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Primetime
Short Answer
List all the factors for the number.
1.
24
2.
31
3.
List all the proper factors of 35.
4.
Which number(s) 24, 31, 35 are prime numbers? Explain why.
5.
Which number(s) 24, 31, 35 are composite numbers? Explain why.
6.
a. You are playing the Product Game on a game board like the one shown. One of the paper
clips is on 7. What products can you make by moving the other paper clip?
b. List two multiples of 7 that are not on the game board.
7.
Why isn’t the number 13 on the Product Game board?
8.
Which of these numbers are square numbers? Explain.
25
36
48
9.
a. List the factors of 16 and the factors of 28.
b. Complete the Venn diagram.
c. What is the greatest common factor of 16 and 28?
10.
a. List the first five multiples of 15 and the first five multiples of 12.
b. Complete the Venn diagram.
c. What is the least common multiple of 15 and 12?
d. Find a common multiple of 15 and 12 that is not in your lists.
11.
Jill says 6 is a common factor of 56 and 36. Is she correct? Explain your reasoning.
12.
Evonne and Dolphus found a new Product Game board. Three of the factors and one of the
products were not filled in.
a. What are the other three factors you would need in order to play the game using this board?
b. What product is missing?
13.
Terrapin Crafts wants to rent between 35 and 40 square yards of space for a big crafts show.
The space must be rectangular, and the side lengths must be whole numbers. Find the number(s) between 35
and 40 with the most factor pairs that gives the greatest number of rectangular arrangements to choose from.
14.
Two radio stations are playing the #1 hit song “2 Nice 2 B True” by Anita and the Goody-2Shoes. WMTH plays the song every 18 minutes. WMSU plays the song every 24 minutes. Both stations play
the song at 3:00 P.M. When is the next time the stations will play the song at the same time?
15.
Judith is planning a party for her younger brother. She has 36 prizes and 24 balloons. How
many children can she have at the party so that each child gets an equal number of prizes and an equal number
of balloons? Explain your answer.
16.
Find three different ways to show factorizations (strings of factors) of the number 16. Do not
use 1 as a factor.
17.
Find the prime factorization of the following two numbers. Show your work.
a.
72
b.
132
18.
reasoning.
19.
A number that is less than 85 has 26 and 6 as factors. Find the number and explain your
What number has the prime factorization ? Show how you found the number.
20.
Find the dimensions of all of the rectangles that can be made from 48 square tiles. Explain
how you found your answers.
21.
“Sam” and “Martha” are the local names for two lighthouses that guard a particularly
dangerous part of the coast. Sam blinks every 12 seconds and Martha blinks every 8 seconds. They blink
together at midnight. How many seconds will pass before they blink together again?
22.
Carlos is packing sacks for treats at Halloween. Each sack has to have exactly the same stuff
in it or the neighborhood kids complain. He has on hand 96 small candy bars and 64 small popcorn balls.
a. What is the greatest number of treat sacks he can make?
b. How many of each kind of treat is in one sack?
23.
a. What is the greatest common factor of 30 and 42?
b. Give a different common factor of 30 and 42.
c. What is the least common multiple of 30 and 42?
d. Give an additional common multiple of 30 and 42.
24.
Dawson wrote the factorization . Without finding the actual number, how can Dawson tell if
the number is even or odd?
25.
Scarlett and Rhett were playing the Factor Game when Ashley looked over and saw that the
numbers 1 to 15 were all circled. Ashley immediately said, “Oh, I see that your game is over.” Is Ashley
correct? Explain.
Describe how you can tell whether a given number is a multiple of the number shown.
26.
2
27.
3
28.
5
29.
List all multiples of 6 between 1 and 100. What do these numbers have in common?
30.
Mr. Matsumoto said, “I am thinking of a number. I know that to be sure I find all of the factor
pairs of this number, I have to check all the numbers from 1 through 15.”
a. What is the smallest number he could be thinking of? Explain your answer.
b. What is the greatest number he could be thinking of? Explain your answer.
31.
What is the mystery number?
Clue 1
My number is between the square numbers 1 and 25.
Clue 2
My number has exactly two factors.
Clue 3
Both 66 and 605 are multiples of my number.
32.
Use concepts you have learned in this unit to make a mystery number question. Each clue
must contain at least one word from your vocabulary list.
33.
a.
List the first ten square numbers.
b.
Give all the factors for each number you listed in part (a).
c.
Which of the square numbers you listed have only three factors?
d.
If you continued your list, what would be the next square number with only three factors?
34.
A mystery number is greater than 50 and less than 100. You can make exactly five different
rectangles with the mystery number of tiles. Its prime factorization consists of only one prime number. What
is the number?
35.
A number has 4 and 5 as factors.
a. What other numbers must be factors? Explain.
b. What is the smallest the number could be?
36.
Chairs for a meeting are arranged in six rows. Each row has the same number of chairs.
a. What is the minimum possible number of chairs that could be in the room?
b. Suppose 100 is the maximum number of chairs allowed in the meeting room. What other numbers of
chairs are possible?
37.
Gloomy Toothpaste comes in two sizes: 9 ounces for $0.89 and 12 ounces for $1.15.
a. Ben and Aaron bought the same amount of toothpaste. Ben bought only 9-ounce tubes, and Aaron
bought only 12-ounce tubes. What is the smallest possible number of tubes each boy bought? (Hint: Use
your knowledge of multiples to help you.)
b. Which size tube is the better buy?
38.
Circle the letter(s) of the statements that are always true about any prime number.
a. It is divisible by only itself and 1.
b. It is a factor of 1.
c. It is divisible by another prime number.
d. It is always an odd number.
39.
Tyrone claims that the longest string of factors for 48 is 48 = . Ian says there is a longer
string. He wrote 48 = . Who is correct? Why?
40.
What is the smallest number divisible by the first three prime numbers and the first three
composite numbers? Explain.
41.
Suppose you are playing the Factor Game on the 30-board. Your opponent goes first and
chooses 29, giving you only 1 point. It is now your turn to choose a number. Which number would be your
best move? Why?
42.
Suppose the person who sits next to you was absent the day you played the Factor Game. On
the back of this paper, write a note to him or her explaining the strategies you have discovered for winning the
Factor Game. Include a description of how you decide which move to make when it is your turn.
