Divisibility and Factors

Pre-AlgebraPre-Algebra

Divisibility and FactorsDivisibility and Factors

Lesson 4-1

Objectives: 1. to use divisibility tests

2. to find factors



Lesson 4-1

Tips: the product of two integers is an integer, and both integers are factors of the product. Moreover, both integers divide the product, and the

product is said to be divisible by each integer.


Is the first number divisible by the second?


Lesson 4-1

a. 1,028 by 2

Yes; 1,028 ends in 8.

b. 572 by 5

No; 572 doesn’t end in 0 or 5.

c. 275 by 10

No; 275 doesn’t end in 0.


Is the first number divisible by the second?


Lesson 4-1

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.


Ms. Washington’s 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?


Lesson 4-1

1 • 35, 5 • 7 Find pairs of factors of 35.

There can be 5 rows of 7 students, or 7 rows of 5 students.


ExponentsExponents

Lesson 4-2

Objectives: 1. to use exponents

2. to use the order of operations with exponents


ExponentsExponents

Lesson 4-2

Tips: the exponent is placed to the upper right of the base, and it only applies to that base. If an exponent has as its base an expression, that expression must be written in parentheses.


Write using exponents.

ExponentsExponents

Lesson 4-2

b. –5 • x • x • y • y • x

a. (–11)(–11)(–11)(–11)

–5 • x • x • x • y • y Rewrite the expression using the Commutative and Associative Properties.

–5x3y2 Write x • x • x and y • y using exponents.

(–11)4 Include the negative sign within parentheses.


Suppose a certain star is 104 light-years from Earth.

How many light-years is that?

ExponentsExponents

Lesson 4-2

104 = 10 • 10 • 10 • 10 The exponent indicates that the base 10 is used as a factor 4 times.

= 10,000 light-years Multiply.

The star is 10,000 light-years from Earth.


ExponentsExponents

Lesson 4-2

a. Simplify 3(1 + 4)3.

b. Evaluate 7(w + 3)3 + z, for w = –5 and z = 6.

3(1 + 4)3 = 3(5)3 Work within parentheses first.

= 3 • 125 Simplify 53.

= 375 Multiply.

= 7(–2)3 + 6 Work within parentheses.

Replace w with –5 and z with 6.7(w + 3)3 + z = 7(–5 + 3)3 + 6

= 7(–8) + 6 Simplify (–2)3.

= –56 + 6 Multiply from left to right.

= –50 Add.


Prime Factorization and Greatest Common FactorPrime Factorization and Greatest Common Factor

Lesson 4-3

Objectives: 1. to find the prime factorization of a number

2. to find the greatest common factor (GCF) of two or more numbers

Tips: remember a factor is a number that divides evenly into another number with a remainder of zero.



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.



Lesson 4-3

Use a factor tree to write the prime factorization of

273.

273

273 = 3 • 7 • 13

Prime Start with a prime factor.Continue branching.

3 • 91

7 • 13Prime Stop when all factors are prime.

3 • 7 • 13 Write the prime factorization.



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.

24 = 23 • 3 Write the prime factorizations.30 = 2 • 3 • 5

GCF = 2 • 3= 6

Use the lesser power of the common factors.

GCF = 32 • b= 9b

Use the lesser power of thecommon factors.

36ab2 = 22 • 32 • a • b2 Write the prime factorizations.81b = 34 • b

Find the common factors.

Find the common factors.


Simplifying FractionsSimplifying Fractions

Lesson 4-4

Objectives: 1. to find equivalent fractions

2. to write fractions in simplest for

Tips: always check your answers and the steps involved in finding the answers



Lesson 4-4

Find two fractions equivalent to .1821

a. 1821

b. 1821

18 • 221 • 2=

18 ÷ 321 ÷ 3

=

=3642

67

=

The fractions and are both equivalent to .67

3642

1821



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.

2128

The GCF of 21 and 28 is 7.

2128

21 ÷ 728 ÷ 7

= Divide the numerator and denominator by the GCF, 7.

