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Prime Factorization, Greatest Common Factors, Least Common Multiples

Prime Factorization, Greatest Common Factors, Least Common Multiples

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## Prime Factorization, Greatest Common Factors, Least Common Multiples

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**Prime Factorization, Greatest Common Factors,Least Common**Multiples**What numbers can be multiplied together to get 24?**The number 24 is called the product because it is the result of multiplying two or more numbers together. 1 x 24 2 x 12 3 x 8 The numbers 1, 2, 3, 4, 6, 8, 12, and 24 are called factors. These are the numbers being multiplied together. 4 x 6 What are the factors of seven? 1 x 7 Composite numbers – are numbers which have two or more factors like the number 24. Prime numbers – are numbers having exactly two factors: one and itself. Seven is a prime number.**Prime Factorization**Factor 56 using only prime factors. To do this, we can make a factor tree as follows. 56 28 is composite so we can factor it further. 2 is prime. 2 • 28 14 is composite, so we can factor it further. 2 is prime. 2 • 14 2 is prime. 7 is prime. • 2 7 The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7.**Prime Factorization**or We can make a factor tree using different starting factors. 56 14 is composite so we can factor it further. 4 is composite so we can factor it further. 4 • 14 • 2 • 2 2 7 2 is prime. 7 is prime. The prime factorization of 56 is 2 • 2 • 2 • 7 or 23 • 7. The prime factorization of every number is unique. No matter what factors you choose, the prime factorization will still be the same for that number.**Greatest Common Factor (GCF)**• The greatest common factor is the largest factor that two numbers share. • Let’s find the GCF of 12 and 42. First, we need to make a list of factors for each number.**12**42 1 x 12 1 x 42 Factors of 12: 1, 2, 3, 4, 6,12 2 x 21 2 x 6 3 x 14 3 x 4 4 x ?? 4 x 3 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42 5 x ?? 6 x 7 7 x 6 Common Factors: 1, 2, 3, 6 Greatest Common Factor: 6**What is the GCF of 18 and 27?**18 27 Factors of 18: 1, 2, 3, 6, 9, 18 1 x 18 1 x 27 2 x ? 2 x 9 3 x 9 3 x 6 Factors of 27: 1, 3, 9, 27 4 x ? 5 x ? 4 x ? 6 x ? 5 x ? 7 x ? Common Factors: 1, 3, 9 8 x ? 6 x 3 9 x 3 GCF: 9**What is the GCF of 48 and 60?**Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 48 60 1 x 48 1 x 60 2 x 30 2 x 24 3 x 20 3 x 16 4 x 15 Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 4 x 12 5 x 12 6 x 8 6 x 10 Common Factors: 1, 2, 3, 4, 6, 12 GCF: 12**Least Common Multiple (LCM)**• The least common multiple is the smallest number that is common between two lists of multiples.**EXAMPLE: Find the LCM of 12 and 18**• The multiples of 12: • 12 x 1 = 12 • 12 x 2 =24 • 12 x 3 = 36 • 12 x 4 = 48 • 12 x 5 =60 • The multiples of 18: • 18 x 1 = 18 • 18 x 2 = 36 • 18 x 3 = 54 • 18 x 4 = 72 • 18 x 5 = 90**12, 24, 36, 48, 60**18, 36, 54, 72, 90 The first number you see in both lists is 36. The least common multiple of 12 and 18 is 36.**Example 2: Find the LCM of 9 and 10**9, 18, 27, 36, 45, 54, 63, 72 81, 90, 99 10, 20, 30, 40, 50, 60, 70, 80 90, 100, 110 If you don’t see a common multiple, make each list go further. The LCM of 9 and 10 is 90**Example 3:Find the LCM of 4 and 12**4, 8, 12, 16 12, 24, 36 Answer: 12**Example 4:Find the LCM of 5 and 8**5, 10, 15, 20, 25, 30 35, 40 8, 16, 24, 32, 40, 48 Answer: 40**Example 5:Find the LCM of 6 and 20**6, 12, 18, 24, 30, 36 42, 48, 54, 60 20, 40, 60, 80, 100, 120 Answer: 60