Fields Medal Prize Winners (1998)


Adding and Subtracting Monomials
Solving Quadratic Equations by Using the Quadratic Formula
Addition with Negative Numbers
Solving Linear Systems of Equations by Elimination
Rational Exponents
Solving Quadratic Inequalities
Systems of Equations That Have No Solution or Infinitely Many Solutions
Dividing Polynomials by Monomials and Binomials
Polar Representation of Complex Numbers
Solving Equations with Fractions
Quadratic Expressions Completing Squares
Graphing Linear Inequalities
Square Roots of Negative Complex Numbers
Simplifying Square Roots
The Equation of a Circle
Fractional Exponents
Finding the Least Common Denominator
Simplifying Square Roots That Contain Whole Numbers
Solving Quadratic Equations by Completing the Square
Graphing Exponential Functions
Decimals and Fractions
Adding and Subtracting Fractions
Adding and Subtracting Rational Expressions with Unlike Denominators
Quadratic Equations with Imaginary Solutions
Graphing Solutions of Inequalities
FOIL Multiplying Polynomials
Multiplying and Dividing Monomials
Order and Inequalities
Exponents and Polynomials
Variables and Expressions
Multiplying by 14443
Dividing Rational Expressions
Division Property of Radicals
Equations of a Line - Point-Slope Form
Rationalizing the Denominator
Imaginary Solutions to Equations
Multiplying Polynomials
Multiplying Monomials
Adding Fractions
Rationalizing the Denominator
Rational Expressions
Ratios and Proportions
Rationalizing the Denominator
Like Radical Terms
Adding and Subtracting Rational Expressions With Different Denominators
Percents and Fractions
Reducing Fractions to Lowest Terms
Subtracting Mixed Numbers with Renaming
Simplifying Square Roots That Contain Variables
Factors and Prime Numbers
Rules for Integral Exponents
Multiplying Monomials
Graphing an Inverse Function
Factoring Quadratic Expressions
Solving Quadratic Inequalities
Factoring Polynomials
Multiplying Radicals
Simplifying Fractions 1
Graphing Compound Inequalities
Rationalizing the Denominator
Simplifying Products and Quotients Involving Square Roots
Standard Form of a Line
Multiplication by 572
Adding and Subtracting Fractions
Multiplying Polynomials
Factoring Trinomials
Solving Exponential Equations
Solving Equations with Fractions
Simplifying Complex Fractions
Multiplying and Dividing Fractions
Mathematical Terms
Solving Quadratic Equations by Factoring
Factoring General Polynomials
Adding Rational Expressions with the Same Denominator
The Trigonometric Functions
Solving Nonlinear Equations by Factoring
Solving Systems of Equations
Midpoint of a Line Segment
Complex Numbers
Graphing Systems of Equations
Reducing Rational Expressions
Rewriting Algebraic Fractions
Rationalizing the Denominator
Adding, Subtracting and Multiplying Polynomials
Radical Notation
Solving Radical Equations
Positive Integral Divisors
Solving Rational Equations
Rational Exponents
Mathematical Terms
Rationalizing the Denominator
Subtracting Rational Expressions with the Same Denominator
Axis of Symmetry and Vertex of a Parabola
Simple Partial Fractions
Simplifying Radicals
Powers of Complex Numbers
Fields Medal Prize Winners (1998)

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Factors and Prime Numbers

What Factors Mean and Why They Are Important

Recall that in a multiplication problem, the whole numbers that we are multiplying are called factors. For instance, 2 is said to be a factor of 8 because 2 · 4 = 8. Likewise, 4 is a factor of 8.

Another way of expressing the same idea is in terms of division: We say that 8 is divisible by 2, meaning that there is a remainder 0 when we divide 8 by 2.

Note that 1, 2, 4, and 8 are all factors of 8.

Although we factor whole numbers, a major application of factoring involves working with fractions, as we demonstrate in the next section.

Finding Factors

To identify the factors of a whole number, we divide the whole number by the numbers 1, 2,3, 4, 5, 6, and so on, looking for remainders of 0.


Find all the factors of 6.


Starting with 1, we divide each whole number into 6.

So the factors of 6 are 1, 2, 3, and 6. Note that

  • 1 is a factor of 6 and that
  • 6 is a factor of 6.
  • We did not need to divide 6 by the numbers 7 or greater. The reason is that no numberlarger than 6 could divide evenly into 6, that is, divide into 6 with no remainder.


For any whole number, both the number itself and 1 are always factors. Therefore all whole numbers (except 1) have at least two factors.

When checking to see if one number is a factor of another, it is generally faster to use the following divisibility tests than to divide.

The number is divisible by


2 the ones digit is 0, 2, 4, 6, or 8, that is, if the number is even.
3 the sum of the digits is divisible by 3.
4 the number named by the last two digits is divisible by 4.
5 the ones digit is either 0 or 5.
6 the number is even and the sum of the digits is divisible by 3.
9 the sum of the digits is divisible by 9.
10 the ones digit is 0.

Note that divisibility by 6 is equivalent to divisibility by both 2 and 3.