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Prime Factorization, Greatest Common Factor, & Least Common Multiple. EDTE 203. Introduction. Determining Prime Factorization Determining the Greatest Common Factor (GCF) Determining the Least Common Multiple (LCM). Introduction.

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## Prime Factorization, Greatest Common Factor, & Least Common Multiple

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**Prime Factorization,Greatest Common Factor, &Least Common**Multiple EDTE 203**Introduction**• Determining Prime Factorization • Determining the Greatest Common Factor (GCF) • Determining the Least Common Multiple (LCM)**Introduction**• The facts you will learn will give you a variety of information about prime factorization, GCF, and LCM. • This lesson will show you different ways to calculate the prime factors of composite numbers. • This lesson will show you how to use the prime factors to calculate the GCF and LCM of two composite numbers. • You will learn how prime factorization equates to everyday life.**Essential Question**The Essentials We Hope To Discover**2**5 2 3 3 3 5 6 2 10 18 180 The Essential Question • How do prime factorization, greatest common factor, and least common multiple help you to understand the world?**Background Information**The Basic facts you need you to know about prime factorization, GCF, and LCM**History of Prime Factorization, Greatest Common Factor, &**Least Common Multiple • Originated around 300 B.C. through the “Theorem of (unique) prime factorization” • Started with Euclid’s “Property of Natural Numbers” (e.g., 24= 2∙2∙2∙3)**History of Prime Factorization, Greatest Common Factor, &**Least Common Multiple cont. • The Theorem of Prime Factorization was further proven through the work of Gauss and Ernst Eduard Kummer • Prime Factorization is the foundation for finding the Greatest Common Factor and the Least Common Multiple**Solving for Prime Factorization, GCF, and LCM.**• There are 2 Ways determine the prime factors • Factor Tree Method • Stacked Method • Determining the GCF and LCM • GCF • LCM**Factor Tree Method**96 • × 12 4 × 22 × 6 2 × 22 × 3 2×2×2×2×2×3 =96 • The CORRECT answer: • must be only PRIME numbers • must multiply to give the specified quantity**Factor Tree Method cont.**There is more than one way to solve the same problem 96 4 × 24 2 × 26 × 4 2 × 3 2 × 2 96= 2×2×2×2×2×3 96 • × 12 4 × 22 × 6 2 × 22 × 3 96= 2×2×2×2×2×3**The Stacked Method**• Begin by dividing the specified quantity by any PRIME number that divides equally, (hint; if it is even try dividing by 2) • Reduce the quotient, dividing again by a PRIME number • Continue reducing the quotient until both the divisor and the quotientare prime numbers. • Re-write the prime numbers as a multiplication problem. (if the final quotient is 1 it doesn’t need included in the answer) • The CORRECT answer: • must be only prime numbers • must multiply to give the specified quantity**Determining the Greatest Common FactorOf Two Composite**Numbers**Solving for the Greatest Common Factor**Find the prime factorization of the given quantities Determine what factors they have in common. 36 3 × 12 3 × 4 2 × 2 2 × 2 × 3 × 3 = 36 54 6 × 9 3 × 23 × 3 2 × 3 × 3 × 3 =54**Determining the Least Common MultipleOf Two Composite**Numbers**Solving for the Least Common Multiple**36 3 × 12 3 × 4 2 × 2 2 × 2 × 3 × 3 = 36 54 6 × 9 2 × 33 × 3 2 × 3 × 3 × 3 =54**Finding the Greatest Common Factor**of Two Numbers We are looking for a factor. The factor must be common to both numbers. We need to pick the greatest of such common factors.**The GCF of 36 and 90**Method 1 1) List the factors of each number. 36: 1 2 3 4 6 36 18 24 9 90: 1 2 3 5 6 9 90 45 30 18 15 10 2) Circle the common factors. 3) The greatest of these will be your Greatest Common Factor: 18**The GCF of 36 and 90**Method 2 1) Prime factor each number. 36 = 2 ● 2 ● 3 ●3 90 = 2 ● 3 ● 3 ●5 2) Circle each pair of common prime factors. 3) The product of these common prime factors will be 2 ● 3 ● 3 =18 the Greatest Common Factor:**Finding the Least Common Multiple**of Two Numbers We are looking for a multiple. The multiple must be common to both numbers. We need to pick the least of such common multiples.**The LCM of 12 and 15**Method 1 1) List the first few multiples of each number. 12: 12 24 36 48 60 72 84 90 108 120 15: 15 30 45 60 75 90 105 120 135 2) Circle the common multiples. 3) The least of these will be your Least Common Multiple: 60**The LCM of 12 and 15.**Method 2 1) Prime factor each number. 12 = 2 ● 2 ● 3 15 = 5 ● 3 2) Circle each pair of common prime factors. 3) Circle each remaining prime factor. 4) Multiply together one factor from each circle to get the 3 ● 2 ● 2 ● 5=60 Least Common Multiple : Note that the common factor, 3, was only used once.**Method 3: Find both GCF and LCM at Once.**The GCF and LCM of 72 and 90 1) Make the following table. 9 8 2 10 4 5 2) Divide each number by a common factor. 3) Divide the new numbers by a common factor. Repeat this process until there is no longer a common factor. The product of the factors on the left is the GCF: The product of the factors on the left AND bottom is the LCM: 9 ● 2= 18 9 ● 2● 4● 5 = 360**Journal & Summary**• Nine people plan to share equally 24 stamps from one set and 36 stamps from another set. Explain why 9 people cannot share the stamps equally. • What's is the LCM for two numbers that have no common factors greater than 1? Explain your reasoning.

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