GCF and LCM Lesson 3.01

After completing this lesson, you will be able to say: I can find the least common multiple of two whole numbers. I can find the greatest common factor of two whole numbers. I can use the distributive property to rewrite the sum of two numbers using the greatest common factor.

Key Words Prime number: Composite number: A whole number greater than one that has exactly two factors, the number 1 and itself. The numbers 2, 3, 5, 7, 11, 13, and 17 are all examples of prime numbers. Composite number: A number with more than two factors. The number 12 is a prime number because we can break it into more than 2 factors: 2 x 6, 3 x 4, 1 x 12

Prime factorization Prime factorization: Writing any composite number as a product of its prime factors. Every number can be written as a product of its factors. You can also write any composite number as a product of its prime factors, which is called prime factorization. When you write the prime factorization, it is always a good idea to write the factors in order from least to greatest.

Prime factorization If you’ve done factorization before, you may have used factor trees to find the prime factorization of a number Because division is the opposite of multiplication, you can start the factorization process by asking, “What two numbers multiply to equal 36?” 36 Now ask yourself, "Can 12 or 3 break down into smaller numbers, or are they prime?” 3 is a prime number so let’s circle it What about the number 12? Can we break it down into smaller parts? 12 3 Okay, 12 can be broken down into 6 times 2. Are either of those numbers prime? 2 is prime so let’s circle it How can we break down 6? 2 6 Finally, 6 can be factored into 3 × 2. Both numbers will go into our prime factorization of 36. Prime factorization of 36: 2 x 2 x 3 x 3

Try it! Create a factor tree, and write the prime factorization of 64

Check you work 64 4 16 2 2 4 4 Prime factorization of 64: 2 x 2 x 2 x 2 x 2 x 2

Least Common Multiple Multiple: A number that is created when it is multiplied by other numbers. Least common multiple: The smallest of the common multiples between two or more numbers, also known as the LCM.

Finding the Least Common Multiple Finding the LCM from a list Find the LCM of 2 and 3 Step 1: Identify the multiples of the two numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 3, 6, 9, 12, 15, 18, 21, 24 Step 2: Identify the common multiples Step 3: Identify the least common multiple (LCM) 6 is the smallest and is the LCM.

Finding the Least Common Multiple Find the LCM using prime factorization Find the LCM of 4 and 6 Find the prime factorization of 4 and 6 Identify the factors in prime factorization that they have common. 4 = 2 × 2 6 = 2 × 3 Calculate the LCM Both numbers have one 2 in common, so it will be used only one time. There is also an extra factor of 2 and an extra factor of 3 that do not appear in both prime factorizations. Multiply the common factor with all extra factors. LCM = 2 × 2 × 3 = 12 The LCM of 4 and 6 is 12. Both 4 and 6 have a factor of 2 in common.

Try It What is the LCM of 8 and 10?

Check your work What is the LCM of 8 and 10?

Greatest Common Factor The largest factor shared in common by two or more numbers, also known as the GCF.

Finding the GCF Finding the GCF using a list Find the GCF of 12 and 18 Step 1: Identify the factors of the two numbers 12 : 1, 2, 3, 4, 6, 12 18: 1, 2, 3, 6, 9, 18 Step 2: Identify the common factors 12: 1, 2, 3, 4, 6, 12 Step 3: Identify the Greatest Common Factor (GCF) 6 is the largest and is the GCF

Finding the GCF 36 = 2 x 2 x 3 x 3 54 = 2 x 3 x 3 x 3 Finding the GCF using prime factorization The GCF of 36 and 54 Find the prime factorization of 36 and 54 Identify the factors in prime factorization that they have common. 36 = 2 x 2 x 3 x 3 2 x 3 x 3 x 3 54 = 2 x 3 x 3 x 3 The greatest common factor is the product of all of the common prime factors. GCF = 2 × 3 × 3 = 18

Try It Find the GCF of 48 and 96

48 = 2 x 2 x 2 x 2 x 3 96 = 2 x 2 x 2 x 2 x 2 x 3 Check your work GCF = 2 x 2 x 2 x 2 x 3 GCF = 48

Using the GCF with the Distributive Property One use of the GCF is with the distributive property. First, recall the distributive property with this visual below to see how 4 x (2 + 6) is the same as 8 + 24. The distributive property shows the 4 rows of 2 yellow boxes plus 6 green boxes is the same as 4 rows of 2 yellow boxes plus 4 rows of 6 green boxes. What is happening is that the 4 is being distributed by multiplying it with each number being added in the parentheses to get 4 × 2 + 4 × 6

Using the GCF with the Distributive Property Use the greatest common factor and the distributive property to express 44 + 16 in a different way First, identify the GCF of both numbers by listing the factors The factors of 44: 1, 2, 4, 11, 22, 44 The factors of 16: 1, 2, 4, 8, 16 The GCF is 4. Then rewrite the problem using the distributive property Since the GCF is 4, you can divide both 44 and 16 by 4. When you do this, place the GCF on the outside and the two quotients remain in parentheses. 44 + 16 = 4 × (11 + 4) Also, 4 × (11 + 4) can be written as 4(11 + 4). So how do you know this is correct? Just calculate the sum. 44+16 = 4(11+4) 44+16 = 4(15) 60 = 60 Both sides are equal.

Try It How can you rewrite the sum of 18 + 48 using the GCF?

Check your work How can you rewrite the sum of 18 + 48 using the GCF? First, identify the GCF of both numbers by listing the factors The factors of 18: 1, 2, 3, 6, 9, 18 The factors of 48: 1, 2, 3, 4, 6, 8, 12,16, 24, 48 The GCF is 6. Then rewrite the problem using the distributive property Since the GCF is 6, you can divide both 18 and 48 by 6. When you do this, place the GCF on the outside and the two quotients remain in parentheses. 18 + 48 = 6 × (3 + 8) Also, 6 × (3 + 8) can be written as 6 (3 + 8)

Now that you completed this lesson, you should be able to say: I can find the least common multiple of two whole numbers. I can find the greatest common factor of two whole numbers. I can use the distributive property to rewrite the sum of two numbers using the greatest common factor.