Fundamentals of Mathematics, 2008a

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CHAPTER 3.
EXPONENTS, ROOTS, AND FACTORIZATION OF WHOLE
NUMBERS
4. Perform all additions and subtractions, moving left to right.
One Number as the Factor of Another (Section 3.4)
A rst number is a factor of a second number if the rst number divides into the second number a whole
number of times.
Prime Number (Section 3.4)
A whole number greater than one whose only factors are itself and 1 is called a prime number. The whole
number 1 is not a prime number. The whole number 2 is the rst prime number and the only even prime
number.
Composite Number (Section 3.4)
A whole number greater than one that is composed of factors other than itself and 1 is called a composite
number.
Fundamental Principle of Arithmetic (Section 3.4)
Except for the order of factors, every whole number other than 1 can be written in one and only one way as
a product of prime numbers.
Prime Factorization (Section 3.4)
The prime factorization of 45 is 3 · 3 · 5. The numbers that occur in this factorization of 45 are each prime.
Determining the Prime Factorization of a Whole Number (Section 3.4)
There is a simple method, based on division by prime numbers, that produces the prime factorization of a
whole number. For example, we determine the prime factorization of 132 as follows.
The prime factorization of 132 is 2 · 2 · 3 · 11 = 2 2 · 3 · 11.
Common Factor (Section 3.5)
A factor that occurs in each number of a group of numbers is called a common factor. 3 is a common
factor to the group 18, 6, and 45
Greatest Common Factor (GCF) (Section 3.5)
The largest common factor of a group of whole numbers is called the greatest common factor.
example, to nd the greatest common factor of 12 and 20,
For
12 = 2 · 2 · 3 = 2 2 · 3
1. Write the prime factorization of each number.
60 = 2 · 2 · 3 · 5 = 2 2 · 3 · 5
2. Write each base that is common to each of the numbers: 2 and 3
3. The smallest exponent appearing on 2 is 2.
The smallest exponent appearing on 3 is 1.
4. The GCF of 12 and 60 is the product of the numbers 2 2 and 3. 2 2 · 3 = 4 · 3 = 12
Thus, 12 is the largest number that divides both 12 and 60 without a remainder.
Finding the GCF (Section 3.5)
There is a simple method, based on prime factorization, that determines the GCF of a group of whole
numbers.
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