Transcript To find the greatest common factor using prime factorization

3.5
Prime Numbers and GCF
Objectives
1.
2.
3.
4.
5.
6.
Determine if a number is prime, composite, or neither.
Find the prime factorization of a given number.
Find all factors of a given number.
Find the greatest common factor of a given set of
numbers by listing.
Find the greatest common factor of a given set of
numbers using prime factorization.
Find the greatest common factor of a set of
monomials.
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Objective 1
Determine if a number is
prime, composite, or neither.
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Definition
Prime number: A natural number that has
exactly two different factors, 1 and itself.
Is 1 prime? No because a prime must have exactly two
different factors and 1’s factors are the same 1 and 1.
Is 2 prime? Yes, because 2 is a natural number whose
only factors are 1 and 2 (itself).
Is 3 prime? Yes, because 3 is a natural number whose
only factors are 1 and 3 (itself).
Is 4 prime? No, because 4 has factors other than 1
and 4 (2 is also a factor).
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If we continue this line of thinking, we get the
following list of prime numbers…
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, …
What about some of the numbers that are not in
this list…what are they called?
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Definition
Composite number: A natural number that
has factors other than 1 and itself.
Is 1 composite? No, because its only factor is itself.
The number 1 is neither prime or composite.
The first composite number is 4. Here is a short list of
composite numbers.
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25,…
Remember…every composite number is divisible by at
least one prime number.
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How do we determine if a number is prime or
composite when it isn’t immediately and obviously
apparent? Here’s an example…
Is 91 prime or composite?
Is 91 divisible by 2?
Is 91 divisible by 3?
Is 91 divisible by 5?
Is 91 divisible by 7?
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Is 127 prime or composite?
Divide by each of the prime numbers…
Divisible by 2?
Divisible by 3?
Divisible by 5?
Divisible by 7?
What about 11?
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Procedure
To determine if a given number
is prime or composite, divide it by the prime
numbers in ascending order and consider the
results.
1. If the given number is divisible by the prime, stop.
The given number is a composite number.
2. If the given number is not divisible by the prime,
consider the quotient.
a. If the quotient is greater than the current
divisor, repeat the process with the next prime
on the list of prime numbers.
b. If the quotient is equal to or greater than the
current prime divisor, stop. The given number
is prime.
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Example 1
Determine if 157 is prime or composite.
Solution: Divide by the list of prime numbers.
Is 157 divisible by 2?
Is 157 divisible by 3?
Is 157 divisible by 5?
Is 157 divisible by 7?
s 157 divisible by 11?
s 157 divisible by 13?
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Objective 2
Find the prime factorization
of a given number.
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Definition
Prime factorization: A product written with
prime factors only.
For example, the prime factorization of 20 is 2 • 2 • 5,
which we can write in exponential form as 22 • 5.
20
2
20
10
2
4
or
5
2
5
2
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Procedure
To find the prime factorization
of a composite number using a factor tree:
1. Draw two branches below the number.
2. Find two factors whose product is the given
number and place them at the end of the
two branches.
3. Repeat steps 1 and 2 for every composite
factor.
4. Write a multiplication sentence containing
all the prime factors.
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Example 2
We could have started a factor tree for 84 many
different ways and still ended up with the same result.
84
84
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84
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Example 2
b. 96
Find the prime factorization. Write the
answer in exponential form.
c. 3500
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Objective 3
Find all factors of a given
number.
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Example 3
List all factors of 24.
List all factors of 45.
List all factors of 60.
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Objective 4
Find the greatest common
factor of a given set of
numbers by listing.
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Definition
Greatest common factor: The greatest
number that divides all given numbers with
no remainder.
The greatest common factor is sometimes called
the greatest common divisor.
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Procedure
To find the greatest common factor by
listing:
1. List all factors for each given
number.
2. Search the lists for the greatest
factor common to all lists.
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Example 4
Find the GCF of 24 and 60.
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Example 4
Find the GCF of 48 and 72.
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Objective 5
Find the greatest common
factor of a given set of
numbers by using prime
factorization.
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Procedure
To find the greatest common
factor using prime factorization:
1. Write the prime factorization of each
number in exponential form.
2. Create a factorization for the GCF that
contains only those prime factors
common to all factorizations, each raised
to the least of its exponents.
3. Multiply.
Note: If there are no common prime factors, the GCF is 1.
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Example 5
Find the GCF of 3024 and 2520.
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Example 6
The floor of a 42 foot by 30 foot room
will be covered with colored squares like a
checkerboard. All the squares must be equal sized
with whole number dimensions and may not be cut
or overlapped to fit inside the room. Find the
dimensions of the largest possible square that can
be used.
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Objective 6
Find the greatest common
factor of a set of monomials
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Example 7
Find the GCF of the monomials.
a. 18x4 and 12x3
b. 24n2 and 40n5
c. 14t4 and 45
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