Transcript Factors and Multiples

Warm-Up #2
• Use the following terms to describe your
number: divisible, prime and composite.
Be sure to explain how your number does
or doesn’t fit these categories.
• Be sure to underline the terms in your
journal.
Factors and Multiples
Number Theory GONE WILD!
Factors “Fit” into Families
Multiples Multiply like
Rabbits!
What am I Learning Today?
Prime Factorization
How will I show that I learned it?
Decompose numbers into ONLY prime factors
using the factor tree
Prove that all numbers have a unique string of
prime number
Vocabulary
Prime Factorization: A number written as the
product of its prime factors.
Fundamental Theorem of Arithmetic: All
positive numbers greater than ONE can be
decomposed into a unique string of prime
numbers.
Visual Vocabulary
An exponent tells how many times
a number called the base is used
as a factor.
A number is in exponential form
when it is written with a base and
an exponent.
Exponent
Base
3
=
7

7

7=
343
7
You can use factors to write a number
in different ways.
Factorization of 12
1 • 12
2•6
3•4
3•2•2
Notice that
these factors
are all prime.
What do you notice about how the
last set of factors are written?
The prime
factorization of a
number is the number
written as the product
of its prime factors.
Questions
What is the term for
decomposing a
number?
How do I write the prime
factorization of a
number?
How do I decompose a
number into its prime
factors?
How do I make a factor
tree?
How do I use the ladder
method?
Answers
Factoring
As the product of prime numbers ONLY
Using a factor tree or a ladder
1. Write your number.
2. Choose any two factors of this number and
attach them to the original number with
“branches.”
3. If one of these numbers is prime, circle it.
4. Continue decomposing numbers until only
prime numbers are left.
Division using an upside down layer cake
Write the prime factorization of 24
(using a factor tree)
Choose any two factors of 24 to begin. Keep finding
factors until each branch ends at a prime factor.
24
2
•
24
2
•
6
12
•
6
2
•
3
3
•
2
4
2
•
2
24 = 3 • 2 • 2 • 2
24 = 2 • 2 • 2 • 3
The prime factorization of 24 is 2 • 2 • 2 • 3
Write the prime factorization of 24
(using a ladder)
Choose a prime factor of 24 to begin. Keep dividing by
prime factors until the quotient is 1.
2
24
2
3
12
2 8
2 6
3
24
2 4
3
1
24 = 2 • 2 • 2 • 3
2 2
1
24 = 3 • 2 • 2 • 2
The prime factorization of 24 is 2 • 2 • 2 • 3
Paired Discussion
Turn to a partner and discuss the
following:
When decomposing a number, will the same
prime factors result even when you start
with different factor pairs? Explain.
YES! There is only one way to write the prime factorization
of a number: Fundamental Theorem of Arithmetic
Prime factors may be written in a different order, but they
are still the same factors.
Fundamental Theorem of
Arithmetic
Factors of 360:
2 x 180, 3 x 120, 4 x 90, 5 x 72,
6 x 60, 8 x 45, 9 x 40, 10 x 36, 12 x 30,
15 x 24, 18 x 20
Paired Discussion
Turn to a partner and discuss the
following:
The prime factorization for 81 is 3 • 3 • 3 • 3.
Is there any easier way to write this?
Explain.
You can use exponents to write prime
factorizations.
34 = 3 • 3 • 3 • 3
Using exponents, can you..?
Shorten the following words :
Mississippi: m • i4 • s4 • p2
Mathematician: m2 • a3 • t2 • h • e • i2 • c • n
Factorization: f • a2 • c • t2 • o2 • r • i2 • z • n
How does this work for numbers?
3  3  3  3  3 = 35
3 is a factor 5 times.
This DOES NOT mean 3 x 5 = 15
Fun Factor Trees
Try these on your own.
1) List all the factors of the following numbers.
Make sure you use the divisibility rules so you
don’t miss any factors. Remember BFF.
2) Find the prime factorization using both tree
and ladder methods.
3) Make sure you use exponential form, where
applicable.
1) 49
2) 76
4) 94
3) 132
5) 249