SECTION 5.1: FACTORS
Excerpts from Keith Devlin, Thursday April 26, 2001 © Guardian Newspapers Limited
Kurt Gödel and Our Loss of Innocence
Born in what was then Austria, on April 28 1906, Gödel died in Princeton, New Jersey on January 14 1978, having developed a paranoia that he was being poisoned and, as a result, starving himself to death (an altogether odd end for one of the greatest logicians the world has ever known).
Gödel is best known for his discovery, in 1931, of the famous Gödel Incompleteness Theorem. In everyday terms, this says that no matter how hard you try, you will never be able to reduce all of mathematics to the application of fixed rules. Regardless of how many rules and procedures you write down, there will always be some true facts that you can't prove. His Incompleteness Theorem says that you can never find enough axioms. No matter how carefully you try to make sure you have written down all the basic assumptions, there will always be some questions that you can't answer. Mathematical knowledge is destined to remain forever incomplete.
In fact, the situation is even worse. Gödel went on to show that one of the questions you cannot answer is whether your chosen set of axioms is consistent or not. You can never be sure that, in writing down your axioms, you haven't made a mistake and introduced some subtle conflict.
In that sense, Gödel's discovery can be regarded as the end of the age of innocence in the field of mathematics.
The concept of a "factor" is quite simple to understand but often understated in its importance when it comes to simplifying very complex mathematical expressions. Factors are the foundation of the cancelling rule of fractions which will be talked about in a later chapter.
There are two definitions of the word "factor."
Definition 1) When two or more numbers (or math expressions) are multiplied, each number (or math expression) that is multiplied is called a factor.
ex.1) 5×7×8: 5 is a factor, 7 is a factor and 8 is a factor.
ex. 2) (x)(y)(z): x is a factor, y is a factor and z is a factor.
ex. 3) (x+6)(x3): (x+6) is a factor and (x3) is a factor.
Definition 2) If a number (or math expression) A divides into another number B(or math expression) with a remainder of 0, then A is called a factor of B.
ex. 1) (27)/3)=9: 3 is a factor of 27 since 3 divides into 27 with a remainder of 0.
ex. 1) (100)/50)=2: 50 is a factor of 100 since 50 divides into 100 with a remainder of 0.
SECTION 5.2: FACTORING
"Factoring" means to express a number, or a "math expression," as the product of its factors.
ex. 1) If you rewrite 6 as 3 x 2, then you are factoring 6.
ex. 2) If you rewrite 20 as 5 x 4, then you are factoring 20.
In the later chapters you will be doing more sophisticated form of factoring expressions such as x^{2 }+5x+6.
SECTION 5.3: WHAT ARE COMMON FACTORS?
If you write down all the factors of a number A and all the factors of a number B, then the factors that are in both A and B are called "common factors".
ex. 1) The factors of 12 are 2, 3, 4, 6, and 12; and the factors of 30 are 2, 3, 5, 6, 10, 15 and 30.
2, 3, and 6 are the common factors of 12 and 30 because they are the same factors in 12 and 30.
ex. 2) The factors of 20 are 2, 4, 5, 10, and 20; and the factors of 40 are 2, 4, 5, 8, 10, 20 and 40.
2, 4, 5, 10 and 20 are the common factors of 20 and 40 because they are the same factors in 20 and 40.
SECTION 5.4: THE GREATEST COMMON FACTOR AND THE LEAST COMMON FACTOR
To get the "greatest common factor" of two numbers, write down all the factors of each number. The largest factor in common is called the "greatest common factor".
ex. 1)
Given 12: 2, 3, 4, 6, and 12 are factors of 12
Given 30: 2, 3, 5, 6, 10, 15, and 30 are factors of 30
The greatest common factor of 12 and 30 is 6.
To get the "least common factor" of two numbers, write down all the factors of each number. The smallest factor in common is called the "least common factor."
ex. 2)
Given 12: 2, 3, 4, 6, and 12 are factors of 12
Given 30: 2, 3, 5, 6, 10, 15, and 30 are factors of 30
The least common factor of 12 and 30 is 2.



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
