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Identify the greatest common factor of 50, a, y, z50ayz and 20, a, w, to the power 4 , z, to the power 420aw
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Answer
Attempt 3 out of 5

Answer 1

To find the greatest common factor (GCF) of the expressions 50ayz and 20aw⁴z⁴, we'll start by breaking down both expressions into their factors.

1. **Factorize the numerical coefficients:**
- The prime factorization of 50 is \(2 \times 5^2\).
- The prime factorization of 20 is \(2^2 \times 5\).

2. **Factoring both expressions:**
- For \(50ayz\):
- The factors are \(2 \times 5^2 \times a \times y \times z\).

- For \(20aw⁴z⁴\):
- The factors are \(2^2 \times 5 \times a \times w^4 \times z^4\).

3. **Identify the common factors:**
- **Numerical Coefficients:**
- From \(50\) and \(20\):
- The common numerical factors are \(2^1\) (the minimum power of 2) and \(5^1\) (the minimum power of 5).
- Thus, the GCF for the coefficients is \(2^1 \times 5^1 = 10\).

- **Variable Factors:**
- **a:** The minimum power of \(a\) is \(a^1\) (present in both).
- **y:** This variable only appears in \(50ayz\) (so it contributes nothing to GCF).
- **w:** This variable only appears in \(20aw⁴z⁴\) (so it contributes nothing to GCF).
- **z:** The minimum power of \(z\) is \(z^0\) (because \(z\) is not in the second expression, it doesn’t contribute).

4. **Combine the common factors:**
The GCF from both parts is:
- For the numerical part: \(10\)
- For the variable part: \(a^1\)

Thus, the GCF of \(50ayz\) and \(20aw⁴z⁴\) is:
\[
\text{GCF} = 10a
\]

So, the greatest common factor of \(50ayz\) and \(20aw⁴z⁴\) is \(10a\).