6 a factor of 72?
Think: If 72 is divisible by 6, then 6 is a factor of 72.
Test for divisibility by 6:
Is 72 even? _
Number
Divisibility Rule
2
The number is even.
3
The sum of the digits
is divisible by 3.
5
The last digit is 0 or 5.
6
The number is even
and divisible by 3.
9
The sum of the digits is
divisible by 9.
What is the sum of the digits of 72?
_+_=_
Is the sum of the digits divisible by 3?
Math Talk
_______
Mathematical Processes
How are divisibility and
factors related? Explain.
72 is divisible by _ .
So, 6 is a factor of 72.
Share
Share and
and Show
Show
1. Is 4 a factor of 28? Draw a model to help.
Think: Can you make a rectangle with 28 squares in 4 equal rows?
Is 5 a factor of the number? Write yes or no.
2. 27
_
292
3. 30
_
4. 36
_
5. 53
_
© Houghton Mifflin Harcourt Publishing Company
4 _ a factor of 28.
Name
Problem
Problem Solving
Solving
List all the factor pairs in the table. Use a model or paper and pencil to help.
6.
Factors of 24
7.
Factors of 39
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
List all the factor pairs for the number. Make a table to help.
8. 56
10.
9. 64
What’s the Error? George said if 2 and 4 are factors of
a number, then 8 is a factor of the number. Is he correct? Explain.
Problem
Problem Solving
Solving
Use the table to solve 11–12.
11. Multi-Step Dirk bought a set of stamps. The number of stamps
in the set he bought is divisible by 2, 3, 5, 6, and 9. Which set is it?
© Houghton Mifflin Harcourt Publishing Company
12.
Stamps Sets
Country
Number of stamps
Germany
90
Sweden
78
Japan
63
Canada
25
Multi-Step Geri wants to put 6 stamps
on some pages in her stamp book and 9 stamps on other
pages. Explain how she could do this with the stamp set
for Sweden.
Module 7 • Lesson 1
293
Mathematical Processes
M
Model ¥ Reason ¥ Communicate
M
Daily
Daily Assessment
Assessment Task
Task
Fill in the bubble completely to show your answer.
13. There are 54 people attending a party. Carla is arranging chairs and
tables. How should she arrange the chairs so that there will be an equal
number of people seated at each table?
A
12 chairs at each table
B
8 chairs at each table
C
9 chairs at each table
D
10 chairs at each table
14. Analyze Jake organizes 48 marbles into packs. He places the same
number of marbles into each pack. How could he arrange the marbles?
A
8 or 10 in each pack
B
9 or 12 in each pack
C
10 or 12 in each pack
D
8 or 12 in each pack
15. Multi-Step Megan has 34 rocks in her rock collection. She wants to put
5 rocks in some cases and 7 rocks in other cases. How could she arrange
the rocks?
A
Have 6 cases of 5 rocks, and 1 case of 7 rocks.
B
Have 4 cases of 5 rocks, and 2 cases of 7 rocks.
C
Have 2 cases of 5 rocks, and 4 cases of 7 rocks.
D
Have 5 cases of 5 rocks, and 2 cases of 7 rocks.
TEXAS Test Prep
16. Mrs. Mastrioni bought a set of 80 stamps. She wanted to give all the stamps
294
A
2 or 3 students.
B
2 or 6 students.
C
2, 4, 5, or 8 students.
D
2, 4, 8, or 9 students.
© Houghton Mifflin Harcourt Publishing Company
to her students as a reward. She could give equal numbers of stamps to
H o mewo rk
and Practice
7.1
Algebraic Reasoning—5.4.A
MATHEMATICAL PROCESSES 5.1.C,
5.1.D, 5.1.F
Name
Factors and Divisibility
List all the factor pairs in the table.
1.
Factors of 30
2.
Factors of 15
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
_ × _ = _ _, _
List all the factor pairs for the number. Make a table to help.
3. 18
4. 45
5. 36
6. 63
Problem
Problem Solving
Solving
7. Planners for the city’s botanical garden are
© Houghton Mifflin Harcourt Publishing Company
planting a rose garden. They want to plant
56 rose bushes. Can they arrange the rose bushes
into 4 equal rows? Explain.
8. A marching band with 75 members makes two
rectangular marching formations. One rectangle
has 6 rows and the other rectangle has 9 rows.
Explain how the members can form the two
rectangles.
Module 7 • Lesson 1 295
TEXAS Test Prep
Lesson
Lesson Check
Check
Fill in the bubble completely to show your answer.
She places the same number of favors in each
bag. How could she arrange the favors?
A
60
B
33
A
3 or 6 in each bag
C
46
B
6 or 9 in each bag
D
21
C
7 or 8 in each bag
D
7 or 9 in each bag
11. A package of marbles contains 20 blue
12. The fifth-grade students participating in the
marbles, 38 red marbles, 48 green marbles,
and 16 yellow marbles. Which color of marbles
can be divided evenly between 6 people?
math competition will be evenly divided into
teams of 3 students each. How many students
could be participating?
A
blue
A
25
B
red
B
21
C
green
C
23
D
yellow
D
26
13. Multi-Step The librarian has 94 books to
organize. He wants to put 5 books on some
shelves and 6 books on other shelves. How
could he arrange the books?
A
B
296
10. Jo Beth organizes 63 party favors into gift bags.
Have 8 shelves of 5 books, and 9 shelves
of 6 books.
Have 9 shelves of 5 books, and 8 shelves
of 6 books.
C
Have 5 shelves of 6 books, and 6 shelves
of 5 books.
D
Have 7 shelves of 5 books, and 6 shelves
of 6 books.
14. Multi-Step A group of 72 students went on a
field trip. The students traveled in vans, with
an equal number of girls and boys in each van.
If 42 boys went on the trip, how many vans
were there?
A
9
B
8
C
7
D
6
© Houghton Mifflin Harcourt Publishing Company
9. Which number is divisible by 2, 3, and 6?