43.
Vicente made three dozen cookies for the student council bake sale. He wants to package
them in small bags with the same number of cookies in each bag.
a. List all the ways Vicente can package the cookies.
b. If you were Vicente, how many cookies would you put in each bag? Why?
c. Vicente spent $5.40 on ingredients for the cookies. The student council will pay him back for the money
he spent. For each of the answers in part (a), determine how much the student council should charge for
each bag of cookies so they make a profit yet still get students to buy the cookies.
44.
Marcia has developed a rule for generating a number sequence. The first 6 numbers in her
sequence are 7, 21, 42, 126, 252, 756.
a. What is Marcia’s rule for finding the numbers in her sequence? Explain.
b. What are the next two numbers in Marcia’s sequence?
c. What is the greatest common factor (GCF) of all the terms in Marcia’s sequence? Explain your
reasoning.
45.
a. List two pairs of numbers whose least common multiple (LCM) is the same as their
product. For example, the least common multiple of 5 and 6 is 30 and 5 × 6 = 30.
b. List two pairs of numbers whose least common multiple is smaller than their product. For example, the
least common multiple of 6 and 9 is 18 and 18 is less than 6 × 9.
c. For a given pair of numbers, how can you tell whether the least common multiple will be less than or
equal to their product?
46.
a. Write the prime factorization of 900.
b. From information in the prime factorization of 900, write five sentences about the number 900. Use
vocabulary from the unit in each sentence.
47.
For each of the following, use the set of clues to determine the secret number.
a.
Clue 1
The number has two digits.
Clue 2
The number has 13 as a factor.
Clue 3
The sum of the digits of the number is 11.
b.
Clue 1
The number is prime.
Clue 2
The number is less than 19.
Clue 3
The sum of the digits of the number is greater than 7.
48.
The numbers 10, 20, and 30 on the 30-board in the Factor Game all have 10 as a factor. Does
any number that has 10 as a factor also have 5 as a factor? Explain your reasoning.
49.
The numbers 14, 28, and 42 on the 49-board in the Factor Game all have 7 as a factor and
also have 2 as a factor. Does any number that has 7 as a factor also have 2 as a factor? Explain your
reasoning.
50.
Look carefully at the numbers 1–30 on the 30-board used for playing the Factor Game. Pick
the two different numbers on the 30-board that will give you the largest number when you multiply them
together, and then answer the following questions.
a. What two numbers did you pick? What is the product of the two numbers?
b. Explain why the product of the two numbers you chose is the largest product you can get using two
different numbers from the 30-board.
c. List all the proper factors of the product. Explain how you found the factors.
51.
For each of the following, find three different numbers that can be multiplied together so that
the given number is the product. Do not use 1 as one of the numbers.
a.
150
b.
1,000
c.
24
d.
66
52.
The number sequence 4, 6, 10 is a multiple of the number sequence 2, 3, 5 because the
sequence 4, 6, 10 can be found by multiplying all the numbers in the sequence 2, 3, 5 by 2. That is, 4 = , 6 = ,
10 = .
a. The number sequence 15, 25, 10 is a multiple of what number sequence?
b. Find two different sequences that are multiples of the number sequence 1, 4, 7.
c. Given a number sequence, how many different sets of multiples of that sequence do you think there
are? Explain your reasoning.
53.
Given the following sets of numbers, write as many different multiplication and division
statements as you can. For example, if the numbers are 5, 7, 35, you can write:
a. 6, 4, 24
b. 96, 12, 8, 3, 32
c. 6, 27, 108, 12, 4, 18, 9
d. When is a number called a factor of a number? A divisor of a number?
54.
Alicia has made a rectangle using 24 square tiles. If she adds the length and width of her
rectangle together, she gets 11. What is the length and width of Alicia’s rectangle? Explain your reasoning.
55.
Jennifer has made a rectangle using 48 square tiles. If she adds the length and width of her
rectangle together she gets a prime number. What is the length and width of Jennifer’s rectangle? Explain
your reasoning.
56.
List all of the factor pairs for each of the following numbers.
a. 56
b. 42
c. 31
d. 80
e. 75
f. 108
g. 225
57.
Phillip is thinking of a number that is less than 20 and has three factor pairs. Phillip also says
that if he adds together the factors in the factor pairs he gets 19, 11, and 9. What is Phillip’s number? Explain
how you found your answer.
58.
In each of the rectangles shown below, only the tiles along the length and width are shown.
For each rectangle, explain how many square tiles it would take to make each rectangle.
a.
b.
c.
59.
a. Draw and label a Venn diagram in which one circle represents the factors of 12 and
another circle represents the factors of 13. Place the numbers from 1 to 15 in the appropriate regions of
the diagram.
b. What do you notice about the numbers in the intersection? Why does this happen?
c. What is another set of labels, one for each of the two circles, that gives the same numbers in the
intersection as you found in part (b)? Explain your reasoning.
60.
a. Draw and label a Venn diagram in which one circle represents the multiples of 5 and
another circle represents the multiples of 2. Place the numbers from 1 to 40 in the appropriate regions of
the diagram.
b. What do you notice about the numbers in the intersection? Why does this happen?
c. Where would you place 75 in the diagram? Where would you place 90? Explain your reasoning.
61.
Karl added four numbers together and got an even sum. Three of the numbers are 42, 35, and
77. What can you say about the fourth number? Explain your reasoning.
62.
On Saturdays, the #14 bus makes roundtrips between Susan’s school and the mall, and the
#11 bus makes roundtrips between the mall and the museum. Next Saturday, Susan wants to take the bus from
her school to the museum. A #14 bus leaves Susan’s school every 15 minutes, beginning at 7 A.M. It takes
the bus 30 minutes to travel between the school and the mall. A #11 bus leaves the mall every 12 minutes,
beginning at 7 A.M.
a. If Susan gets on the #14 at 9:30 A.M., how long will she have to wait at the mall for a #11 bus?
Explain your reasoning.
b. If Susan gets on the #11 bus at the museum and arrives at the mall at 11:48 A.M., how long will she
have to wait for the #14 bus? Explain your reasoning.
c. At what times between 9 A.M. and noon are the #14 and #11 buses at the mall at the same time?
