Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Prime Factors

To factor a whole number is to write it as a product of two or more other whole numbers. The individual numbers in this product are called factors of the original number.

A prime number or prime is a whole number which is evenly divisible only by itself and 1. In other words, prime numbers have only 1 and themselves as factors. Whether or not the number 1 should be considered a prime number is a matter of futile debate. Ignoring 1, the first few prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, …

There is no known formula which generates all prime numbers. Instead, to establish that a number is prime, you have to use a method of systematic trial and elimination of potential factors to demonstrate that it has no factors other than itself and 1. Mathematicians have been intrigued by prime numbers for thousands of years, and continue to study their properties enthusiastically today because they have applications in many problems of technology.

In simplifying fractions, it is useful to begin by factoring the numerator and denominator into a product of factors which are all prime numbers – hence the term prime factor . (We also did this earlier in these notes in attempting to simplify square roots of numbers.) The procedure is very straightforward and systematic:

  • start by checking for factors of 2 in the number, and remove as many as possible
  • then check for factors of 3 in what’s left after all factors of 2 have been removed, and remove as many factors of 3 as possible
  • repeat this process on what’s left after previous prime factors have been removed, using the prime numbers in turn: 5, 7, 11, 13, … etc. When it is clear that what’s left to be factored is a prime number itself (and so cannot be factored further), the process is done.



Find the prime factors of 156.


  • 156 is even, so it contains a factor of 2:

  • 78 is even, so it contains a factor of 2:

  • 39 is not even, so there are no further prime factors of 2. So, check if 39 is divisible by 3. Since 39 / 3 = 13, a whole number, we conclude that 3 is a prime factor of 39, and so

Having removed three prime factors ( 2, 2, and 3) from 156, we are left with the factor 13. But 13 is itself a prime number. Thus 156, written as a product of prime factors is

156 = 2 × 2 × 3 × 13

Notice that as we remove prime factors, the number on which we need to focus further attention is always getting smaller and smaller.

We’ll do one more really long example to make sure you clearly understand this systematic approach to determining the prime factors of a number. Usually however, the numbers we need to factor are much smaller (hence the factoring process is much shorter and less tedious) than the one in the next example.



Find the prime factors of 58212.


  • 58212 is even, so it contains a factor of 2:

  • 29106 is even, so it contains a factor of 2:

  • 14553 is not even, so it does not contain a factor of 2. But, since 3 divides evenly into 14553, we have that there is a factor of 3 here:

  • we are now working on factors of 3. Again, 3 divides evenly into 4851, so there is at least one more factor of 3:

  • we are still working on factors of 3. Again, 3 divides evenly into 1617, so there is at least one more factor of 3:

  • we are still working on factors of 3. However, a test calculation shows that 3 does not divide evenly into 539, so we conclude there are no more factors of 3 in the original number. Clearly 539 is not divisible by 5. So the next prime number to try is 7. We find that 539 is evenly divisible by 7, giving

  • We can see that the remaining unfactored part, the 77, is still evenly divisible by 7, so we have:

This is the end, because all of the numbers in the product on the right are prime numbers. Thus, the prime factorization of 58212 is:

58212 = 2 × 2 × 3 × 3 × 3 × 7 × 7 × 11

You can easily verify that multiplying the numbers on the right together does give 58212, and all seven factors shown are prime numbers. Thus, this is the requested solution to the problem.

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