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 MultiplesFactors and MultiplesNumber Theory GONE WILD!

Factors “Fit” into FamiliesMultiples Multiply like

Rabbits!

What am I Learning Today?What am I Learning Today?Prime Factorization

How will I show that I learned it?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

VocabularyVocabularyPrime 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.

A number is in exponential form when it is written with a base and an exponent.

7733Base

Exponent

= 7 7 7= 343

An exponent tells how many times a number called the base is used as a factor.

Visual VocabularyVisual Vocabulary

You can use factors to write a number in different ways.

Factorization of 12

2 • 61 • 12 3 • 4 3 • 2 • 2

The prime factorization of a

number is the number written as the product

of its prime factors.

Notice that these factors are all prime.

What do you notice about how the last set of factors are written?

What is the term for decomposing a number?

Factoring

Questions Answers

How do I write the prime factorization of a number?

As the product of prime numbers ONLY

How do I decompose a number into its prime factors?

Using a factor tree or a ladder

How do I make a factor tree?

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.

How do I use the ladder method? Division using an upside down layer cake

Choose any two factors of 24 to begin. Keep finding factors until each branch ends at a prime factor.

24

2 12•

6

2

2 •

3•

24

6 4•

3 2 2 2

24 = 2 • 2 • 2 • 3

24 = 3 • 2 • 2 • 2

The prime factorization of 24 is 2 • 2 • 2 • 3

• •

Write the prime factorization of 24Write the prime factorization of 24(using a factor tree)(using a factor tree)

Choose a prime factor of 24 to begin. Keep dividing by prime factors until the quotient is 1.

2 24

2

1

12

6

24 = 2 • 2 • 2 • 3

3 24

2

1

8

2 4

24 = 3 • 2 • 2 • 2

The prime factorization of 24 is 2 • 2 • 2 • 3

Write the prime factorization of 24Write the prime factorization of 24

(using a ladder)(using a ladder)

2

33 2 2

Paired DiscussionPaired DiscussionTurn 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 Fundamental Theorem of ArithmeticArithmetic

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

You can use exponents to write prime factorizations.

34 = 3 • 3 • 3 • 3

Paired DiscussionPaired DiscussionTurn 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.

Using exponents, can you..?Using exponents, can you..?

Shorten the following words :Mississippi:

Mathematician:

Factorization:

m • i4 • s4 • p2

m2 • a3 • t2 • h • e • i2 • c • n

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 TreesFun Factor Trees

Try these on your own.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 3) 132 4) 94 5) 249