To find the greatest common factor (GCF) of 180, 225, and 270, we need to determine the largest number that can divide all three numbers evenly.

Prime Factorization:
First, let's determine the prime factorization of each number.

1. Prime factorization of 180:
180 can be factored as 2 × 2 × 3 × 3 × 5, which can be written as 2² × 3² × 5.

2. Prime factorization of 225:
225 can be factored as 3 × 3 × 5 × 5, which can be written as 3² × 5².

3. Prime factorization of 270:
270 can be factored as 2 × 3 × 3 × 3 × 5, which can be written as 2 × 3³ × 5.

Identifying Common Factors:
Now, let's identify the common factors among the prime factorizations.

- The common factors of 180 and 225 are 3² × 5, which is equal to 45.
- The common factors of 45 and 270 are 3² × 5, which is equal to 45.

Finding the Greatest Common Factor:
The GCF is the largest common factor among the given numbers. In this case, the GCF of 180, 225, and 270 is 45 (option C).

Explanation:
The GCF is found by identifying the common factors among the prime factorizations of the given numbers. In this case, both 180 and 225 have a prime factorization of 3² × 5, while 270 has a prime factorization of 2 × 3³ × 5. The common factor among all three numbers is 3² × 5, which is equal to 45. Therefore, the GCF is 45.