Divisibility%20and%20Factors

1
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Karen placed 56 bottles into boxes that each held
6 bottles. How many boxes did she use?
10
4-1
2
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
(For help, go to Skills Handbook p. 760.)
Find each quotient. 1. 480 3 2. 365 5
3. 459 9 4. 288 6 5. 6.
Check Skills Youll Need
4-1
3
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
4-1
4
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Solutions (continued)
4-1
5
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Is the first number divisible by the second?
a. 1,028 by 2
Yes 1,028 ends in 8.
b. 572 by 5
No 572 doesnt end in 0 or 5.
c. 275 by 10
No 275 doesnt end in 0.
Quick Check
4-1
6
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Is the first number divisible by the second?
a. 1,028 by 3
No 1 0 2 8 11 11 is not divisible by 3.
b. 522 by 9
Yes 5 2 2 9 9 is divisible by 9.
Quick Check
4-1
7
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
Ms. Washingtons class is having a class photo
taken. Each row must have the same number of
students. There are 35 students in the class. How
can Ms. Washington arrange the students in rows
if there must be at least 5 students, but no
more than 10 students, in each row?
1 35, 5 7  Find pairs of factors of 35.
There can be 5 rows of 7 students, or 7 rows of 5
students.
Quick Check
4-1
8
Divisibility and Factors
PRE-ALGEBRA LESSON 4-1
State whether each number is divisible by 2, 3,
5, 9, or 10. 1. 18 2. 90 3. 81 4. 25
5. List the positive factors of 36.
2, 3, 9
2, 3, 5, 9, 10
3, 9
5
1, 2, 3, 4, 6, 9, 12, 18, 36
4-1
9
Exponents
PRE-ALGEBRA LESSON 4-2
Which word best completes the statement
sometimes, always, or never?
If the product of two factors is zero, both
factors are zero.
sometimes
4-2
10
Exponents
PRE-ALGEBRA LESSON 4-2
(For help, go to Lesson 1-9.)
Find each product. 1. 3 3 3 3 2. 12
(12) 3. (4)(4)(4) 4. 10 10 10
10
Check Skills Youll Need
4-2
11
Exponents
PRE-ALGEBRA LESSON 4-2
Solutions 1. 3 3 3 3 81 2. 12 (12)
144 3. (4)(4)(4) 64 4. 10 10 10
10 10,000
4-2
12
Exponents
PRE-ALGEBRA LESSON 4-2
Write using exponents.
a. (11)(11)(11)(11)
b. 5 x x y y x
Quick Check
4-2
13
Exponents
PRE-ALGEBRA LESSON 4-2
Suppose a certain star is 104 light-years from
Earth. How many light-years is that?
Quick Check
4-2
14
Exponents
PRE-ALGEBRA LESSON 4-2
a. Simplify 3(1 4)3.
b. Evaluate 7(w 3)3 z, for w 5 and z 6.
Quick Check
4-2
15
Exponents
PRE-ALGEBRA LESSON 4-2
Write using exponents. 1. x y z x z
2. a b b b 3 3. Simplify 5(2
4)2. 4. Evaluate (g3 7)2 5 4, for g
3.
x2yz2
3ab3
180
2,004
4-2
16
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Find three integers whose sum is 12 and whose
product is 42.
2, 3, 7
4-3
17
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
(For help, go to Lesson 4-1.)
List the positive factors of each
number. 1. 15 2. 35 3. 7 4. 20
5. 100 6. 121
Check Skills Youll Need
18
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Solutions 1. 1 15, 3 5 1, 3, 5,
15 2. 1 35, 5 7 1, 5, 7, 35 3. 1
7 1, 7 4. 1 20, 2 10, 4 5 1, 2,
4, 5, 10, 20 5. 1 100, 2 50, 4 25, 5
20, 10 10 1, 2, 4, 5, 10, 20, 25, 50,
100 6. 1 121, 11 11 1, 11, 121
19
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
State whether each number is prime or composite.
Explain.
a. 46
Composite 46 has more than two factors, 1, 2,
23, and 46.
b. 13
Prime 13 has exactly 2 factors, 1 and 13.
Quick Check
20
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Use a factor tree to write the prime
factorization of 273.
