Factors and Greatest Common Factors

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Lesson 9-1

• Factors and Greatest Common Factors

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Objectives

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

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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

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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?

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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

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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

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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.
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Example 2
Find the prime factorization of 84.
Method 1
All of the factors in the last row are prime.
Method 2 Use a factor tree.
84
All of the factors in the last branch of the
factor tree are prime.
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Example 3
Find the prime factorization of 132.
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Example 4
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Example 5
A. Find the GCF of 12 and 18.
Circle the common prime factors.
The integers 12 and 18 have one 2 and one 3 as
common prime factors. The product of these common
prime factors, 2?3 or 6, is the GCF.
Answer The GCF of 12 and 18 is 6.
Circle the common prime factors.
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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.
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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.
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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