Step 2: Determine the common factors. $$ \begin{array}{l}27=3 \cdot 3 \cdot 3 \ 45=3 \cdot 3 \cdot 5\end{array} $$

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Great job on **Step 2: Determining the Common Factors**! You've correctly factored both numbers into their prime factors:

$$
\begin{array}{l}
27 = 3 \times 3 \times 3 \
45 = 3 \times 3 \times 5
\end{array}
$$

### Identifying the Common Factors

To find the **Greatest Common Factor (GCF)** of 27 and 45, follow these steps:

1. **List the Prime Factors:**
   - **27:** $$3 \times 3 \times 3$$ (which can be written as $$3^3$$)
   - **45:** $$3 \times 3 \times 5$$ (which can be written as $$3^2 \times 5$$)

2. **Identify the Common Prime Factors:**
   - Both numbers have the prime factor **3** in common.
   - The **lowest power** of the common prime factor is $$3^2$$.

3. **Calculate the GCF:**
   - Multiply the common prime factors using the lowest powers found.
   $$
   \text{GCF} = 3^2 = 9
   $$

### Summary

- **Common Prime Factors:** $$3 \times 3$$
- **Greatest Common Factor (GCF):** **9**

### Moving Forward

If you're working on finding the **Least Common Multiple (LCM)** or solving another part of a problem related to these numbers, here’s how you can proceed:

#### Finding the Least Common Multiple (LCM)

1. **List All Prime Factors:**
   - **27:** $$3^3$$
   - **45:** $$3^2 \times 5$$

2. **Take the Highest Power of Each Prime Factor:**
   - $$3^3$$ (from 27)
   - $$5$$ (from 45)

3. **Multiply Them Together:**
   $$
   \text{LCM} = 3^3 \times 5 = 27 \times 5 = 135
   $$

- **Least Common Multiple (LCM):** **135**

#### Application Example

- **Simplifying Fractions:**
  - To add or subtract fractions with denominators 27 and 45, you can use the LCM (135) as the common denominator.

- **Solving Diophantine Equations:**
  - Understanding GCF and LCM can help in solving equations where you need integer solutions.

Feel free to let me know if you need further assistance with the next steps or if you have any other questions!
The common factors of 27 and 45 are 3 and 3, so the Greatest Common Factor (GCF) is 9.

Step 2: Determine the common factors.

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