To find the factors of \(108\), we need to determine all the whole numbers that can divide \(108\) without leaving a remainder. Here's how you can systematically find them:

### Steps to Find Factors of 108

1. **Start from 1**: The number 1 is always a factor of any integer.

2. **Check divisibility**: Start dividing 108 by integers starting from 1 up to \(\sqrt{108}\) (approximately 10.39). If the result is an integer, both the divisor and the quotient are factors.

3. **List all factors**: Record both the divisor and the quotient if the division results in a whole number.

Let's do this step by step:

#### Factors Calculation

1. **1**:
   \[
   108 \div 1 = 108 \quad \text{(Both 1 and 108 are factors)}
   \]

2. **2**:
   \[
   108 \div 2 = 54 \quad \text{(Both 2 and 54 are factors)}
   \]

3. **3**:
   \[
   108 \div 3 = 36 \quad \text{(Both 3 and 36 are factors)}
   \]

4. **4**:
   \[
   108 \div 4 = 27 \quad \text{(Both 4 and 27 are factors)}
   \]

5. **5**: 
   \[
   108 \div 5 = 21.6 \quad \text{(5 is not a factor)}
   \]

6. **6**:
   \[
   108 \div 6 = 18 \quad \text{(Both 6 and 18 are factors)}
   \]

7. **7**: 
   \[
   108 \div 7 \approx 15.43 \quad \text{(7 is not a factor)}
   \]

8. **8**: 
   \[
   108 \div 8 = 13.5 \quad \text{(8 is not a factor)}
   \]

9. **9**:
   \[
   108 \div 9 = 12 \quad \text{(Both 9 and 12 are factors)}
   \]

10. **10**: 
   \[
   108 \div 10 = 10.8 \quad \text{(10 is not a factor)}
   \]

### List of Factors

By identifying the whole number results from the division, we find that the factors of 108 are:

\[ 
1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 
\]

Thus, \(108\) has a total of 12 factors.

### Prime Factorization

For further verification, we can use prime factorization:

1. **Divide by smallest prime (2):**
   \[
   108 \div 2 = 54 \quad (2)
   \]
   \[
   54 \div 2 = 27 \quad (2)
   \]

2. **Divide by next smallest prime (3):**
   \[
   27 \div 3 = 9 \quad (3)
   \]
   \[
   9 \div 3 = 3 \quad (3)
   \]
   \[
   3 \div 3 = 1 \quad (3)
   \]

The prime factorization of 108 is \(2^2 \times 3^3\), confirming our list of factors.

### Verification with Prime Factorization

From the prime factorization \(2^2 \times 3^3\), we can generate all factors by combining the prime factors in different ways:

- \(2^0 \times 3^0 = 1\)
- \(2^1 \times 3^0 = 2\)
- \(2^2 \times 3^0 = 4\)
- \(2^0 \times 3^1 = 3\)
- \(2^1 \times 3^1 = 6\)
- \(2^2 \times 3^1 = 12\)
- \(2^0 \times 3^2 = 9\)
- \(2^1 \times 3^2 = 18\)
- \(2^2 \times 3^2 = 36\)
- \(2^0 \times 3^3 = 27\)
- \(2^1 \times 3^3 = 54\)
- \(2^2 \times 3^3 = 108\)

This confirms that the factors are indeed \[1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\].

### Summary

- **Factors of 108**: \[1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108\]

If you have any questions or need further assistance, please let me know. Here are some related questions you might be interested in:

1. What are the factors of 144?
2. How do you determine if a number is prime?
3. What are the factors of 72?
4. How do you find the least common multiple of two numbers?
5. What is the greatest common divisor of 108 and 144?
6. How does prime factorization help in simplifying fractions?
7. Can you explain the difference between factors and multiples?
8. What is the prime factorization of 360?

**Tip:** When finding factors, remember to check divisibility using small primes first. This can simplify the process and help in finding factors quickly.

Learn how to find all factors of 108 and its prime factorization with a step-by-step guide. This problem covers key number theory concepts, including factorization and prime factorization. Suitable for students in Grades 5-8.

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