273
273 3 7 13 
Quick Check
21
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Find the GCF of each pair of numbers or
expressions.
a. 24 and 30
The GCF of 24 and 30 is 6.
b. 36ab2 and 81b
The GCF of 36ab2 and 81b is 9b.
Quick Check
22
Prime Factorization and Greatest Common Factor
PRE-ALGEBRA LESSON 4-3
Tell whether each number is prime or
composite. 1. 123 2. 47 Write the prime
factorization for each number. 3. 64 4. 45
Find the GCF for each pair. 5. 80 and 120
6. 62b3c2d and 31b2c3d
composite
prime
26
32 5
40
31b2c2d
23
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Find the GCF for each pair of numbers. a. 12 and
18 b. 15 and 60
c. 54 and 60
6
15
6
24
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
(For help, go to Lesson 4-3.)
Find each GCF. 1. 14, 21 2. 48, 60
3. 5mn, 15m2n 4. 63r2, 48s3
Check Skills Youll Need
25
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Solutions
1. 14, 21 14 2 7 21 3 7 GCF 7
2. 48, 60 48 24 3 60 22 3 5
GCF 22 3 12
3. 5mn, 15m2n 5mn 5 m n 15m2n 3 5
m2 n GCF 5 m n 5mn
4. 63r2, 48s3 63r2 32 7 r2 48s3 24
3 s3 GCF 3
26
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
18 21
Find two fractions equivalent to .
18 21
a. 
18 21
b. 
Quick Check
27
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
You learn that 21 out of the 28 students in a
class, or , buy their lunches in the
cafeteria. Write this fraction in simplest form.
The GCF of 21 and 28 is 7.
Quick Check
28
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Write in simplest form.
p 2p
a.
29
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
(continued)
14q2rs3 8qrs2
b.
Quick Check
30
Simplifying Fractions
PRE-ALGEBRA LESSON 4-4
Find two fractions equivalent to each
fraction. 1. 2. Write in simplest form.
3. 4.
31
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Choose the symbol lt, ,or gt that makes each
statement true. a. 2 1 2 ? 4 1
b. 3 2 ? 4 3

lt
32
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(For help, go to Skills Handbook p. 775.)
Compare. Use gt to lt to complete each
statement. 1. 3 0 2. 16 25 3. 0
1 4. 30 20
Check Skills Youll Need
33
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(For help, go to Skills Handbook p.775.)
Solutions 1. 3 gt 0 2. 16 gt 25 3. 0 lt
1 4. 30 lt 20
34
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Aaron, Chris, Maria, Sonia, and Ling are on a
class committee. They want to choose two members
to present their conclusions to the class. How
many different groups of two members can they
form?
35
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
(continued)
Each successive tree has one less branch.
Quick Check
There are 10 different groups of two committee
members.
36
Problem Solving Strategy Solve a Simpler Problem
PRE-ALGEBRA LESSON 4-5
Solve each problem. 1. Twelve people are at a
party. Each person greets each of the other
persons exactly once. How many greetings will
there be in all? 2. How many different
pairs of classmates can you choose from six
classmates? 3. Each small box is a
square. What is the number of different squares
shown?
66
15 pairs
17
37
Rational Numbers
PRE-ALGEBRA LESSON 4-6
29,716 52,003
Write in simplest form.
38
Rational Numbers
PRE-ALGEBRA LESSON 4-6
(For help, go to Lesson 4-4.)
Write in simplest form. 1. 2. 3. 4.
Check Skills Youll Need
39
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Solutions 1. 2. 3. 4.
40
Rational Numbers
PRE-ALGEBRA LESSON 4-6
2 3
Write two lists of fractions equivalent to .
Quick Check
41
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Graph each rational number on a number line.
3 4
a.

b. 0.5
c. 0
1 3
d.
Quick Check
42
Rational Numbers
PRE-ALGEBRA LESSON 4-6
A fast sports car can accelerate from a stop to
90 ft/s in 5 seconds. What is its acceleration in
feet per second per second (ft/s2)? Use the
formula a , where a is acceleration, f
is final speed, i is initial speed, and t is time.
f i t
The cars acceleration is 18 ft/s2.
Quick Check
43
Rational Numbers
PRE-ALGEBRA LESSON 4-6
Write three fractions equivalent to the given
fraction. 1. Graph each rational number on a
number line. 2. a. b. c. 1.5 d.
