To find the Highest Common Factor (HCF) of 12 and 66, we need to determine the greatest number that divides both 12 and 66 without leaving a remainder. This can be done using the prime factorization method or the Euclidean algorithm. Here, we'll use the prime factorization method.

1. Prime factorization of 12:

   • 12 can be factored into $2^2 \times 3$.

2. Prime factorization of 66:

   • 66 can be factored into $2 \times 3 \times 11$.

3. Identify the common prime factors:

   • The common prime factors of 12 and 66 are 2 and 3.

4. Find the highest power of these common factors that divides both numbers:

   • The highest power of 2 that divides both numbers is $2^1$.
   • The highest power of 3 that divides both numbers is $3^1$.

5. Multiply these common prime factors together:

   • HCF = $2^1 \times 3^1 = 2 \times 3 = 6$.

Therefore, the HCF of 12 and 66 is 6.

Would you like any further details or have any questions?

Here are 5 more questions you might want to ask:

1. What is the LCM of 12 and 66?
2. How do you find the HCF of more than two numbers?
3. What is the prime factorization of 144?
4. Can you explain the Euclidean algorithm for finding the HCF?
5. How do you find the HCF using the division method?

Tip: When finding the HCF of two numbers, the Euclidean algorithm is often the quickest method, especially for larger numbers.