Greatest Common Factor (GCF) of 28 and 24 Using the Distributive Property

When solving problems in mathematics, the greatest common factor (GCF) is a useful tool. It’s particularly helpful when simplifying fractions, factoring expressions, or working through number theory problems. The GCF of two numbers is the largest number that divides both without leaving a remainder. In this article, we’ll explore how to find the GCF of 28 and 24 using the distributive property.

Step 1: Understanding the Prime Factorization of 28 and 24

Before diving into the distributive property, it’s essential to understand the prime factorizations of 28 and 24. This will give us a foundation to determine the GCF.

• The prime factorization of 28 is:
  
  28 = 2² × 7
  
• The prime factorization of 24 is:
  
  24 = 2³ × 3
  

Now that we have the prime factorizations, let’s identify the common factors between the two numbers.

• Common prime factor: 2
• The smallest power of 2 common to both is (2²).

Thus, the GCF of 28 and 24 is: GCF = 2² = 4

Step 2: Using the Distributive Property to Find the GCF

The distributive property states that:

a(b + c) = ab + ac

This property can be applied in reverse to factor out the GCF. Essentially, we use the GCF to rewrite the sum (or difference) of numbers in a factored form. Here’s how we apply it to the numbers 28 and 24.

We already know the GCF of 28 and 24 is 4. Now, let’s express both 28 and 24 as multiples of 4:

28 = 4 ×7


24 = 4 × 6

Using the distributive property, we can factor out the common factor (4) from both terms:

28 + 24 = (4 × 7 + 4 × 6)

= 4 × (7 + 6)

= 4 × 13

Step 3: Verifying the Result

By factoring out the GCF (4), we’ve simplified the sum of 28 and 24 as:

28 + 24 = 52


4 × 13 = 52

Since both methods give us the same result, we’ve successfully used the distributive property to solve the problem and confirm that the GCF is correct.

Why Is This Useful?

Using the distributive property in this way is more than just an academic exercise. It provides a deeper understanding of how common factors work and helps with simplifying algebraic expressions and solving equations more effectively. By breaking numbers down to their GCF, you can factor them out to make complex operations much simpler.