Finding GCF’s and LCM’s Grade 6

Study Island Responders Do Now (Pre-Test) Study Island Responders 5 minutes

Learning Objective Students will be able to identify factors and multiples of a positive integer, common factors, common multiples, the greatest common factor, and the least common multiple of a set of positive integers. 5N13 - Calculate multiples of a whole number and the least common multiple of two numbers 5N14 - Identify the factors of a given number 5N15 - Find the common factors and the greatest common factor of two numbers

Vocabulary GCF – the factor with the greatest value that is shared by all numbers. Multiples – the product of a number and a non zero number. LCM – the multiple with the least value that is shared by all numbers.

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

12 = 2 ● 2 ● 3 15 = 5 ● 3 3 ● 2 ● 2 ● 5 = 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. 72 90 9 2 8 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

Method 3: Find both GCF and LCM at Once. One more example: The GCF and LCM of 96 and 144 1) Make the following table. Note that you can pick any common factor to start and any remaining common factor for each step. Try starting by dividing by 3 to see that this is so. 96 144 2 6 48 72 4 8 12 2 3 2) Divide each number by a common factor. 3) Divide the new numbers by a common factor. 4) 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: 2 ● 6 ● 4 = 48 2 ● 6 ● 4 ● 2 ● 3 = 288

Independent Practice Group 1 Group 2 Group 3

Quick Quiz Study Island (Responders) “Games”

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. Nine is not a common factor of 24 and 36. The LCM will be the product of the two numbers.