Greatest Common Factor (GCF): Fifth Grade Mathematics

Finding the Greatest Common Factor (GCF) allows you to simplify fractions to lowest terms and divide objects into the largest possible equal groups.

Defining the Greatest Common Factor

The Greatest Common Factor (GCF) of two or more whole numbers is the largest whole number that divides evenly into each of the numbers with no remainder.

+--------------------------------------------------------------------------+
|                  TWO METHODS FOR FINDING THE GCF                         |
+---------------------+----------------------------------------------------+
| Method              | Step-by-Step Procedure                             |
+---------------------+----------------------------------------------------+
| List All Factors    | List all factors of each number, identify common   |
|                     | factors, and select the greatest value.            |
| Prime Factorization | Find prime factorizations; identify all shared     |
| (Venn Diagram)      | prime factors; multiply the shared primes together.|
+---------------------+----------------------------------------------------+

Method 1: Listing Factors

To find the GCF of 24 and 36:
- Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}

- Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}

- Common Factors: {1, 2, 3, 4, 6, 12}

- Greatest Common Factor: 12.

Method 2: Prime Factorization Overlap

Decompose both numbers into prime factorizations:
- 24 = 2 * 2 * 2 * 3

- 36 = 2 * 2 * 3 * 3

- Identify shared prime factors: both numbers share two 2s and one 3.

- Multiply the shared prime factors: GCF = 2 * 2 * 3 = 12.

Venn Diagram Representation of Prime Factors:
         24                Overlap (GCF)              36
   +---------------+     +---------------+     +---------------+
   |               |     |               |     |               |
   |       2       | <-> |    2, 2, 3    | <-> |       3       |
   |               |     |               |     |               |
   +---------------+     +---------------+     +---------------+
                          GCF = 2 * 2 * 3 = 12

Real-World Applications of GCF

GCF is used whenever you need to divide items into equal groups with nothing left over, or when finding the largest square tiles that can cover a rectangular floor.

Exercises and Solutions

Exercise 1: Prime Factorization Method for GCF. Find the Greatest Common Factor (GCF) of 45 and 75 using prime factorization.

Solution 1: Find the prime factorization of 45: 45 = 3 * 3 * 5 = 3^2 * 5. Find the prime factorization of 75: 75 = 3 * 5 * 5 = 3 * 5^2. Identify the prime factors shared by both numbers: both share one 3 and one 5. Multiply the shared prime factors: GCF = 3 * 5 = 15. The GCF of 45 and 75 is 15.

Exercise 2: Party Favors Word Problem. An event coordinator has 48 glow sticks and 64 temporary tattoos. The coordinator wants to create identical party favor bags with no items left over. What is the greatest number of identical bags that can be created, and how many of each item will be in each bag?

Solution 2: Find the GCF of 48 and 64. Prime factorization of 48 = 2^4 * 3. Prime factorization of 64 = 2^6. The shared prime factors are four 2s: 2 * 2 * 2 * 2 = 16. Thus, the GCF is 16. The greatest number of identical bags that can be created is 16 bags. To find the contents of each bag: divide 48 glow sticks by 16 = 3 glow sticks per bag; divide 64 tattoos by 16 = 4 tattoos per bag.