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Rotating a Parabola
Multiplying Fractions
Finding Factors
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Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
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Simplifying Complex Fractions
Decimals and Fractions
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The Square of a Binomial
Factoring Trinomials
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Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Algebra
Order of Operations
Dividing Complex Numbers
Polynomials
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Factoring
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
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Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Fractions
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Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Exponents
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Radicals
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
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Product of a Sum and a Difference
Solving First Degree Inequalities
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Roots and Powers
Multiplying Numbers
   
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Finding Factors

Examples with Solutions

EXAMPLE 1

What are the factors of 45?

Solution

Let’s see if 45 is divisible by 1, 2, 3, and so on, using the divisibility tests wherever they apply.

Is 45 divisible by

Answer

1? Yes, because 1 is a factor of any number; , so 45 is also a factor.
2? No, because the ones digit is not even.
3? Yes, because the sum of the digits, 4 + 5 = 9, is divisible by 3; , so 15 is also a factor.
4? No, because 4 will not divide into 45 evenly.
5? Yes, because the ones digit is 5; , so 9 is also a factor.
6? No, because 45 is not even.
7? No, because 45 ÷ 7 has remainder 3.
8? No, because 45 ÷ 8 has remainder 5.
9? We already know that 9 is a factor.

The factors of 45 are therefore 1, 3, 5, 9, 15, and 45.

Note that we really didn’t have to check to see if 9 was a factor—we learned that itwas when we checked for divisibility by 5. Also, because the factors were beginning torepeat with 9, there was no need to check numbers greater than 9.

EXAMPLE 2

Identify all the factors of 60.

Solution

Let’s check to see if 60 is divisible by 1, 2, 3, 4, and so on.

Is 60 divisible by

Answer

1? Yes, because 1 is a factor of all numbers; , so 60 is also a factor.
2? Yes, because the ones digit is even; , so 30 is also a factor.
3? Yes, because the sum of the digits, 6 + 0 = 6, is divisible by 3; , so 20 is also a factor.
4? Yes, because 4 will divide into 60 evenly; , so 15 is also a factor.
5? Yes, because the ones digit is 0; , so 12 is also a factor.
6? Yes, because the ones digit is even and the sum of the digits is divisibleby 3; , so 10 is also a factor.
7? No, because 60 ÷ 7 has remainder 4.
8? No, because 60 ÷ 8 has remainder 4.
9? No, because the sum of the digits, 6 + 0 = 6, is not divisible by 9.
10? We already know that 10 is a factor.

The factors of 60 are therefore 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. Can you explain how we knew that 10 was a factor of 60 when we checked for divisibility by 6?

EXAMPLE 3

A presidential election takes place in the United States every year thatis a multiple of 4. Was there a presidential election in 1866?

Solution

The question is: Does 4 divide into 1866 evenly? Using the divisibility test for 4, we check whether 66 is a multiple of 4.

Because has remainder 2, 4 is not a factor of 1866. So there was nopresidential election in 1866.

 
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