34

= Simplify.

34

of the students in the class buy their lunches in the cafeteria.



Lesson 4-4

Write in simplest form.

a.p

2p

p2p

= p1

2p1

Divide the numerator and denominator by the common factor, p.

12

= Simplify.



Lesson 4-4

(continued)

b. 14q2rs3

8qrs2

= Write as a product of prime factors.

14q2rs3

8qrs2

2 • 7 • q • q • r • s • s • s2 • 2 • 2 • q • r • s • s

21 • 7 • q1 • q • r1 • s1 • s1 • s21 • 2 • 2 • q1 • r1 • s1 • s1

Divide the numerator and denominator by the common factors.

=

Simplify.7 • q • s

2 • 2=

Simplify.7 • q • s

4=

7qs4=


Problem Solving Strategy: Solve a Simpler ProblemProblem Solving Strategy: Solve a Simpler Problem

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?

Additional Examples


Problem Solving Strategy: Solve a Simpler ProblemProblem Solving Strategy: Solve a Simpler Problem

Lesson 4-5

(continued)

Each successive tree has one less branch.

There are 10 different groups of two committee members.

First, pair Aaron with each of the four other committee members.

Aaron MariaChris

SoniaLing

Sonia LingMaria SoniaLing

Since Aaron and Chris have already been paired, you don’t need to count them again. Repeat for the rest of the committee members.

Next, pair Chris with each of the three members left.

Chris SoniaMaria

Ling

Additional Examples


Rational NumbersRational Numbers

Lesson 4-6

Objectives: 1. to identify and graph rational numbers

2. to evaluate fractions containing variables

Tips: the quotient of two integers with the same sign is positive



Lesson 4-6

Write two lists of fractions equivalent to .23

23

46

69

= = = … Numerators and denominators are positive.

= = = … Numerators and denominators are negative.23

–2–3

–4–6



Lesson 4-6

Graph each rational number on a number line.

a.

b. 0.5

c. 0

d.

34–

13

Additional Examples



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 – it

a = f – i

t Use the acceleration formula.

The car’s acceleration is 18 ft/s2.

90 – 05

Substitute.=

905 Subtract.=

18= Write in simplest form.


Exponents and MultiplicationExponents and Multiplication

Lesson 4-7

Objectives: 1. to multiply powers with the same base

2. to find a power of a power



Lesson 4-7

Tips: when in doubt, write it out.



Lesson 4-7

Simplify each expression.

a. 52 • 53

= 55

b. x5 • x7 • y2 • y

52 • 53 = 52 + 3 Add the exponents of powers with the same base.

= 3,125 Simplify.

x5 • x7 • y2 • y = x5 + 7 • y2 + 1 Add the exponents of powers with the same base.

= x12y3 Simplify.



Lesson 4-7

Simplify 3a3 • (–5a4).

3a3 • (–5a4) = 3 • (–5) • a3 • a4 Use the Commutative Property of Multiplication.

Add the exponents.= –15a3 + 4

Simplify.= –15a7



Lesson 4-7


a. (23)3

= (2)9 Simplify the exponent.

(23)3 = (2)3 • 3 Multiply the exponents.

= 512 Simplify.

b. (g5)4

(g5)4 = g5 • 4 Multiply the exponents.

= g20 Simplify the exponent.


Exponents and DivisionExponents and Division

Lesson 4-8

Objectives: 1. To divide expressions containing exponents

2. To simplify expressions with integer exponents



Lesson 4-8

Tips: Sometimes students think an expression with a negative exponent makes the expression negative, that is not true. A negative exponent makes the number smaller (unless the base is less than one but greater than zero).



Lesson 4-8


a.

b.w18

w13

= 44 Simplify the exponent.

= 256 Simplify.

= w5 Simplify the exponent.

412

48

412

48 = 412 – 8 Subtract the exponents.

= w18 – 13w18

w13 Subtract the exponents.



