To generate the prime factorization for the number 36, we follow these steps:

1. Divide the number by the smallest prime number (2) and continue dividing by 2 until it is no longer divisible by 2.
2. Move to the next smallest prime number (3) and continue dividing by 3 until it is no longer divisible by 3.
3. Repeat this process with the next smallest prime numbers until the quotient is 1.

Let's begin the process for 36:

1. $36 \div 2 = 18$ (36 is divisible by 2, so divide by 2)
2. $18 \div 2 = 9$ (18 is divisible by 2, so divide by 2 again)
3. $9 \div 3 = 3$ (9 is divisible by 3, so divide by 3)
4. $3 \div 3 = 1$ (3 is divisible by 3, so divide by 3 again)

Now we have completely factorized 36 into its prime factors.

Prime Factorization of 36:

$36 = 2^2 \times 3^2$

So, the prime factors of 36 are $2$ and $3$, with the respective powers $2^2$ and $3^2$.

Would you like more details on this process or have any other questions?

Here are some questions you might ask next:

1. How do you find the prime factorization of larger numbers?
2. Can you explain the significance of prime numbers in number theory?
3. How is prime factorization used in finding the LCM and GCD?
4. What are some applications of prime factorization in cryptography?
5. How do you determine if a number is prime?

Tip: When performing prime factorization, always start with the smallest prime number and work your way up to ensure a thorough factorization process.