Download Number Operations and Integers

Number Operations and Integers
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
____
1. Order the numbers from least to greatest.
1,994
1,984
1,910
____
2.
____
3.
____
4.
____
5.
____
6.
____
7.
____
8.
____
9.
a. 1,984; 1,910; 1,994
c. 1,910; 1,984; 1,994
b. 1,994; 1,984; 1,910
Evaluate 2 • 9 • 5.
a. 90
c. 180
b. 10
d. 45
Use the Distributive Property to find the product of 3 • 68.
a. 216
c. 102
b. 204
d. 408
List all of the factors of 18.
a. 1, 18
c. 1, 2, 3, 6, 9, 18
b. 1, 2, 2, 3, 6, 18
d. 2, 3, 6, 9
List all of the factors of 13.
a. 1, 13
c. 1, 2, 3, 4, 6, 13
b. 1, 2, 6, 13
d. 1, 3, 4, 13
Find the GCF of 45, 60, and 18.
a. 3
c. 2
b. 4
d. 6
What is the least common multiple of the numbers 12 and 9?
a. 36
c. 48
b. 27
d. 34
What is the least common multiple of the numbers 4, 5, and 10?
a. 20
c. 16
b. 42
d. 15
Order the integers 3, –7, 6, and 0 from least to greatest, and then plot each of them on a number line.
a. 6, 3, 0, –7
–10 –8
–6
–4
–2
0
2
4
6
8
10
–4
–2
0
2
4
6
8
10
–4
–2
0
2
4
6
8
10
–4
–2
0
2
4
6
8
10
b. –7, 0, 3, 6
–10 –8
–6
c. 3, –7, 6, 0
–10 –8
–6
d. –7, 0, 3, 6
–10 –8
–6
____ 10. Find the product of –5 (–6).
____ 11.
____ 12.
____ 13.
____ 14.
a. –11
c. 1
b. 30
d. –30
Find the quotient of 66 ÷ 11.
a. 77
c. –6
b. 6
d. 7
Find the greatest common factor (GCF) of 21, 56, and 49.
a. The GCF is 7.
c. The GCF is 3.
b. The GCF is 8.
d. The GCF is 6.
Find the least common multiple (LCM) of 3, 8, and 20.
a. The LCM is 160.
c. The LCM is 24.
b. The LCM is 120.
d. The LCM is 20.
Write the integers –1, –7, 6, –6, 8, and 0 in order from least to greatest, and then plot each of them on a
number line.
a. –7, –6, –1, 0, 6, 8
–10 –8
–6
–4
–2
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
b. –1, –7, 6, –6, 8, 0
–10 –8
–6
–4
–2
c. –7, –6, –1, 0, 6, 8
–10 –8
–6
–4
–2
d. 8, 6, 0, –1, –6, –7
–10 –8
–6
–4
–2
____ 15. Add.
–34 + (–35)
a. 69
b. –69
____ 16. Subtract.
–25 26
c. 1
d. –1
a. 1
b. –1
____ 17. Multiply.
4 • (–6)
c. 51
d. –51
a. 2
b. –24
____ 18. Find the quotient.
–102 (–3)
c. –2
d. 24
a. –99
b. –105
Short Answer
c. –34
d. 34
19. Order the following numbers from least to greatest. Show your work.
1,482; 1,827; 1,431; 1,542; 1,772
20. Use the Distributive Property to find the product of 7  46. Show your work.
21. Consider the following integers: 7, –4, 0, –1, and 3.
a. Order the numbers from least to greatest.
b. Graph the integers on the same number line.
22. There are two ways to find the greatest common factor (GCF) of 32 and 40. Find the GCF of 32 and 40 by
using both of the following methods.
a.
Find the GCF by using a list of all factors for each number.
b.
Find the GCF by using the prime factorization of each number.
Number Operations and Integers
Answer Section
MULTIPLE CHOICE
1. ANS: C
Graph the three numbers on a number line like the following. The numbers are then in the correct order when
the number line is read from left to right.
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
KEY: number line, compare, order, whole numbers
NOT: /A/Did you start at the left of each number and compare the digits in the same place-value position?
/B/Did you sort the numbers from greatest to least? /C/Correct!
2. ANS: A
The first and third numbers make a product that ends in zero. Multiply these values together first, then use
mental math to get the answer.