Lesson 4-8


a. (–12)73

(–12)73

= 1

= (–12)0 Simplify.

b. 8s20

32s20

(–12)73

(–12)73 = (–12)73 – 73 Subtract the exponents.

14= • 1 Simplify s0.

14= Multiply.

832

8s20

32s20

14= Subtract the exponents. Simplify .s0



Lesson 4-8


a. 612

614

b. z4

z15

= 6–2

612

614 = 612 – 14 Subtract the exponents.

= 162 Write with a positive exponent.

136= Simplify.

= z–11

z4

z15= z4 – 15 Subtract the exponents.

= 1z11

Write with a positive exponent.



Lesson 4-8

Write without a fraction bar.a2b3

ab15

a2b3

ab15 = a2 – 1b3 – 15 Use the rule for Dividing Powers with the Same Base.

= ab–12 Subtract the exponents.


Scientific NotationScientific Notation

Lesson 4-9

Objectives: 1. to write and evaluate numbers in scientific notation

2. to calculate with scientific notation

Tips: be careful with which way you move the decimal, think if the number is suppose to be smaller or larger



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

6.3 Drop the zeros after the 3.

6.3 x 106 You moved the decimal point 6 places. The original number is greater than 10. Use 6 as the exponent of 10.

Move the decimal point to get a decimal greater than 1 but less than 10.

6 places



Lesson 4-9

Write 0.00037 in scientific notation.

0.00037 Move the decimal point to get a decimal greater than 1 but less than 10.

4 places

3.7 x 10–4 You moved the decimal point 4 places. The originalnumber is less than 1. Use –4 as the exponent of 10.

3.7 Drop the zeros before the 3.



Lesson 4-9

Write each number in standard notation.

a. 3.6 x 104

3.6000 Write zeros while moving the decimal point.

36,000 Rewrite in standard notation.

b. 7.2 x 10–3

007.2

0.0072 Rewrite in standard notation.

Write zeros while moving the decimal point.



Lesson 4-9

Write each number in scientific notation.

a. 0.107 x 1012

0.107 x 1012 = 1.07 x 10–1 x 1012 Write 0.107 as

1.07 10–1.

= 1.07 x 1011 Add the exponents.

b. 515.2 x 10–4

515.2 x 10–4 = 5.152 x 102 x 10–4 Write 515.2 as 5.152 102.

= 5.152 x 10–2 Add the exponents.



Lesson 4-9

Write each number in scientific notation.

0.035 x 104 0.69 x 102

3.5 x 102

710 x 10–1

7.1 x 10 6.9 x 10

Order the powers of 10. Arrange the decimals with the same power of 10 in order.

6.9 x 10 7.1 x 10 3.5 x 102

Write the original numbers in order.

0.69 x 102, 710 x 10–1, 0.035 x 104

Order 0.035 x 104, 710 x 10–1, and 0.69 x 102 from least to greatest.



Lesson 4-9

Multiply 4 x 10–6 and 7 x 109. Express the result in scientific notation.

Use the Commutative Property of Multiplication.

(4 x 10–6)(7 x 109) = 4 x 7 x 10–6 x 109

= 28 x 10–6 x 109 Multiply 4 and 7.

= 28 x 103 Add the exponents.

= 2.8 x 101 x 103 Write 28 as 2.8 101.

= 2.8 x 104 Add the exponents.



Lesson 4-9

In chemistry, one mole of any element contains

approximately 6.02 x 1023 atoms. If each hydrogen atom weighs

approximately 1.67 x 10–27 kg, approximately how much does one mole of hydrogen atoms weigh?

(6.02 x 1023)(1.67 x 10–27) Multiply number of atoms by weight of each.

= 6.02 x 1.67 x 1023 x 10–27 Use the Commutative Property of Multiplication.


= 1.01 x 101 x 10–4 Write 10.1 as 1.01 101.


Multiply 6.02 and 1.67.10.1 x 1023 x 10–27

One mole of hydrogen atoms weighs approximately 1.01 x 10–3 kg.

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Divisibility and Factors