Algebra Honors (Period 6 & 7)

Factoring Integers 5-1

When we write 56= 8⋅7 or 56 = 4⋅14 we have factored 56
to factor a number over a given set, you write it as a product of integers in that set ( the factor set).
When integers are factored over the set of integers, the factors are called integral factors.

We used the T- charts ( students learned in 6th grade) to first find the positive integer factors
56 = 1, 2, 4, 7, 8, 14, 28, 56

A prime number is an integer greater than 1 that has no positive integral factors other than itself and 1.
The first ten prime numbers are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29

To find prime factorization of a positive integer, you express it as a product of primes. We used inverted division (again taught in 6th grade)

504
Try to find the primes in order as divisors.
Divide each prime as many times as possible before going on to the next prime
we found 504 - 2⋅2⋅2⋅3⋅3⋅7
which we write as 2³⋅3²⋅7
Exponents are generally used for prime factors
The prime factorization is unique--> and the order should be from the smallest prime to the largest.
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.

Find the GCF(882, 945)
First find the prime factorization of each integer Then form product of the smaller powers of each common prime factor.
The GCF is only the primes (and the powers) that they SHARE!!
882 = 2⋅3²⋅7²
945 = 3³⋅5⋅7
The common factors are 3 and 7
The smaller powers of 3 and 7 are 3² and 7
You combine these as a PRODUCT and get

the GCF(882, 945) = 3²⋅7 = 63

We also talked about listing ALL pairs of factors--> thus including negative integers
For example:
List all the pairs of factors of 20
(1)(20) but also (-1)(-20)
(2)(10) and (-2)(-10)
(4)(5) and (-4)(-5)

Listing all the factors of -20, we discovered
(1)(-20) but also (-1)(20)
(2)(-10) and (-2)(10)
(4)(-5) and (-4)(5)