To find the Highest Common Factor (HCF) of 27, 63, and 207, we can follow these steps:

Step 1: Prime Factorization

1. 27:
   $27 = 3 \times 3 \times 3 = 3^3$

2. 63:
   $63 = 3 \times 3 \times 7 = 3^2 \times 7$

3. 207:
   $207 = 3 \times 3 \times 23 = 3^2 \times 23$

Step 2: Identify Common Factors

• The only common prime factor among 27, 63, and 207 is $3$.

Step 3: Find the Lowest Power of the Common Factor

• The lowest power of $3$ common to all three numbers is $3^2$.

Step 4: Calculate the HCF

• HCF = $3^2 = 9$

Thus, the HCF of 27, 63, and 207 is 9.

Would you like more details, or do you have any questions?

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Here are some related questions you might be interested in:

1. How to find the HCF of multiple numbers using the Euclidean algorithm?
2. What is the difference between HCF and LCM?
3. How do prime factorization and division methods for finding HCF compare?
4. Can the HCF of two numbers ever be greater than either number?
5. How to calculate the HCF of large numbers efficiently?
6. How is HCF used in simplifying fractions?
7. What are some real-life applications of HCF?
8. How does the HCF relate to the concept of divisibility?

Tip: When working with large numbers, the Euclidean algorithm can be a faster way to find the HCF compared to prime factorization.