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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.

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**Number Theory GONE WILD!**

Factors and Multiples Number Theory GONE WILD! Factors “Fit” into Families Multiples Multiply like Rabbits!

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**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

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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.

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**73 Visual Vocabulary = 7 7 7= 343**

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 73 Base = 7 7 7= 343

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

You can use factors to write a number in different ways. Factorization of 12 Notice that these factors are all prime. 3 • 4 3 • 2 • 2 1 • 12 2 • 6 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.

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**Questions Answers Factoring As the product of prime numbers ONLY**

What is the term for decomposing a number? Factoring 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 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 make a factor tree? How do I use the ladder method? Division using an upside down layer cake

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**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 24 2 • 12 6 • 4 2 • 6 3 • 2 2 • 2 2 • 3 24 = 3 • 2 • 2 • 2 24 = 2 • 2 • 2 • 3 The prime factorization of 24 is 2 • 2 • 2 • 3

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**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 3 24 2 12 2 8 2 6 2 4 3 3 2 2 1 1 24 = 2 • 2 • 2 • 3 24 = 3 • 2 • 2 • 2 The prime factorization of 24 is 2 • 2 • 2 • 3

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**Turn to a partner and discuss the following:**

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.

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**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

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**Turn to a partner and discuss the following:**

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

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**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 = is a factor 5 times This DOES NOT mean 3 x 5 = 15

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Fun Factor Trees

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**Try these on your own. 1) 49 2) 76 3) 132 4) 94 5) 249**

List all the factors of the following numbers. Make sure you use the divisibility rules so you don’t miss any factors. Remember BFF. Find the prime factorization using both tree and ladder methods. Make sure you use exponential form, where applicable. 1) 49 2) 76 3) ) ) 249

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Number Theory and the Real Number System

Number Theory and the Real Number System

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