Step 1: Understanding the Question:
The question requires us to find the Highest Common Factor (HCF) of three numbers: 36, 54, and 90. We must use the specific method of prime factorization.
Step 2: Key Formula or Approach:
The prime factorization method for finding HCF involves these steps:
1. Express each number as a product of its prime factors.
2. Identify the prime factors that are common to all the numbers.
3. The HCF is the product of these common prime factors, with each raised to the lowest power it appears in any of the factorizations.
Step 3: Detailed Explanation:
First, we find the prime factorization for each of the three numbers.
For 36:
\[
36 = 4 \times 9 = 2^2 \times 3^2
\]
For 54:
\[
54 = 2 \times 27 = 2^1 \times 3^3
\]
For 90:
\[
90 = 9 \times 10 = 3^2 \times (2 \times 5) = 2^1 \times 3^2 \times 5^1
\]
Now, we list the factorizations:
\[
36 = 2^2 \times 3^2
\]
\[
54 = 2^1 \times 3^3
\]
\[
90 = 2^1 \times 3^2 \times 5^1
\]
Next, we identify the common prime factors. The factors that appear in all three lists are 2 and 3.
Then, we find the lowest power for each common factor:
For the prime factor 2, the powers are 2, 1, and 1. The lowest power is 1. So we take \(2^1\).
For the prime factor 3, the powers are 2, 3, and 2. The lowest power is 2. So we take \(3^2\).
The HCF is the product of these terms:
\[
\text{HCF}(36, 54, 90) = 2^1 \times 3^2 = 2 \times 9 = 18
\]
Step 4: Final Answer:
The HCF of 36, 54, and 90 is 18.
\[
\boxed{18}
\]