Lesson 9-1

Factors and Greatest Common Factors

Objectives

• Find prime factorizations of integers and monomials

• Find the Greatest Common Factor (GCF) of integers and monomials

Vocabulary

• Prime number – a whole number, greater than 1, whose only factors are 1 and itself

• Composite number – a whole number, greater than 1, that has more than two factors

• Prime factorization - a whole number expressed as a product of factors that are all prime numbers

• Factored form – when a monomial is expressed as a product of prime numbers and variables and no variable has an exponent greater than 1.

• Greatest common factor (GCF) – the product of the prime factors common two or more integers

Four Step Problem Solving Plan

• Step 1: Explore the Problem

– Identify what information is given (the facts)

– Identify what you are asked to find (the question)

• Step 2: Plan the Solution

– Find an equation the represents the problem

– Let a variable represent what you are looking for

• Step 3: Solve the Problem

– Plug into your equation and solve for the variable

• Step 4: Examine the Solution

– Does your answer make sense?

– Does it fit the facts in the problem?

Prime and Composite Numbers

• Prime Number– A whole number, greater than 1, whose only factors

are 1 and itself– Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23

• Composite Number– A whole number, greater than 1, that has more than

two factors– Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22

Greatest Common Factor

• The greatest common factor (GCF) of two or more integers is the product of the prime factors common to the integers

• The GCF of two or more monomials is the product of their common factors when each monomial is in factored form

• If two or more integers or monomials have a GCF of 1, then the integers or monomials are said to be relatively prime

Example 1

A. Factor 22. Then classify it as prime or composite.

To find the factors of 22, list all pairs of whole numbers whose product is 22.

Answer: Since 22 has more than two factors, it is a composite number. The factors of 22, in increasing order, are 1, 2, 11, and 22.

B. Factor 31. Then classify it as prime or composite.

The only whole numbers that can be multiplied together to get 31 are 1 and 31.

Answer: The factors of 31 are 1 and 31. Since the only factors of 31 are 1 and itself, 31 is a prime number.

Example 2

Find the prime factorization of 84.

Method 1The least prime factor of 84 is 2.

The least prime factor of 42 is 2.

The least prime factor of 21 is 3.

All of the factors in the last row are prime.

Method 2 Use a factor tree.84

21 4

3 7 2 2 and

All of the factors in the last branch of the factor tree are prime.Answer: Thus, the prime factorization of 84 is or

Example 3

Find the prime factorization of –132.

Express –132 as –1 times 132.

Answer: The prime factorization of –132 isor

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Example 4A. Factor completely.

Answer: in factored form is

B. Factor completely.

Answer: in factored form is

Express –26 as –1 times 26.

Example 5A. Find the GCF of 12 and 18.

The integers 12 and 18 have one 2 and one 3 as common prime factors. The product of these common prime factors, 23 or 6, is the GCF.

Factor each number.

Circle the common prime factors.

Answer: The GCF of 12 and 18 is 6.

Factor each number.

Circle the common prime factors.

B. Find the GCF of .

Answer: The GCF of and is .

Example 6

Crafts Rene has crocheted 32 squares for an afghan. Each square is 1 foot square. She is not sure how she will arrange the squares but does know it will be rectangular and have a ribbon trim. What is the maximum amount of ribbon she might need to finish an afghan?

Find the factors of 32 and draw rectangles with each length and width. Then find each perimeter.The factors of 32 are 1, 2, 4, 8, 16, 32.

Example 6 cont

The greatest perimeter is 66 feet. The afghan with this perimeter has a length of 32 feet and a width of 1 foot.

Answer: The maximum amount of ribbon Rene will need is 66 feet.

Summary & Homework

• Summary:– The greatest common factor (GCF) of two

or more monomials is the product of their common prime factors

• Homework: – pg. 477 20-58 even