KEY: addition, multiplication, properties, whole numbers
NOT: /A/Correct! /B/Look at the ones digits. Do any of these digits multiplied together produce a product
that is a multiple of 10? /C/Look at the ones digits. Do any of these digits multiplied together produce a
product that is a multiple of 10?/D/Look at the ones digits. Do any of these digits multiplied together produce
a product that is a multiple of 10?
3. ANS: B
Multiply the first number by the ones digit of the second number. Use mental math to multiply the first
number by ten times the tens digit of the second number. Use mental math to add these two products.
KEY: distributive property, multiplication
NOT: /A/Multiply the first number by each digit in the second number, then add the two products.
/B/Correct! /C/Multiply the first number by each digit in the second number, then add the two products.
/D/Multiply the first number by each digit in the second number, then add the two products.
4. ANS: C
Factors are whole numbers that are multiplied together to find a product. Begin listing factors in pairs. When
the pairs of factors begin to repeat, then you have found all of the factors.
KEY: factor, prime
NOT: /A/Is this a prime number? /B/Are there any factors here that don't belong? /C/Correct! /D/Did you
remember to include 1 and the number itself in the list of factors?
5. ANS: A
13 is a prime number. The only factors of a prime number are 1 and the number itself.
KEY: factor, prime
NOT: /A/Correct! /B/Is this number divisible by 2? /C/Is this number divisible by 2 or 3? /D/Is this number
divisible by 3?
6. ANS: A
List all the factors of each number, and find the factors that are common to all three. The greatest one is the
GCF.
KEY: GCF, greatest common factor, factor
NOT: /A/Correct! /B/Are you sure that this number is a factor of any of the numbers listed? If so, is it a
factor of all three numbers? /C/Is this number a common factor of all three numbers? If so, are there any
common factors greater than 2? /D/Is this number a factor of all three numbers?
7. ANS: A
There are two ways to find the least common multiple.
For example:
Find the LCM of 12 and 20.
A list can be used to find the LCM:
12: 12, 24, 36, 48, {60}, 72, 84
20: 20, 40, {60}
List some multiples of each number.
Choose the least value that is in both lists.
The LCM is 60.
Another way to find the LCM is to find the prime factorization of each number.
12 = { • 2} • 3
20 = {2 • 2} •5
{2 • 2} • 3 • 5, or 60
once.
Note the prime factors that are common to
both numbers.
Multiply all of the prime factors, using those that are common to both numbers only
The LCM is 60.
KEY: LCM, least common multiple
NOT: /A/Correct! /B/Is this the least multiple of both numbers? /C/Are both numbers factors of this number?
/D/Is this a multiple of either number?
8. ANS: A
There are two ways to find the least common multiple.
For example:
Find the LCM of 12 and 20.
A list can be used to find the LCM:
12: 12, 24, 36, 48, {60}, 72, 84
20: 20, 40, {60}
List some multiples of each number.
Choose the least value that is in both lists.
The LCM is 60.
Another way to find the LCM is to find the prime factorization of each number.
12 = { • 2} • 3
20 = {2 • 2} •5
{2 • 2} • 3 • 5, or 60
once.
Note the prime factors that are common to
both numbers.
Multiply all of the prime factors, using those that are common to both numbers only
The LCM is 60.
KEY: LCM, least common multiple
NOT: /A/Correct! /B/Is this a multiple of any of the numbers? /C/Is this a multiple of all the numbers? /D/Do
all the numbers divide evenly into this number?
9. ANS: D
Order the integers from least to greatest, then and plot each integer on a number line. Remember that negative
numbers are to the left of 0.
KEY: integer, order
NOT: /A/The integers must be in order from least to greatest. /B/Be sure to plot each integer on the number
line. /C/The integers must be in order from least to greatest. /D/Correct!
10. ANS: B
Multiply the two integers. If the signs are the same, the product will be positive; if the signs are different, the
product will be negative.
KEY: integer, multiplication
NOT: /A/Be sure to multiply the integers. /B/Correct! /C/Be sure to multiply the integers. /D/If the signs of
the two integers are the same, the product will be positive. If the signs are different, the product will be
negative.
11. ANS: B
Remember that division is the inverse of multiplication. Think of the multiplication problem that is related to
this division problem.
KEY: integer, division
NOT: /A/Remember to divide the integers. /B/Correct! /C/If the signs of the two integers are the same, the
sign of the quotient will be positive; if they are different, the sign of the quotient will be negative. /D/Doublecheck your division.
12. ANS: A
Example: Find the GCF of 18, 24, and 48.
Method 1: Use lists.
