Download 2.4 Factors Numbers that are multiplied together are called factors

2.4 Factors
Numbers that are multiplied together are called factors.
Factors of a number, a, are numbers that when multiplied
together produce a product of a.
The number 12 has 6 possible factors:
1 x 12 = 12
2 x 6 = 12
3 x 4 = 12
So the factors are 1, 2, 3, 4, 6, and 12.
A factor pair table can be written for 12 to find all the factors.
Just start from 1 and keep incrementing and checking if 12 is divisible by
it. When a factor you choose has already appeared in the table. You
know you are finished.
12
1
12
2
6
3
4
4
3
3 was already
in the table.
Note that a number is always a factor of itself because a x 1 = a
When two whole numbers, m and n, multiply to get a product, p (m x n =
p), then we can say these facts:
1) p is a multiple of m and n
2) p is divisble by m and n
3) m and n divide evenly into p.
4) m and n are factors of p.
Example: 12 is a multiple of 4 and a multiple of 3
12 is divisible by 3 and 4
3 and 4 divide evenly into 12
3 and 4 are factors of 12.
Prime Numbers
A prime number is a whole number, greater than 1, that has only
1 an itself as factors.
Composite Numbers
A composite number is a whole number, greater than 1, that are
not prime.
Prime Factorization
To find the prime factorization of a whole number means to write it as
the product of only prime numbers.
This is can be useful when finding things like the Greatest Common Factor or
Least Common Multiple between two numbers (we’ll get into that later).
Example: Factor 90 into its prime factors.
Choose any two
factors of 90
(besides 1 and
90)
Then do the
same with each
of those factors.
Keep going until
you have only
prime factors as
the bottom “roots”
of the “factor
tree.”
90
9
3
10
3
2
5
90 = 3 3 2 5
Putting these factors in numerical order and
then combing like terms into exponents gives:
90 = 2 32 5
Theorem:
Any composite number has exactly one set of prime factors.
Example 5
Find the prime factorization of 210
First, pick any two factors of 210.
For instance 21 and 10.
We could have also picked 7 and 30 as the
factors.
210
21
3
210
10
7
2
7
30
5
Notice that either method gives us
210 = 2 3 5 7
6
3
5
2
Alternate Factoring Method –
I call it the “Pyramid Division Method”
For more info:
http://www.purplemath.com/modules/factnumb.htm
With this method, you start and the bottom of the pyramid
and move up, so you have to leave lots of room at the top
of your problem.
Example: Find the prime factorization of 90.
Start with the first prime factor you can think of that goes
into 90. We can’t choose 9 or 10 because they aren’t
prime. Since 90 is even, start with the factor 2.
3
39
5 45
is 3 prime? Yes! You are done.
is 9 prime? No. Keep dividing.
is 45 prime? No. Keep dividing.
2 90
Prime Factorization of 90 is 2x5x3x3
Write the factors in numerical order and use exponents where factors are repeated
=
2 32 5
Upside-Down Division Method
Example: Find the prime factorization of 60
2
60
3
30 Is 30 prime? No.
2
10 Is 10 prime? No.
5 Is 5 prime? Yes. Stop
Prime Factorization of 60 is
22 3 5