Identify the greatest common factor of $$ 20 z $$ and $$ 40 a^{3} z^{3} $$

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UpStudy AI · Accepted Answer

To find the greatest common factor (GCF) of the expressions $$ 20z $$ and $$ 40a^{3}z^{3} $$, we will follow these steps:

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

### Step 1: Factor each expression

- For $$ 20z $$:
  - The prime factorization of $$ 20 $$ is $$ 2^2 \times 5 $$.
  - Therefore, $$ 20z = 2^2 \times 5 \times z $$.

- For $$ 40a^{3}z^{3} $$:
  - The prime factorization of $$ 40 $$ is $$ 2^3 \times 5 $$.
  - Therefore, $$ 40a^{3}z^{3} = 2^3 \times 5 \times a^{3} \times z^{3} $$.

### Step 2: Identify the common factors

Now we will identify the common factors in both expressions:

- The common prime factors are $$ 2 $$ and $$ 5 $$.
- For $$ z $$, the minimum power in both expressions is $$ z^1 $$ (from $$ 20z $$).

### Step 3: Multiply the common factors

Now we can find the GCF by multiplying the common factors:

- For $$ 2 $$: The minimum power is $$ 2^2 $$ (from $$ 20z $$).
- For $$ 5 $$: The minimum power is $$ 5^1 $$.
- For $$ z $$: The minimum power is $$ z^1 $$.

Thus, the GCF is:

$$
GCF = 2^2 \times 5^1 \times z^1 = 4 \times 5 \times z = 20z
$$

### Final Answer

The greatest common factor of $$ 20z $$ and $$ 40a^{3}z^{3} $$ is $$ 20z $$.
The greatest common factor is $$ 20z $$.

Identify the greatest common factor of \( 20 z \) and \( 40 a^{3} z^{3} \)

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