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To find the greatest common factor (GCF) of 27 and 36, we can break down each number into its prime factors.

1. **Prime factorization of 27:**
   - 27 = 3 × 3 × 3 = \( 3^3 \)

2. **Prime factorization of 36:**
   - 36 = 2 × 2 × 3 × 3 = \( 2^2 \times 3^2 \)

To find the GCF, we take the lowest power of the common prime factors. The common prime factor between 27 and 36 is 3, and the lowest power of 3 in both factorizations is \( 3^2 \).

Thus, the GCF of 27 and 36 is:

\[
\text{GCF} = 3^1 = 3
\]

### The Greatest Common Factor (GCF) of 27 and 36 is 3.

Would you like a further breakdown of this process? Also, here are some related questions for further exploration:

1. What is the least common multiple (LCM) of 27 and 36?
2. How do you find the GCF of three numbers instead of two?
3. Can the GCF of two numbers ever be larger than either of the numbers?
4. How does the Euclidean algorithm help in finding the GCF?
5. What is the relationship between the GCF and LCM of two numbers?

**Tip:** When finding the GCF, always look for the prime factors that both numbers share, and pick the smallest power of those shared factors.

Find the greatest common factor (GCF) of 27 and 36.

Learn how to find the greatest common factor (GCF) of 27 and 36 using prime factorization. This solution explains the process step-by-step, including the prime factorization of both numbers and the method to determine the GCF. Ideal for students in Grades 6-8 studying number theory.

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