Explain your reasoning.
63.
Kyong has built two rectangles. Each has a width of 7 tiles.
a. If each rectangle is made with an even number of tiles that is greater than 40 but less than 60, how many
tiles does it take to make each rectangle? Explain your reasoning.
b. What is the length of each of Kyong’s rectangles? Explain your reasoning.
c. Without changing the number of tiles used to make either rectangle, Kyong rearranges the tiles of each
rectangle into different rectangles. What is a possibility for the length and width of each of Kyong’s new
rectangles? Explain your reasoning.
64.
Jack plays on a basketball team after school (or on the weekend) every third day of the
month. He babysits his younger brother after school every seventh day of the month. How many times during
a 30-day month, if any, will Jack have a conflict between basketball and babysitting? Explain your reasoning.
65.
Suppose you have two different numbers which are both prime.
a. What is the least common multiple of the numbers? Explain your reasoning.
b. What is the greatest common factor? Explain your reasoning.
66.
Find the least common multiple and the greatest common factor for each pair of numbers:
a. 8 and 12
b. 7 and 15
c. 11 and 17
d. 36 and 108
e. For which pairs in parts (a) – (d) is the least common multiple the product of the two numbers? Why is
this so? What is special about the numbers in these pairs?
67.
Find the greatest common factor of each pair of numbers:
a. 4 and 12
b. 5 and 15
c. 10 and 40
d. 25 and 75
e. When is the greatest common factor of two numbers one of the two numbers? Explain your
reasoning.
68.
Find the prime factorization for each of the numbers below.
a. 630
b. 144
c. 1,011
d. 133
e. 23
69.
Solve each of the multiplication mazes given below. Record your solution for each maze by
copying the maze on your paper and then tracing out the path through the maze.
a.
b.
c.
d.
70.
For each of the pairs of numbers given below, find the greatest common factor and the least
common multiple.
a. 25 and 105
b. 27 and 81
c. 36 and 63
71.
An odd number that is less than 160 has exactly three different prime factors. What is the
number? Explain your reasoning.
72.
What number has the prime factorization ?
73.
a. Name a pair of numbers whose greatest common factor is the same as one of the numbers.
b. Name another pair of numbers whose greatest common factor is the same as one of the numbers.
c. Make a conjecture about what must be true about the least common multiple of any number pairs in which
one number is the greatest common factor of the other number.
74.
a. Are 45 and 64 relatively prime? Explain your reasoning.
b. Are 25 and 36 relatively prime? Explain your reasoning.
c. Is it possible for two numbers that are both even to be relatively prime? Why or why not?
d. How can you choose one number so that it will be relatively prime to any other number?
75.
Find all of the numbers less than 1,000 that have 3 as their only prime factor. Explain your
strategy for finding all of these numbers.
76.
A number sequence is an ordered series of numbers that follow a pattern or rule. Jason has
developed a secret rule for generating his own number sequence. Here are the first five terms in the sequence:
3, 15, 45, 225, 675, ... and so on. Use Jason’s sequence to answer the following questions.
a. What is Jason’s rule for finding the numbers in his number sequence? Explain how you found your
answer.
b. What are the next two terms in Jason’s number sequence?
c. What is the greatest common factor of all the terms in Jason’s sequence, no matter how many new
numbers he adds to the sequence? Explain your reasoning.
77.
In the 1,000-locker problem, which students touched the lockers indicated?
a. both lockers 13 and 19
b. lockers 12, 16, and 20
78.
In the 1,000-locker problem, what was the last locker touched by the students indicated?
a. both students 20 and 25
b. both students 13 and 19
c. all three students 3, 4, and 5
d. all three students 30, 40, and 50
79.
A set of consecutive numbers that contains no prime numbers is called a prime desert. For
example, the set {14, 15, 16} is a prime desert because it is a set of consecutive numbers and none of the
numbers are prime. Find the prime desert that has the most numbers in it where all the numbers are less than
50.
80.
For each of the sets of clues below, find the secret number.
a.
Clue 1
The number is less than 130.
Clue 2
The number ends in a 5.
Clue 3
The number is a multiple of a prime that is greater than 20, but less than 30.
b.
Clue 1
The number ends in a 0.
Clue 2
The number is a multiple of 21.
Clue 3
The number is less than 400.
81.
Find the prime factorization of each of the following numbers.
a. 190
b. 319
c. 255
d. 406
82.
Test 71,094 for divisibility by 5.
83.
Tell whether 5,136 is divisible by 2, 3, 4, 5, 9, or 10. Write none of these if applicable.
84.
Write the prime factorization of 160. Use exponents where possible.
85.
Mr. Wolfe wants to divide a 6th grade class of 40 students into equal groups. He wants the
number of students in each group to be a prime number.
a.
In how many different ways can Mr. Tindell divide the class?
b.
How many students will be in each group?
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
a.
2
86.
What is the greatest common factor of 16 and 28?
b.
3
c.
4
d.
112
____
a.
8
b.
11
c.
14
d.
24
87.
Which of the following is a factor of 44?
____
a.
1
b.
5
c.
10
d.
30
88.
Which number is a multiple of 15?
____
a.
2
b.
3
c.
4
d.
6
89.
How many different factors does 20 have?
____
a.
11
b.
14
c.
24
d.
56
90.
Which number is a common multiple of 7 and 4?
____
a.
91.
Find the prime factorization of 160
____
a.
2
b.
9
c.
252
d.
504
92.
What is the least common multiple of 18 and 28?
____
a.
93.
Which answer is always odd?
c.
b.
d.
b.
odd + odd
c.
d.
even + even
e.
____
a.
94.
Which string of factors below is not a factorization of 180
95.
What is the least number that has 2, 3, and 4 as factors?
c.
b.
d.
____
a.
6
b.
9
c.
12
d.
18
____ 96.
a.
28 by 7
c.
43 by 9
b.
43 by 4
d.
67 by 8
In which case is the first number divisible by the second? Use mental math.
____ 97.
Test 72,238 for divisibility by 2, 5, or 10.
a.
It is divisible by 2, but not by 5 or 10.
b.
It is divisible by 5 and 10 but not by 2.
c.
It is divisible by 2, 5, and 10.
d.
