FreeAlgebra Tutorials!   Home 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 Powers 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 http: 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 Fractions 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 Adding Quadratic Functions Conjugates Factoring 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 Percents Arithmetics 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 Polynomials Laws of Exponents Multiplying Polynomials Vertical Line Test http: Solving Inequalities with Fractions and Parentheses http: Multiplying Polynomials Fractions Solving Quadratic and Polynomial Equations Extraneous Solutions Fractions

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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.

Example:

Find the prime factors of 156.

Solution:

• 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 Ã— 2 Ã— 3 Ã— 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.

Example:

Find the prime factors of 58212.

Solution:

• 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 Ã— 2 Ã— 3 Ã— 3 Ã— 3 Ã— 7 Ã— 7 Ã— 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.

 All Right Reserved. Copyright 2005-2007