Transcript Factors and Greatest Common Factors

Factors and Greatest
Common Factors
Prime Factorization
A list of prime numbers that when
multiplied together will give your given
number
For example, the prime factorization of 24
is 2 • 2 • 2 • 3 (or 23 • 3).
All are prime numbers
Their product is 24.
Remember that 1 is NOT a prime number!
Finding Prime Factorization
Building Towers
Put the number in a “division house”
Divide by prime numbers stacking division
houses on top of each other.
Keep dividing by primes until you get one.
All the numbers on the side are the prime
factors.
Find the prime factorization of 36
1
3
3
2
2
3
9
18
36
Divide by any prime number that
will go into 36
The prime factorization of 36 is 2 • 2 • 3 • 3 or 22 • 32
GCF
GCF is Greatest Common Factor
This is the largest number (or expression)
that will divide into a set of numbers.
Finding GCF
 Put all the numbers in a single division house
with vertical lines separating the numbers.
 Divide by something that will go into all the
numbers under the division house evenly (it
does NOT have to be prime).
 Continue dividing by an expression that will go
into all the numbers under the division house
evenly until the only thing that will divide into all
of them is 1.
 Multiply all the numbers on the left together.
Find the GCF of 60 and 100
3
2
10
5
6
10
60
100
The GCF is 20.
The only thing that will
Go into both 3 and 5 is
1. Stop here.
Find the GCF of 3x3 and 6x2
x
x
2
x2
2x
x
x3
3
3x3
The GCF is 3x2
2x2
6x2
Find the GCF of
16a6 and 9b
There are no common factors
other than 1.
The GCF of 16a6 and 7b is 1.
Example 1
A cafeteria has 18 chocolate-milk cartons and
24 regular-milk cartons. The cook wants to
arrange the cartons with the same number of
cartons in each row. Chocolate and regular
milk will not be in the same row. How many
rows will there be if the cook puts the greatest
possible number of cartons in each row?
The 18 chocolate and 24 regular milk cartons must
be divided into groups of equal size. The number of
cartons in each row must be a common factor of 18
and 24.
The greatest possible number of milk cartons in
each row is 6. Find the number of rows of each type
of milk when the cook puts the greatest number of
cartons in each row.
When the greatest possible number of types of
milk is in each row, there are 7 rows in total.
Example 2
Adrianne is shopping for a CD storage unit.
She has 36 CDs by pop music artists and 48
CDs by country music artists. She wants to put
the same number of CDs on each shelf without
putting pop music and country music CDs on
the same shelf. If Adrianne puts the greatest
possible number of CDs on each shelf, how
many shelves does her storage unit need?
When the greatest possible number of CD types
are on each shelf, there are 7 shelves in total.
Try these…
Write the prime factorization of each number.
1. 50
2  52
2. 84
22  3  7
Find the GCF of each pair of numbers.
3. 18 and 75 3
4. 20 and 36 4
Try these (cont)…
Find the GCF each pair of monomials.
5. 12x and 28x3 4x
6. 27x2 and 45x3y2 9x2
7. Cindi is planting a rectangular flower bed with 40
orange flowers and 28 yellow flowers. She wants
to plant them so that each row will have the
same number of plants but of only one color. How
many rows will Cindi need if she puts the greatest
possible number of plants in each row?
17