In the realm of mathematics, the concept of greatest common factor (GCF) plays a crucial role in simplifying expressions and understanding the relationship between numbers. The GCF of two numbers represents the largest factor that divides both numbers without leaving any remainder. Determining the GCF is essential for solving various mathematical problems, including simplifying fractions, finding the least common denominator, and working with numbers in general.

In this article, we delve into the fascinating world of GCF by exploring the greatest common factor of 72 and 45. Through a step-by-step approach, we will uncover the mathematical techniques involved in finding the GCF of these two numbers, shedding light on the underlying principles that govern this fundamental concept.

As we embark on this mathematical journey, we encourage you to engage actively with the content, follow the examples, and test your understanding by solving the practice problems provided. By the end of this article, you will have gained a solid grasp of the GCF of 72 and 45, equipping you with the knowledge and skills to tackle a wide range of mathematical challenges with confidence.

## Prime Factorization: Decomposing Numbers into Building Blocks

### Understanding Prime Factorization

Prime factorization is a fundamental technique in number theory that involves expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that are divisible only by 1 and themselves. By breaking down numbers into their prime components, we gain insights into their structure and relationships.

### Prime Factorization of 72 and 45

To find the GCF of 72 and 45, we begin by prime factorizing both numbers:

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- 72 = 2
^{3}x 3^{2} - 45 = 3
^{2}x 5

## Identifying the GCF: Common Factors and the Largest Exponent

### Common Factors

The common factors of 72 and 45 are the factors that divide both numbers without leaving a remainder. By comparing the prime factorizations of 72 and 45, we can identify their common factors:

- 2
- 3
- 3

### Largest Exponent

Once we have identified the common factors, the GCF is determined by taking the largest exponent for each common factor. In this case, both 2 and 3 appear in the prime factorizations, but 3 has a larger exponent in both numbers (2 for 72 and 2 for 45). Therefore, the largest exponent for 3 is 2.

## The GCF of 72 and 45

### Final Answer

Based on the principles of prime factorization and common factors, we can conclude that the GCF of 72 and 45 is:

**GCF(72, 45) = 3 ^{2} = 9**

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