It is divisible by 5, but not by 2 or 10.
Find the LCM of the numbers.
____
a.
32
c.
16
b.
64
d.
48
98.
4, 16
Find the prime factorization of the number.
____
a.
99.
168
c.
b.
d.
____
a.
100.
360
____
a.
45
b.
58
c.
58
d.
108
101.
Which number is divisible by 5?
____
a.
109
b.
135
c.
116
d.
125
102.
Which number is divisible by 3?
____
a.
7
b.
0
c.
6
d.
103.
Find the missing digit to make 27,47_ divisible by 9.
c.
b.
d.
3
____
977
428
2,552
1,800
104.
Which number is divisible by 9 and 3?
a.
428
b.
2,552
c.
1,800
d.
977
____ 105.
2,447 and 5
2,227 and 9
4,366 and 10
1,644 and 2
For which pair of numbers is the first number divisible by the second number?
a.
2,447 and 5
c.
2,227 and 9
b.
4,366 and 10
d.
1,644 and 2
____
1,272
3,786
8,891
9,634
a.
8,891
b.
9,634
c.
3,786
d.
1,272
106.
Which number is divisible by 4?
____
a.
42
b.
3
c.
84
d.
126
107.
Find the LCM of 6 and 21.
____
a.
53
b.
56
c.
57
d.
51
108.
Tell which number is prime: 51, 53, 56, 57.
____
a.
4
b.
3
c.
12
d.
6
109.
Find the GCF of 66 and 87.
____ 110.
The local reader’s club has a set of 28 hardback books and a set of 44 paperbacks. Each set
can be divided equally among the club members. What is the greatest possible number of club members?
a.
308
b.
4
c.
2
d.
8
____
a.
32
c.
111.
List the factors to find the GCF of 32 and 56.
224
b.
24
d.
8
____
a.
3
c.
2
b.
4
d.
1
112.
Use a division ladder to find the GCF of 22 and 50.
Find the GCF of the numbers.
____
a.
11
c.
8
b.
2
d.
1,408
113.
64, 16, 176
____
a.
68
c.
8
b.
4
d.
240
114.
20, 48
____ 115.
Alejandro and Jean are distributing erasers and pencils to the art class. There are 40 erasers
and 25 pencils. Each student receives the same number of pencils and the same number of erasers, and no
supplies are left over. What is the greatest number of students in the class?
a.
10 students
c.
65 students
b.
200 students
d.
5 students
List all the factors for the number.
____ 116.
60
a.
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
c.
1, 2, 3, 4, 7, 14, 15, 20, 30, 60
b.
1, 2, 4, 5, 10, 20
d.
1, 2, 3, 4, 5, 10
____ 117.
Which number is composite?
43, 23, 17, 52, 67
a.
52
b.
23
c.
67
d.
43
Other
118.
Is 3665 divisible by 9? How do you know?
119.
Jennifer and Melissa are both moving and must pack their collections of CDs. Jennifer has 9
boxes and 108 CDs, while Melissa has 5 boxes and 193 CDs. Can each of them divide their CDs evenly
among their boxes? Explain.
Primetime
Answer Section
SHORT ANSWER
1.
ANS:
1, 2, 3, 4, 6, 8, 12, 24
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
factor | proper factors | whole number factors | whole number divisors
2.
ANS:
1, 31
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors
3.
1, 5, 7, 35
PTS:
OBJ:
STA:
KEY:
ANS:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors
4.
ANS:
31; It has exactly two factors, one and the number itself.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
prime number | composite number | abundant numbers | perfect numbers | deficient numbers
5.
ANS:
24, 35; These numbers have more than two factors.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
prime number | composite number | abundant numbers | perfect numbers | deficient numbers
6.
ANS:
a. 7, 14, 21, 28, 35, 49, 56
b. Possible answers: 42, 63
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
multiple | square numbers | divisible by | near-perfect numbers
7.
ANS:
The factors of 13 are 1 and 13. Both of these factors are not listed below the game board.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Check-Up 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
multiple | square numbers | divisible by | near-perfect numbers
8.
ANS:
25 and 36; Each of these numbers can be expressed as a number times itself. 5 × 5 = 25 and 6 × 6 = 36. If you
made a tile model you could arrange each number of tiles into a square.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Check-Up 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor
9.
ANS:
a. Factors of 16: 1, 2, 4, 8, 16
Factors of 28: 1, 2, 4, 7, 14, 28
b.
c. 4
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Check-Up 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.3 Classifying Numbers
KEY: Venn diagram | intersection | factor
10.
ANS:
a. Multiples of 15: 15, 30, 45, 60, 75
Multiples of 12, 24, 36, 48, 60;
b.
c. 60
d. Possible answers: 120, 180, 240, 300
PTS: 1
DIF: L2
REF: Prime Time | Check-Up 2
OBJ: Investigation 2: Whole-Number Patterns and Relationships
NAT: NAEP N5a| NAEP N5b| NAEP N5f
STA:
6PA 2.2.8.B
TOP:
Problem 2.3 Classifying Numbers
KEY: Venn diagram | intersection | multiple
11.
ANS:
Jill is incorrect. In order for 6 to be a common factor, both numbers must be divisible by 6. The number 56
cannot be divided evenly by 6.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Check-Up 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.3 Classifying Numbers
KEY: Venn diagram | intersection | factor | common factors
12.
a. 3, 4, 6
b. 81
PTS:
OBJ:
STA:
KEY:
ANS:
1
DIF: L2
REF: Prime Time | Partner Quiz
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
multiple | square numbers | divisible by | near-perfect numbers
13.
ANS:
36; If we consider an rectangle to be different from a rectangle, of the numbers 36, 37, 38, and 39, the
number 36 gives nine choices of rectangles.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Partner Quiz
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions
14.
ANS:
In 72 minutes, or at 4:12 P.M.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Partner Quiz
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.1 Finding Common Multiples | Problem 3.2 Finding the Least Common Multiple
common multiples | least common multiple | common factors | greatest common factor
15.
ANS:
Judith could have 1, 2, 3, 4, 6, or 12 children at the party. These numbers are common factors of 24 and 36.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Partner Quiz
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.3 Finding Common Factors | Problem 3.4 Finding the Greatest Common Factor
common factors | common multiples | greatest common factor | least common multiple
16.