0.4 3. A car can accelerate from 0 to 70
ft/s in 5 s. What is the acceleration of the car
in feet per second per second (ft/s2)?
14 ft/s2
44
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
23 ? 32 is the prime factorization for __.
?
72
45
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
(For help, go to Lesson 4-2.)
Write using exponents. 1. k k k k 2. m
n m n 3. 2 2 2 2 4. 5 5 5
Check Skills Youll Need
46
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Solutions 1. k k k k k4 2. m n m
n m m n n m2n2 3. 2 2 2 2
24 4. 5 5 5 53
47
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression.
a. 52 53
55
b. x5 x7 y2 y
Quick Check
48
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify 3a3 (5a4).
Quick Check
49
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression.
a. (23)3
b. (g5)4
Quick Check
50
Exponents and Multiplication
PRE-ALGEBRA LESSON 4-7
Simplify each expression. 1. 22 23 2. g2
h2 h4 h 3. 4r s 7r s5 4. (22)5 5.
(v3)8
32
g2 h7
6
3
1,024
v24
51
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Mari can package 14 seashells in a box. She has
360 seashells. How many full boxes does she have?
How many shells are left over after the boxes are
filled?
25 10
52
Exponents and Division
PRE-ALGEBRA LESSON 4-8
(For help, go to Lesson 4-4.)
Write in simplest form. 1. 2. 3. 4.
Check Skills Youll Need
53
Exponents and Division
PRE-ALGEBRA LESSON 4-8
(For help, go to Lesson 4-4.)
54
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
a.
w18 w13
b.
Quick Check
55
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
(12)73 (12)73
a.
1
8s20 32s20
b.
Quick Check
56
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression.
612 614
a.
z4 z15
b.
Quick Check
57
Exponents and Division
PRE-ALGEBRA LESSON 4-8
a2b3 ab15
Write without a fraction bar.
Quick Check
58
Exponents and Division
PRE-ALGEBRA LESSON 4-8
Simplify each expression. 1. 2. 3. Writ
e without a fraction bar.
49
2n10
59
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Estimate in years the age of someone who is one
million minutes old.
about 2 years
60
Scientific Notation
PRE-ALGEBRA LESSON 4-9
(For help, go to Lesson 4-7.)
Write each expression with a simple
exponent. 1. 103 105 2. 107 109 3. 105
103 4. 106 103
Check Skills Youll Need
61
Scientific Notation
PRE-ALGEBRA LESSON 4-9
(For help, go to Lesson 4-7.)
Solutions
1. 103 105 1035 2. 107 109 1079
108 1016
3. 105 103 1053 4. 106 103
1063 102 103
62
Scientific Notation
PRE-ALGEBRA LESSON 4-9
About 6,300,000 people visited the Eiffel Tower
in the year 2000. Write this number in scientific
notation.
6,300,000
Quick Check
63
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write 0.00037 in scientific notation.
0.00037
Quick Check
64
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in standard notation.
a. 3.6 ? 104
3.6000
b. 7.2 ? 103
007.2
Quick Check
65
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in scientific notation.
a. 0.107 ? 1012
b. 515.2 ? 104
Quick Check
66
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Order 0.035 ? 104, 710 ? 101, and 0.69 ? 102
from least to greatest.
Quick Check
67
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Multiply 4 ? 106 and 7 ? 109. Express the
result in scientific notation.
Quick Check
68
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Quick Check
In chemistry, one mole of any element contains
approximately 6.02 ? 1023 atoms. If each hydrogen
atom weighs approximately 1.67 ? 1027 kg,
approximately how much does one mole of hydrogen
atoms weigh?
One mole of hydrogen atoms weighs approximately
1.01 ? 103 kg.
69
Scientific Notation
PRE-ALGEBRA LESSON 4-9
Write each number in scientific
notation. 1. 5,400,000 2. 0.0000867 Write
each number in standard notation. 3. 3.45 ?
106 4. 1.99 ? 105 5. Order 7.2 ? 105, 7.2 ?
106, 7.02 ? 106, and 7.1 ? 106 from least to
greatest. 6. Multiply 14 ? 106 and 4 ? 104.
Express the result in scientific notation.
5.4 ? 106
8.67 ? 105
3,450,000
0.0000199
7.1 ? 106, 7.2 ? 105, 7.02 ? 106, 7.2 ? 106
5.6 ? 103