To find the highest common factor (HCF) of 324 and 128, we can use the prime factorization method:

1. Prime Factorization of 324:

   • 324 is an even number, so divide by 2: $324 \div 2 = 162$
   • 162 is also even, so divide by 2: $162 \div 2 = 81$
   • 81 is not even, so divide by 3: $81 \div 3 = 27$
   • 27 can be divided by 3: $27 \div 3 = 9$
   • 9 can be divided by 3: $9 \div 3 = 3$
   • 3 is a prime number.

   So, the prime factorization of 324 is: $324 = 2^2 \times 3^4$

2. Prime Factorization of 128:

   • 128 is an even number, so divide by 2: $128 \div 2 = 64$
   • 64 is even, so divide by 2: $64 \div 2 = 32$
   • 32 is even, so divide by 2: $32 \div 2 = 16$
   • 16 is even, so divide by 2: $16 \div 2 = 8$
   • 8 is even, so divide by 2: $8 \div 2 = 4$
   • 4 is even, so divide by 2: $4 \div 2 = 2$
   • 2 is a prime number.

   So, the prime factorization of 128 is: $128 = 2^7$

3. Finding the HCF:

   • The only common prime factor between 324 and 128 is 2.
   • The lowest power of 2 is $2^2$.

Therefore, the HCF of 324 and 128 is: $\text{HCF} = 2^2 = 4$

Would you like further clarification, or do you have any other questions?

Here are some related questions you might consider:

1. What is the relationship between the HCF and the LCM of two numbers?
2. Can two numbers be coprime if they share any common prime factors?
3. How does the HCF help in simplifying fractions?
4. What is the prime factorization method, and why is it useful?
5. Can the HCF of two numbers be larger than their smallest number?
6. How do you find the HCF of three or more numbers?
7. What is the significance of the HCF in real-world problem-solving?
8. How can you quickly find the HCF without using prime factorization?

Tip: When two numbers are relatively prime (i.e., their HCF is 1), their LCM is simply the product of the two numbers.