18: 1, 2, 3, {6}, 9, 18
24: 1, 2, 3, 4, {6}, 8, 12, 24
48: 1, 2, 3, 4, {6}, 8, 12, 16, 24, 48
The GCF is 6.
List all the factors of each number.
Choose the greatest factor that is in all the lists.
Method 2: Use prime factorization.
The GCF is 6.
KEY: GCF, greatest common factor
NOT: /A/Correct! /B/Check to see whether this number divides into all three of the numbers./C/Are you sure
that this is a factor of all three of the numbers? /D/Is this a factor of all three numbers?
13. ANS: B
Example:
Find the LCM of 112, 96 and 144
A list can be used to find the LCM:
16: 16, 32, {48}, 64, 80, 96, 112
24: 24, {48}, 72, 96
48: {48}, 96, 144
List some multiples of each number.
Choose the least value that is in all the lists.
The LCM is 48.
Another way to find the LCM is to find the prime factorization of each number.
16 = {  2  2}  2
24 = {2  2  2} 3
48 = {2  2  2}  2  3
{2  2  2}  2  3 or 48.
Note the prime factors that are common to all three numbers
Multiply all of the prime factors, using those that are common
to all three numbers only once.
The LCM is 48.
KEY: LCM, least common multiple
NOT: /A/Is this a multiple of the least number? If so, check to see whether there is a lesser multiple common
to all three of the numbers. /B/Correct! /C/Is this a multiple of the greatest number? If so, check to see
whether there is a lesser multiple common to all three./D/Check to be sure this is a multiple common to all
three of the numbers.
14. ANS: C
Put the integers in ascending order and plot each integer on the number line.Remember that negative numbers
are to the left of zero on the number line.
KEY: integers, order, number line
NOT: /A/Be sure to plot all of the given integers on the number line. /B/Use the number line to help put the
integers in order from least to greatest./C/Correct! /D/The integers must be in order from least to greatest.
15. ANS: B
To add two integers with the same sign, find the sum of their absolute values and use the sign of the two
integers. To add two integers with different signs, find the difference of their absolute values and use the sign
of the integer with the greater absolute value.
KEY: addition, integers
NOT: /A/Double-check the sign of your answer. /B/Correct! /C/When adding integers with different signs,
do you add or subtract their absolute values? /D/When adding integers with different signs, do you add or
subtract their absolute values?
16. ANS: D
To subtract an integer, add its opposite.
KEY: integers, subtraction
NOT: /A/Subtracting a number is the same as adding its opposite. What is the opposite of the number being
subtracted? /B/When subtracting integers, for which integer do you use the opposite? /C/Subtracting a number
is the same as adding its opposite. When adding integers with different signs, how do you find the sign of the
answer? /D/Correct!
17. ANS: B
Find the product of the two integers. If the signs are the same, the product will be positive. If the signs are
different, the product will be negative.
KEY: integers, multiplication
NOT: /A/Remember to multiply the integers together. /B/Correct! /C/Remember to multiply the integers
together. /D/Remember the rules about the sign of a product.
18. ANS: D
Find the quotient of the two integers. If the signs are the same, the quotient will be positive. If the signs are
different, the quotient will be negative.
KEY: integers, division
NOT: /A/Remember to divide the integers. /B/Remember to divide the integers./C/Remember the rules about
the sign of a quotient. /D/Correct!
SHORT ANSWER
19. ANS:
1,431; 1,482; 1,542; 1,772; 1,827
1431 1482
1542
1772 1827
1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850
Graph the numbers on a number line.
The numbers are then in the correct order when the number line is read from left to right.
KEY: compare, order, whole numbers
20. ANS:
322
7 • 46 = 7 • (40 + 6)
7 • (40 + 6) = (7 • 40) + (7 • 6)
(7 • 40) + (7 • 6) = 280 + 42
280 + 42 = 322
KEY: mental math
21. ANS:
a.
–4, –1, 0, 3, 7
b.
–8
–6
–4
–2
0
2
KEY: integer, compare, order
22. ANS:
a.
32: 1, 2, 4, {8}, 16, 32
40: 1, 2, 4, 5, {8}, 10, 20, 40
The GCF of 32 and 40 is 8.
b.
32 = {2} • {2} • {2} • 2 • 2
4
6
8
40 = {2} • {2} • {2} • 5
2•2•2=8
The GCF of 32 and 40 is 8.
KEY: GCF, greatest common factor, Performance Assessment, prime factorization