ANS:
Possible answers: ; ;
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.1 Finding Factor Strings
prime factorization | factor tree | factoring
17.
ANS:
a. 72:
b. 132:
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String
factor tree | exponent | Fundamental Theorem of Arithmetic
18.
ANS:
The number is 78 = .
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String | Problem 4.3 Using Prime Factorizations
factor tree | exponent | Fundamental Theorem of Arithmetic | relatively prime
19.
ANS:
The number is 2,100. = = 2,100
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String
factor tree | exponent | Fundamental Theorem of Arithmetic
20.
ANS:
, , , , . List all the factor pairs, starting with 1 and 48, 2 and 24, etc., until you come to one that you have
already used. For example, after matching 6 with 8, you move to 7 which is not a factor. The next number is 8
and you have already used it with 6, so you are finished.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions
21.
24 seconds
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
ANS:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.1 Finding Common Multiples
common multiples | least common multiple | common factors | greatest common factor
22.
ANS:
a. Carlos can make 32 sacks.
b. Each sack would contain 3 small candy bars and 2 small popcorn balls.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.3 Finding Common Factors
common multiples | least common multiple | common factors | greatest common factor
23.
ANS:
a. 6
b. Possible answers: 2 or 3
c. 210
d. Possible answers: 420, 630, 840
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | factor tree | prime factorization
24.
ANS:
Odd; possible answers: is equivalent to or a string of odd factors. An odd number times an odd number is an
odd number. This means that 32 is odd and 52 is odd, and when these two odd products are multiplied the final
product will be odd.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Unit Test
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String
factor tree | exponent | Fundamental Theorem of Arithmetic | odd numbers
25.
ANS:
Yes; the factors of numbers greater than 16 on the Factor Game board are between 1 and 15, so any number
greater than 16 would be an illegal move because its factors are already circled.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors
26.
ANS:
Possible answer: It is an even number. It can be divided by 2 without a remainder.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors | multiple
27.
ANS:
Possible answer: It can be divided by 3 without a remainder.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors | multiple
28.
ANS:
Possible answer: It can be divided by 5 without a remainder. It ends in 0 or 5.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors | divisibility
29.
ANS:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96; Possible answers: They can be divided by 6
without a remainder. They have 6 as a factor. They are divisible by 2 and 3.
PTS: 1
DIF: L2
REF: Prime Time | Extra Questions
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: prime number | composite number | abundant numbers | perfect numbers | deficient numbers |
common multiples | multiple
30.
ANS:
a. 225; to find all the factors of a number, you must check every whole number less than or equal to the
square root of the number. If Mr. Matsumoto must check the numbers from 1 through 15, the number must
be greater than or equal to , or 225, and less than , or 256.
b. 255; as mentioned in the answer to part (a), the number must be less than , or 256. The greatest it could
be is 255.
PTS:
OBJ:
STA:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: prime number | composite number | abundant numbers | perfect numbers | deficient numbers | factor
31.
ANS:
11
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions
32.
ANS:
Answers will vary.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions
33.
ANS:
a. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
b.
c. 4, 9, 25, 49 (the squares of primes)
d. 121
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns
factor pair | rectangular model | dimensions | square numbers | factor
34.
ANS:
81
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns
factor pair | rectangular model | dimensions | prime factorization | prime number
35.
ANS:
a. 1, 2, 10, 20; If a number has 4 and 5 as factors, it must have the factors of 4 and 5 as factors, namely 1,
2, 4, and 5. It must also have the products of 2 and 5 and of 4 and 5 as factors, since these pairs of factors
do not have any common factors.
b. The smallest number is 20, because 4 and 5 do not have any common factors.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor
36.
ANS:
a. 6
b. 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, and 96
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA:
Problem 2.1 Finding Patterns
factor pair | rectangular model | dimensions | factor | multiple
6PA 2.2.8.B
37.
ANS:
a. Ben bought four 9-ounce tubes. Aaron bought three 12-ounce tubes.
b. The 12-ounce tube is the better buy at 9.6 cents per ounce. The 9-ounce tube cost .9 cents per ounce.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.1 Finding Common Multiples
common multiples | least common multiple | common factors | greatest common factor
38.
ANS:
a
PTS: 1
DIF: L2
REF: Prime Time | Extra Questions
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: divisible by | prime number | composite number | abundant numbers | perfect numbers | deficient
numbers | divisibility
39.
ANS:
Both are correct but Tyrone's is the accepted form. When we make a factor string, we use only prime factors.
Otherwise, the strings could go on forever.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.1 Finding Factor Strings
strings of factors | factorization | prime factorization
40.
ANS:
120; The first three prime numbers are 2, 3, and 5. The first three composite numbers are 4 = 2 multiply 2, 6 =
2 multiply 3, and 8 = 2 multiply 2 multiply 2. The shortest string that contains the factors of all these numbers
is 2 multiply 2 multiply 2 multiply 3 multiply 5. The smallest number that is divisible by all the numbers is
the product of this string, which is 120.
PTS: 1
DIF: L2
REF: Prime Time | Extra Questions
OBJ: Investigation 2: Whole-Number Patterns and Relationships
NAT: NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
TOP: Problem 2.1 Finding Patterns
KEY: factor pair | rectangular model | dimensions | prime number | divisible by | divisibility | composite
number
41.
ANS:
The best move in this case would be 25, which gives your opponent only 5 points. Note: A prime number
would be a bad move, since its only proper factor, 1, has already been circled.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
whole number factors | whole number divisors | factor | proper factors
42.
ANS:
Possible response: We played the Factor Game today. I discovered that it is best to go first and choose 29, the
highest prime on the board, as your first move. After the first move, choose numbers like 25 that leave your
opponent a small number of factors. Stay away from numbers like 30, which have many factors, until most of
the factors are already circled.
PTS:
OBJ:
STA:
KEY:
factors
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
composite number | prime number | whole number factors | whole number divisors | factor | proper
43.
