Find the GCF (greatest common factor) of the following terms. $$ \{27 x y, 81,54 x y\} $$

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To find the greatest common factor (GCF) of the terms $$27xy$$, $$81$$, and $$54xy$$, we will follow these steps:

1. **Factor each term into its prime factors.**
2. **Identify the common factors.**
3. **Multiply the common factors to find the GCF.**

### Step 1: Factor each term

- **For $$27xy$$**:
  $$
  27 = 3^3 \quad \text{so,} \quad 27xy = 3^3 \cdot x \cdot y
  $$

- **For $$81$$**:
  $$
  81 = 3^4
  $$

- **For $$54xy$$**:
  $$
  54 = 2 \cdot 3^3 \quad \text{so,} \quad 54xy = 2 \cdot 3^3 \cdot x \cdot y
  $$

### Step 2: Identify the common factors

Now we have:
- $$27xy = 3^3 \cdot x \cdot y$$
- $$81 = 3^4$$
- $$54xy = 2 \cdot 3^3 \cdot x \cdot y$$

The common factors among these terms are:
- The factor $$3$$ appears in all terms.
- The lowest power of $$3$$ is $$3^3$$.
- The variables $$x$$ and $$y$$ are only present in $$27xy$$ and $$54xy$$, so they are not common to all three terms.

### Step 3: Multiply the common factors

The GCF is:
$$
GCF = 3^3 = 27
$$

Thus, the greatest common factor of the terms $$27xy$$, $$81$$, and $$54xy$$ is $$\boxed{27}$$.
The greatest common factor (GCF) of $$27xy$$, $$81$$, and $$54xy$$ is **27**.

Find the GCF (greatest common factor) of the following terms. \( \{27 x y, 81,54 x y\} \)

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