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## Using Prime Factorizations:

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**Using Prime Factorizations:**How can prime factorizations help us find the GCF and LCM of two numbers????**Prime Factorization Defn:**• The prime factorization of a number is a string of factors made up of only prime numbers.**The GCF:**• Find the Prime Factorization of 24 and 60. • 24 = and 60 = • Work with your partner: • What would be the GCF of 24 and 60?**The GCF would be the product of the longest string of prime**factors that the numbers have in common. For example, the longest string of factors 24 and 60 have in common is 2 x 2 x 3. So, the GCF of 24 and 60 is : 2 x 2 x 3 = 12**Now you try….**• Using the idea of prime factorization, find the GCF for: • 48 and 72 • 30 and 54**The LCM:**• 24 = 2 x 2 x 2 x 3 and 60 = 2 x 2 x 3 x 5 • Work with your partner to come up with a way to find the LCM of 24 and 60 using their prime factorizations.**The least common multiple of two numbers is the product of**the shortest string that contains the prime factorizations of BOTH numbers. • For example: the shortest string that contains the prime factorizations of 24 and 60 is: 2 x 2 x 2 x 3 x 5**Now you try….**• Find the least common multiple for: • 48 and 72 • 30 and 54**Follow-up:**• The GCF of 25 and 12 is 1. Find two other pairs of numbers with a GCF of 1. Such pairs of numbers are said to be relatively prime.**The LCM of 6 and 5 is 30. Find two other pairs of numbers**for which the LCM is the product of the numbers.**Find two pairs of numbers for which the LCM is small than**the product of the two numbers. For example, the product of 6 and 8 is 48; the LCM is 24.**How can you tell from the prime factorization whether the**LCM of two numbers is the product of the two numbers or is less than the product of the two numbers? EXPLAIN