ANS:
a.
b. Possible answer: Two cookies in a bag would be affordable and is a number a student would
typically eat. This would also allow more students to buy cookies.
c. Possible answer: Each cookie cost $0.15 to make. They could be sold at $0.25 per cookie. So, a bag
of one would cost $0.25, a bag of two would cost $0.50, …
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.4 Finding the Greatest Common Factor
factor | greatest common factor | least common multiple | multiple
44.
ANS:
a. Alternate multiplying of terms by 2 and 3 to generate the next term.
b. 1,512 and 4,536
c. 7, since it is the only prime number in the sequence and is a factor of all the other terms in the sequence
no matter how many terms are added.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | greatest common factor | prime factorization | factor
45.
ANS:
a. Any two numbers that do not share a common factor (relatively prime numbers) will work. Examples
are 3 and 4, 11 and 12, 15 and 8.
b. Any two numbers that share a common factor will work. Examples are 15 and 9, 10 and 25, 18 and 48,
45 and 8l.
c. If the numbers do not have a common factor, their least common multiple will be equal to their product.
If the numbers have a common factor, their least common multiple will be less than their product.
PTS:
OBJ:
NAT:
STA:
TOP:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding Least Common Multiple KEY: least common multiple | multiple
46.
ANS:
a. , or written as
b. Answers will vary. Some possible answers: It is an even number. It is not an odd number. Nine is one
of its factors. It is divisible by 15. It is not prime. It is composite because it has more than 2 factors. This
is its one unique string of factorization (Fundamental Theorem of Arithmetic). It is a square number
because you could group its prime factors to represent two of the same numbers multiplying themselves:
multiplied by is the same as saying or 302, which is 900.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | Extra Questions
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String
factor tree | exponent | Fundamental Theorem of Arithmetic
47.
a. 65
ANS:
b. 17
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
prime number | whole number factors | whole number divisors | factor | proper factors
48.
ANS:
Yes; since 2 and 5 are factors of 10, any number that has 10 as a factor must also have 5 as a factor.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
factor | prime number | composite number | abundant numbers | perfect numbers | deficient numbers
49.
ANS:
No; for example, the number 35 has 7 as a factor, but since it is an odd number it does not have 2 as a factor.
2 and 7 are both prime factors of 14; for a number to be a factor of 2 and 7, it would also be a factor of 14.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
factor | prime number | composite number | abundant numbers | perfect numbers | deficient numbers
50.
ANS:
a. 29 and 30, which have a product of .
b. These two numbers give the largest product because they are the largest numbers.
c. 1, 2, 3, 5, 6, 10, 15, 29, 30, 58, 87, 145, 174, 290, 435
One way to find the factors is to test factors below 29 to identify factor pairs. We know that so the
middle factor pair is .
PTS: 1
DIF: L2
REF: Prime Time | AP Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: factor pair | factor | prime number | composite number | abundant numbers | perfect numbers |
deficient numbers
51.
ANS:
Answers will vary, but the following are examples of correct answers:
a.
b.
c.
d.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
factorization | factoring | factor | multiple | square numbers | divisible by | near-perfect numbers
52.
a. 3, 5, 2
ANS:
b. possible answer: 2, 8, 14 and 3, 12, 21
c. infinitely many
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
multiple | square numbers | divisible by | near-perfect numbers
53.
a. ; ; ;
ANS:
b. ; ; ; ; ; ; ;
c. ; ; ; ; ; ; ; ; ; ; ;
d. A number is called a factor when it is multiplied by another number to find a product. A number is
called a divisor of a number when it divides the dividend evenly to find a quotient.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.3 Finding Multiples
factor | multiple | square numbers | divisible by | near-perfect numbers
54.
ANS:
The dimensions are . The possible dimensions are , , , and . Only the rectangle has dimensions with a sum of
11.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor | factor pair | rectangular model | dimensions
55.
ANS:
The dimensions are . The possible dimensions are , , , , and . Only the rectangle has dimensions with a sum
that is a prime number.
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns
factor pair | rectangular model | dimensions | prime number | factor
56.
ANS:
a. , , ,
b. , , ,
c.
d. , , ,
e. , ,
f. , , , , ,
g. , , ,
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor
57.
ANS:
Phillip’s number is 18 since the factor pairs , , and have the required sums and 18 < 20.
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor
58.
ANS:
In order to determine the number of tiles in each of the rectangles, multiply the tiles along the length by the
tiles on the width.
a. 60
b. 40
c. 40
PTS:
OBJ:
NAT:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.1 Finding Patterns KEY: factor pair | rectangular model | dimensions | factor
59.
ANS:
a.
b. 1 is the only number in the intersection since 13 is a prime number.
c. Answers will vary; examples include factors of two different prime numbers (e.g., 5 and 13) or
factors of two different relatively prime numbers (e.g., 11 and 14).
PTS: 1
DIF: L2
REF: Prime Time | AP Investigation 2
OBJ: Investigation 2: Whole-Number Patterns and Relationships
NAT: NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
TOP: Problem 2.3 Classifying Numbers
KEY: prime number | composite number | factor | Venn diagram | intersection | greatest common factor |
least common multiple
60.
ANS:
a.
b. The numbers in the intersection are the multiples of 10, which is 5 × 2. Every number that is a multiple
of 10 must have 5 and 2 as factors since 5 × 2 = 10.
c. 75 would be placed with the multiples of 5 since it is a multiple of 5 but not a multiple of 2. 90 would
be placed in the intersection since it is a multiple of both 5 and 2.
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.3 Classifying Numbers
multiple | Venn diagram | intersection | greatest common factor | least common multiple
61.
ANS:
The fourth number is even. Since the first three numbers were even, odd, and odd, respectively, the sum of
these three will be even. Thus, an even number must be added to this even sum to produce an even number.
PTS:
OBJ:
NAT:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 2
Investigation 2: Whole-Number Patterns and Relationships
NAEP N5a| NAEP N5b| NAEP N5f
STA: 6PA 2.2.8.B
Problem 2.2 Reasoning with Even and Odd Numbers
even numbers | odd numbers | conjecture
62.
