As developers of mathematical tools, we understand that finding the Greatest Common Factor (GCF) of two numbers can be challenging for many users. Today, we’ll explore how to determine the GCF of **192** and **72** using three different methods: **Prime Factorization**, **Listing Factors**, and the **Euclidean Algorithm**. By the end of this lesson, you’ll be equipped with multiple strategies to tackle similar problems with confidence.

## What is the Greatest Common Factor (GCF)?

The **Greatest Common Factor (GCF)** of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. In other words, it’s the biggest number that both numbers share as a factor.

**Example:**

For the numbers **8** and **12**, the factors of **8** are **1, 2, 4, 8**, and the factors of **12** are **1, 2, 3, 4, 6, 12**. The common factors are **1, 2, 4**, so the GCF is **4**.

## Methods to Find the GCF of 192 and 72

Below, we provide three different methods to find the GCF. You can also determine the GCF using our free online GCF Calculator

### Method 1: Prime Factorization

Prime factorization involves breaking down each number into its prime components. Once we have the prime factors, the GCF is the product of the smallest powers of all common primes.

**Step 1: Prime Factorization of Each Number**

**Factorizing 192:**

**192 ÷ 2 = 96**

**96 ÷ 2 = 48**

**48 ÷ 2 = 24**

**24 ÷ 2 = 12**

**12 ÷ 2 = 6**

**6 ÷ 2 = 3**

**3 ÷ 3 = 1**

**192**can be expressed as:

**192= 2**

^{6}×3^{1}**Factorizing 72:**

**72 ÷ 2 = 36**

**36 ÷ 2 = 18**

**18 ÷ 2 = 9**

**9 ÷ 3 = 3**

**3 ÷ 3 = 1**

**72**can be expressed as:

**72=2**

^{3}×3^{2}**Step 2: Identify Common Prime Factors**

**Common primes**: 2 and 3**Smallest exponents**:- For
**2**: min(6, 3) = 3 - For
**3**: min(1, 2) = 1

- For

**Step 3: Calculate the GCF**

**GCF=2 ^{3}×3^{1}= 8×3=24**

**Therefore, the GCF of 192 and 72 is 24.**

### Method 2: Listing All Factors

This method involves listing all factors of each number and identifying the largest common one.

**Step 1: List the Factors**

**Factors of 192:**1,2,3,4,6,8,12,16,24,32,48,64,96,192**Factors of 72:**1,2,3,4,6,8,9,12,18,24,36,72

**Step 2: Identify Common Factors**

**Common factors**: 1, 2, 3, 4, 6, 8, 12, 24

**Step 3: Determine the GCF**

- The
**largest common factor**is**24**.

**Therefore, the GCF of 192 and 72 is 24.**

### Method 3: The Euclidean Algorithm

The Euclidean Algorithm is an efficient method for finding the GCF of two numbers, especially large ones. It uses a series of division steps based on the principle that the GCF of two numbers also divides their difference.

**Step 1: Apply the Euclidean Algorithm**

Given numbers: a=192, b=72

**Step 1:** 192**÷**72 = 2 with a remainder of 48 ⇒ 192=72×2+48

**Step 2:** 72**÷**48 = 1 with a remainder of 24 ⇒ 72 = 48×1+24

**Step 3:** 48**÷**24 = 2 with a remainder of 0 ⇒ 48 = 24×2+0

**Step 2: Identify the GCF**

- The algorithm stops when the remainder is
**0**. - The
**last non-zero remainder**is**24**, which is the GCF.

**Therefore, the GCF of 192 and 72 is 24.**

## Conclusion

Regardless of the method you choose, the **Greatest Common Factor of 192 and 72 is 24**. Here’s a quick recap of the three methods:

**Prime Factorization**: Break down each number into prime factors and multiply the smallest powers of common primes.**Listing Factors**: List all factors of each number and identify the largest one they share.**Euclidean Algorithm**: Use division steps to find the last non-zero remainder.

Each method has its own advantages. **Prime Factorization** is great for understanding the underlying structure of numbers, **Listing Factors** is straightforward but can be time-consuming for large numbers, and the **Euclidean Algorithm** is the most efficient for large or multiple numbers.

**Practice Problem:**

*Find the GCF of 84 and 30 using all three methods discussed above and tell us in the comment section*

## Additional Tips

**Prime Factorization**is particularly useful when dealing with multiple numbers, as it allows you to identify all common prime factors simultaneously.- The
**Euclidean Algorithm**is highly efficient for larger numbers and is widely used in computational applications. - Understanding all three methods enriches your number sense and provides flexibility in problem-solving.