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INVESTIGATION 4.3: USING PRIME FACTORIZATIONS Learning Targets I can find common factors, greatest common factors, common multiples, and least common multiples using prime factorizations Lets Revisit some material What are the common factors of 24 and 60? What are the common multiples of 24 and 60? Prime Factorization Using either the division method or the factor tree method, find the prime factorization of 24 and 60. 60 10 5 x x 6 2 3 x 2 24 2 12 2 60 5 3 2 2 2 6 3 24 3 2 2 2 Common Factors Here we have a Double Bubble Map to find the common factors of 24 and 60. Let’s pick a common factor of 24 and 60. 1 24 2 5 10 15 3 24 8 60 20 4 6 30 12 60 24 3 2 2 2 60 5 3 2 2 12 3 2 2 What does the factorization of 12 have in common with the factorizations of 24 and 60? Find the prime factorization of 12 with your strategy of choice and write the prime factorization below the prime factorizations of 24 and 60 without exponents. 24 3 2 2 2 60 5 3 2 2 6 3 2 What does the factorization of 6 have in common with the factorizations of 24 and 60? Lets pick another common factor. Find the prime factorization of 6 with your strategy of choice and write the prime factorization below the prime factorizations of 24 and 60 without exponents. Our Conjecture Can we make any conjectures about the prime factorization of any common factors of 24 and 60? The prime factorizations of common factors are “substrings” of the factorizations of both 24 and 60. Common Multiples For some kinds of problems, we need to find common factors. For others we need to find common multiples. Let’s think about common multiples and see whether we can find a way to use prime factorization to help us find them. Common Multiples What are some common multiples of 24 and 60? Common Multiples Questions to ponder If we find the product of two numbers, for example 24 x 60, will the product be a common multiple? If so, why? Is this product a multiple of both 24 and 60? How do you know? Common Multiples Questions to ponder If so, what do you multiply 24 by to get 3 Multiply by 5 x 3 x 2 x 2 or 60. What do you multiply 60 by to get 3 x 2 x 2 x 2 x 5 x 3 x 2 x 2? x 2 x 2 x 2 x 5 x 3 x 2 x 2? Multiply 60 by 3 x 2 x 2 x 2 or 24. Common Multiples This means that both 24 and 60 “live” in the product of the two numbers. Can you circle prime factors from the product that will give you 24? 24 60 : 3 2 2 2 5 3 2 2 Can you circle prime factors from the product that will give you 60? Let’s use these ideas to explore the questions in Problem 4.3 Problem 4.3 You may work with a shoulder partner 1) 2) 3) Write the prime factorizations of 72 and 120. What is the longest string common to both factorizations? What is the greatest common factor of 72 and 120? How do you know? What is the shortest string of factors that includes the prime factorizations of both 72 and 120? Can you find a smaller common multiple of 72 and 120? Why or why not? Homework Investigation 4.3 ACE (19-22, 32)