ANS:
a. Susan shouldn’t have to wait at all. The #14 bus should arrive at the mall at 10 A.M. and the #11 bus
should leave the mall for the museum at about 10 A.M. (since the #11 bus runs every 12 minutes, it leaves
at the top of every hour).
b. Susan will have to wait 12 minutes because the #14 bus should arrive at noon.
c. Both buses are at the mall at 9 A.M., 10 A.M., 11 A.M., and noon because the least common multiple of
15 and 12 is 60.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
factor
1
DIF: L2
REF: Prime Time | AP Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.1 Finding Common Multiples
factor | multiple | common multiples | least common multiple | common factors | greatest common
63.
ANS:
a. One rectangle is made with 42 tiles, and the other is made with 56 tiles. These are the only two even
multiples of 7 between 40 and 60.
b. The rectangle with 42 tiles has a length of 6, and the rectangle with 56 tiles has a length of 8. These
answers are found by finding the other number in the factor pair with 7 for each rectangle.
c. Students’ answers will vary. For 42: , , or .
For 56: , , or .
PTS: 1
DIF: L2
REF: Prime Time | AP Investigation 3
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.1 Finding Common Multiples
KEY: even numbers | factor | multiple | common multiples | least common multiple | common factors |
greatest common factor
64.
ANS:
Jack will only have a conflict one day per month on the 21st. The least common multiple of 3 and 7 is 21. The
next common multiple, 42, is greater than the number of days in a month.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple
factor | multiple | common multiples | least common multiple | common factors
65.
ANS:
a. Since the numbers are prime, they don’t have any proper factors other than 1. Therefore, their least
common multiple would be their product.
b. Since the numbers are prime, the only factors each number has is 1 and itself. Therefore, the greatest
common factor must be 1.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple | Problem 3.4 Finding the Greatest Common Factor
common factors | least common multiple | multiple | prime number | greatest common factor
66.
ANS:
a. LCM: 24; GCF: 4
b. LCM: 105; GCF: 1
c. LCM: 187; GCF: 1
d. LCM: 108; GCF: 36
e. Parts (b) and (c), for part ((b), the two numbers are relatively prime. For part (c), the two numbers are
prime.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple | Problem 3.4 Finding the Greatest Common Factor
factor | multiple | greatest common factor | least common multiple
67.
ANS:
a. 4
b. 5
c. 10
d. 25
e. The greatest common factor of two numbers is one of the two numbers when the smaller number is a
factor of the larger number
PTS:
OBJ:
NAT:
STA:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.4 Finding the Greatest Common Factor
KEY: factor | greatest common factor
68.
ANS:
a.
b.
c.
d.
e. 23 is prime.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.1 Finding Factor Strings
factor | strings of factors | factorization | prime factorization
69.
a.
b.
c.
d.
ANS:
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Finding the Longest Factor String
factor tree | exponent | Fundamental Theorem of Arithmetic
70.
ANS:
a. GCF = 5, LCM = 525
b. GCF = 27, LCM = 81
c. GCF = 9, LCM = 252
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | greatest common factor | least common multiple
71.
ANS:
105; an odd number cannot have a factor of 2, and 3, 5, and 7 are the only three primes with a product less
than 160.
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | odd numbers | prime number | prime factorization
72.
ANS:
1,800
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | prime factorization | exponent
73.
ANS:
a. Possible answer: 6 and 36
b. Possible answer 12 and 60
c. The least common multiple is the other number in the pair.
PTS: 1
DIF: L2
REF: Prime Time | AP Investigation 4
OBJ: Investigation 4: Factorizations: Searching for Factor Strings
NAT: NAEP N5b| NAEP N5d
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 4.3 Using Prime Factorizations
KEY: prime factorization | abundant numbers | factor | relatively prime | greatest common factor | least
common multiple
74.
ANS:
a. Yes, the only common factor they have is 1.
b. Yes, the only common factor they have is 1.
c. No, even numbers always have a factor of 2.
d. Make sure that all the factors of the second number differ from the first (except for 1).
PTS:
OBJ:
NAT:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | factor | common factors
75.
ANS:
Begin with the number 1, repeatedly multiply by 3 until the result exceeds 1,000. The numbers would be 3, 9,
27, 81, 243, 729.
PTS:
OBJ:
STA:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
KEY: factor | prime factor
76.
ANS:
a. Alternate multiplying terms by 5 and then 3. In other words, multiply the first term by 5 to get the
second term, multiply the second term by 3 to get the third, multiply the third term by 5 to get the fourth,
multiply the fourth term by 3 to get the fifth, and so on.
b. 3,375 and 10,125.
c. The greatest common factor is 3, since it is the first term in the sequence and a prime number.
PTS:
OBJ:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
factor | multiple | greatest common factor
77.
a. Student 1
ANS:
b. Students 1, 2, and 4
PTS:
OBJ:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
factor | multiple | greatest common factor | least common multiple
78.
ANS:
a. Locker 1,000
b. Locker 988
c. Locker 960
d. Locker 600
PTS:
OBJ:
STA:
TOP:
KEY:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
factor | multiple | greatest common factor | least common multiple
79.
ANS:
The largest prime desert less than 50 is {24, 25, 26, 27, 28}.
PTS:
OBJ:
STA:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
KEY: factor | multiple | prime number
80.
ANS:
a.
b.
PTS:
OBJ:
STA:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
KEY: factor | multiple | prime number
81.
a.
b.
c.
ANS:
d.
PTS:
OBJ:
STA:
TOP:
1
DIF: L2
REF: Prime Time | AP Investigation 5
Investigation 5: Putting It All Together
NAT: NAEP N5b| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 5.1 Using Multiples and Factors
KEY: factor | prime factorization
82.
ANS:
71,094 is not divisible by 5.
PTS:
OBJ:
STA:
KEY:
1
DIF: L1
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
divisible | divisibility test | factor
83.
ANS:
2, 3, 4
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
divisible | divisibility test | composite number | prime number
84.
ANS:
PTS:
OBJ:
STA:
TOP:
KEY:
1
DIF: L1
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.1 Finding Factor Strings
strings of factors | prime factorization | composite number | prime number | factor tree | factor
a.
85.
2
b.
5 or 2 students in each group
PTS:
OBJ:
STA:
TOP:
KEY:
1
DIF: L2
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.2 Using Prime Factorizations
multi-part question | factor | factoring | prime number | word problem | problem solving
ANS:
MULTIPLE CHOICE
OBJ:
NAT:
STA:
TOP:
86.
