Presentation on theme: "Factors and Greatest Common Factors"— Presentation transcript:
1Factors and Greatest Common Factors Lesson 9-1Factors and Greatest Common Factors
2Objectives Find prime factorizations of integers and monomials Find the Greatest Common Factor (GCF) of integers and monomials
3VocabularyPrime number – a whole number, greater than 1, whose only factors are 1 and itselfComposite number – a whole number, greater than 1, that has more than two factorsPrime factorization - a whole number expressed as a product of factors that are all prime numbersFactored 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
4Four Step Problem Solving Plan Step 1: Explore the ProblemIdentify what information is given (the facts)Identify what you are asked to find (the question)Step 2: Plan the SolutionFind an equation the represents the problemLet a variable represent what you are looking forStep 3: Solve the ProblemPlug into your equation and solve for the variableStep 4: Examine the SolutionDoes your answer make sense?Does it fit the facts in the problem?
5Prime and Composite Numbers Prime NumberA whole number, greater than 1, whose only factors are 1 and itselfExamples: 2, 3, 5, 7, 11, 13, 17, 19, 23Composite NumberA whole number, greater than 1, that has more than two factorsExamples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22
6Greatest Common Factor The greatest common factor (GCF) of two or more integers is the product of the prime factors common to the integersThe GCF of two or more monomials is the product of their common factors when each monomial is in factored formIf two or more integers or monomials have a GCF of 1, then the integers or monomials are said to be relatively prime
7Example 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.
8Example 2 Find the prime factorization of 84. Method 1 The 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.8421 43 72 2andAll of the factors in the last branch of the factor tree are prime.Answer: Thus, the prime factorization of 84 is or
9Example 3 Find the prime factorization of –132. Express –132 as –1 times 132./ \/ \/ \Answer: The prime factorization of –132 is or
10Example 4 A. Factor completely. Answer: in factored form is B. Factor completely.Express –26 as –1 times 26.Answer: in factored form is
11Example 5 A. Find the GCF of 12 and 18. Factor each number. 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, 23 or 6, is the GCF.Answer: The GCF of 12 and 18 is 6.B. Find the GCF of .Factor each number.Circle the common prime factors.Answer: The GCF of and is .
12Example 6Crafts 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.
13Example 6 contThe 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.
14Summary & Homework Summary: The greatest common factor (GCF) of two or more monomials is the product of their common prime factorsHomework:pg even