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Section 2.2 Prime Numbers and Factorization
Basic College Mathematics, Miller/O’Neill/Hyde
Andrea Hendricks
I. Connection
In Chapter 1, we discussed multiplying whole numbers. In this section, we are interested
in given a number, find the numbers that when multiplied form this product. This
process is called factoring. It has an important significance in the study of fractions and
this process also extends to the study of algebra, as you will see in future math courses.
II. Overview – In this section, you will learn about
A. Factors and Factorizations
B. Divisibility Rules
C. Prime and Composite Numbers
D. Prime Factorization
E. Identifying All Factors of a Whole Number
III. Factors and Factorizations
A. Example – We know that 2 ∙ 3 = 6. The numbers 2 and 3 are called factors of 6.
B. Definition – A factor of a number n is a nonzero whole number that divides evenly
into n.
C. Definition – A factorization of a number n is a product of factors that equal n. (In
the above example, 2 ∙ 3 is called a factorization of 6.
IV. Divisibility Rules – These are some rules that allow us to quickly determine whether
one number is divisible by another.
A. Divisibility by 2 – A whole number is divisible by 2 if it is an even number. That
is, the ones-place digit is 0, 2, 4, 6, or 8.
B. Divisibility by 3 - A whole number is divisible by 3 if the sum of its digits is
divisible by 3. (Example: 27 is divisible by 3 since 2 + 7 = 9 is divisible by 3.)
C. Divisibility by 5 – A whole number is divisible by 5 if its ones-place digit is 0 or 5.
D. Divisibility by 10 – A whole number is divisible by 10 if its ones-place digit is 0.
V. Prime and Composite Numbers
A. A prime number is a whole number greater than 1 that has only two factors (itself
and 1). Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …
B. A composite number is a whole number greater than 1 that is not prime. That is, a
composite number will have at least one factor other than 1 and the number itself.
C. Note: The whole numbers 0 and 1 are neither prime nor composite.
VI. Prime Factorization
A. The prime factorization of a number is the factorization in which every factor is a
prime number.
B. Two methods for finding the prime factorization
1) Factor tree
a) Begin with any two factors of the given number.
b) Continue factoring each factor until the branches “end” in prime
numbers.
c) Check your answer by multiplying the prime factors.
2) Using division to find prime factorization
a) Divide the given number by the smallest known prime factor.
b) Continue dividing in this fashion until the quotient is a prime number.
VII. Identify All Factors of a Whole Number
A. To identify all factors of a number, is to list all the whole numbers that evenly
divide the number.
B. Steps to identify all factors
1) List all of the two-number factorizations of the number by systematically
dividing the number by 1, 2, 3, and so on.
2) Continue this process until the two-number factorizations repeat.
3) The list of factors consists of all of the individual factors in the products.
VIII. Summary – You should be able to
A. Identify factors and factorizations of numbers.
B. Use the divisibility rules to determine factors.
C. Determine if a number is prime, composite, or neither.
D. Find the prime factorization of a number.
E. Identify all factors of a whole number.
IX. Next Section
The next section will discuss simplifying fractions to lowest terms. The skills you
learned in this section will be used in the next section.