ANS: C
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.4 Finding the Greatest Common Factor
KEY: factor | greatest common factor
87.
ANS: B
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.3 Finding Common Factors
KEY: factor
OBJ:
NAT:
STA:
TOP:
88.
ANS: D
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.3 Finding Common Factors
KEY: multiple
OBJ:
NAT:
STA:
TOP:
KEY:
89.
ANS: D
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.1 Finding Factor Strings
strings of factors | factorization | prime factorization | factor
90.
ANS: D
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.2 Finding the Least Common Multiple
KEY: multiple | common multiples
OBJ:
NAT:
STA:
TOP:
KEY:
91.
ANS: D
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
relatively prime | prime factorization
OBJ:
NAT:
STA:
TOP:
92.
ANS: C
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple
KEY: multiple | least common multiple
93.
ANS: A
PTS: 1
DIF: L2
REF:
OBJ: Investigation 2: Whole-Number Patterns and Relationships
NAT: NAEP N5a| NAEP N5b| NAEP N5f
STA:
TOP: Problem 2.2 Reasoning with Even and Odd Numbers
KEY: even numbers | odd numbers | conjecture
OBJ:
NAT:
STA:
TOP:
Prime Time | Multiple Choice
6PA 2.2.8.B
94.
ANS: B
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
KEY: relatively prime | prime factorization | factor
OBJ:
NAT:
STA:
TOP:
95.
ANS: C
PTS: 1
DIF: L2
REF: Prime Time | Multiple Choice
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple
KEY: multiple | least common multiple
96.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisible | divisibility test | proper factors
97.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: divisible | divisibility test | factor | composite number | prime number
98.
ANS: C
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.1 Finding Common Multiples | Problem 3.2 Finding the Least Common Multiple | Problem
3.3 Finding Common Factors | Problem 3.4 Finding the Greatest Common Factor
KEY: multiple | least common multiple | prime factorization | factor
99.
ANS: C
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5d
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 4.1 Finding Factor Strings | Problem 4.2 Finding the Longest Factor String | Problem 4.3
Using Prime Factorzations
KEY: prime factorization | prime number | factoring | factor | factor tree | exponent
100.
ANS: C
PTS: 1
DIF: L1
REF: Skills Practice Investigation 4
OBJ: Investigation 4: Factorizations: Searching for Factor Strings
NAT: NAEP N5b| NAEP N5d
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 4.1 Finding Factor Strings | Problem 4.2 Finding the Longest Factor String | Problem 4.3
Using Prime Factorizations
KEY: prime factorization | prime number | factoring | factor | factor tree | exponent
101.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisibility | factor
OBJ:
STA:
102.
ANS: B
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisibility | factor
103.
ANS: A
PTS: 1
DIF: L2
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisibility | factor
104.
ANS: C
PTS: 1
DIF: L2
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisible | divisibility test
105.
ANS: D
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisible | divisibility test
106.
ANS: D
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisible | divisibility test
OBJ:
NAT:
STA:
TOP:
107.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.2 Finding the Least Common Multiple
KEY: multiple | least common multiple
108.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: composite number | prime number
109.
ANS: B
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.4 Finding the Greatest Common Factor
KEY: greatest common factor | prime factorization | factor
OBJ:
NAT:
STA:
TOP:
KEY:
110.
ANS: B
PTS: 1
DIF: L2
REF: Skills Practice Investigation 4
Investigation 4: Factorizations: Searching for Factor Strings
NAEP N5b| NAEP N5d
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 4.3 Using Prime Factorizations
greatest common factor | prime factorization | factor | word problem | problem solving
111.
ANS: D
PTS: 1
DIF: L1
REF:
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
Skills Practice Investigation 3
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.4 Finding the Greatest Common Factor
KEY: greatest common factor | factor | factoring
OBJ:
NAT:
STA:
TOP:
KEY:
112.
ANS: C
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.4 Finding the Greatest Common Factor
division ladder | greatest common factor | factor | factoring
113.
ANS: C
PTS: 1
DIF: L2
REF: Skills Practice Investigation 3
OBJ: Investigation 3: Common Multiples and Common Factors
NAT: NAEP N5b| NAEP N5c| NAEP N5f
STA: 6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
TOP: Problem 3.4 Finding the Greatest Common Factor
KEY: greatest common factor | factor | factoring
OBJ:
NAT:
STA:
TOP:
KEY:
114.
ANS: B
PTS: 1
DIF: L1
REF: Skills Practice Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.4 Finding the Greatest Common Factor
greatest common factor | factor | factoring
OBJ:
NAT:
STA:
TOP:
KEY:
115.
ANS: D
PTS: 1
DIF: L3
REF: Skills Practice Investigation 3
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.3 Finding Common Factors
greatest common factor | factor | factoring | word problem | problem solving
OBJ:
NAT:
STA:
TOP:
116.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 4
Investigation 3: Common Multiples and Common Factors
NAEP N5b| NAEP N5c| NAEP N5f
6PA 2.2.8.B| 6PA M6.A.1.3.1| 6PA M6.A.1.3.2| 6PA M6.A.1.3.3
Problem 3.3 Finding Common Factors
KEY: factor | factoring
117.
ANS: A
PTS: 1
DIF: L1
REF: Skills Practice Investigation 1
OBJ: Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.2 Prime and Composite Numbers
KEY: composite number
OTHER
118.
ANS:
No; the sum of the digits is not divisible by 9.
PTS:
OBJ:
1
DIF: L2
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
STA: 6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
KEY: divisibility | writing in math
119.
ANS:
Since 1 + 0 + 8 = 9, and 9 is divisible by 9, 108 is divisible by 9. So Jennifer can divide her CDs evenly
among her boxes. She can put 12 CDs in each box. Since 193 does not end in 0 or 5, 193 is not divisible by 5.
So Melissa cannot divide her CDs evenly among her boxes.
PTS:
OBJ:
STA:
KEY:
1
DIF: L2
REF: Skills Practice Investigation 1
Investigation 1: Factors and Products
NAT: NAEP N5b| NAEP N5c| NAEP N5d
6PA 2.2.8.B
TOP: Problem 1.1 Finding Proper Factors
divisible | divisibility test | word problem | problem solving | writing in math | reasoning