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Factoring Integers
November 3, 2011
Factoring Integers
Objective To factor integers and to find the
greatest common factor of
several integers.
Factoring Integers
When you write 56 = 8 ∙ 7 or 56 = 4 ∙ 14,
you have factored 56. In the first case the
factors are 8 and 7. In the second
case the
factors are 4 and 14. You could also write
1
56 = ∙ 112 and call ½ and
112 factors of
2
56. Usually, however, you are interested
only in factors that are integers.
Factoring Integers
To factor a number over a given set. you
write it as a product of numbers in that set,
called the factor set. In this class, integers
will be factored over the set of integers
unless some other set is specified. The
factors are then integral factors.
Positive Factors
You can find the positive factors of a given
positive integer by dividing it by positive
integers in order. Record only the integral
factors. Continue until a pair of factors is
repeated.
Example 1
Give all the positive factors of 56.
Solution
Divide 56 by l, 2, 3, 
56 = 156 = 228 = 414 = 78 (= 87)
Stop
the positive factors of 56 are 1, 2, 4, 7, 8,
14, 28, and 56.
Prime Numbers
A prime number, or prime, is an integer
greater than 1 that has no positive integral
factor other than itself and 1.
Is 1 a prime number?
No!
1≯1
The first ten prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Prime Factorization
To find the prime factorization of a positive
integer, you express it as a product of
primes. Example 2 shows a way to organize
your work.
Example 2
Find the prime factorization of 504.
Solution
504 = 2 ∙ 252
Try the primes in
= 2 ∙ 2 ∙ 126
order as divisors.
= 2 ∙ 2 ∙ 2 ∙ 63
Divide by each prime
= 2 ∙ 2 ∙ 2 ∙ 3 ∙ 21
as many times as
possible before going
=2∙2∙2∙3∙3∙7
on to the next prime.
= 23 ∙ 32 ∙ 7
Exponents
Exponents are generally used for prime
factors that occur more than once in a
factorization. The prime factorization of an
integer is unique (there is only one) except
for the order of the factors.
Greatest Common Factor
A factor of two or more integers is called a
common factor of the integers. The greatest
common factor (GCF) of two or more
integers is the greatest integer that is a
factor of all the given integers.
2 is a common factor of 6 and 16.
6 is the greatest common factor of 12 and 18.
Example 3
Find the GCF of 882 and 945.
Solution
First find the prime factorization of each
integer. Then form the product of the
smaller powers of each common prime
factor.
Example 3
Solution
882 = 2 ∙ 32 ∙ 72
945 = 33 ∙ 5 ∙ 7
The common prime factors are 3 and 7.
The smaller powers of 3 and 7 are 32 and 7.
the GCF of 882 and 945 is 327, or 63.
Class work
p 186:
Oral Exercises 1-22
Homework
p 186: 3-39 mult of 3,
41-46,
p 187: Mixed Review