Find the HCF of 96 and 404 by the Prime Factorization Method.

To find the Highest Common Factor (HCF) of 96 and 404 using the Prime Factorization Method, we first need to break down each number into its prime factors. This method is effective because it allows us to see the common prime factors clearly, making it easier to determine the HCF.

Step 1: Prime Factorization of 96

Let's start with 96. We can divide it by the smallest prime number, which is 2, and continue dividing until we reach 1.

• 96 ÷ 2 = 48
• 48 ÷ 2 = 24
• 24 ÷ 2 = 12
• 12 ÷ 2 = 6
• 6 ÷ 2 = 3
• 3 ÷ 3 = 1

Now, we can write the prime factorization of 96 as:

96 = 2^5 × 3^1

Step 2: Prime Factorization of 404

Next, we will factor 404 in a similar manner.

• 404 ÷ 2 = 202
• 202 ÷ 2 = 101
• 101 ÷ 101 = 1

The prime factorization of 404 is:

404 = 2^2 × 101^1

Step 3: Identifying Common Factors

Now that we have the prime factorizations of both numbers, we can identify the common prime factors:

• For 96: 2^5 and 3^1
• For 404: 2^2 and 101^1

The only common prime factor is 2. To find the HCF, we take the lowest power of the common prime factor:

HCF = 2^2

Step 4: Calculating the HCF

Now we calculate the value:

HCF = 2^2 = 4

Final Result

Thus, the Highest Common Factor of 96 and 404 is 4.

This method not only helps in finding the HCF but also reinforces the understanding of prime factorization, which is a fundamental concept in number theory. If you have any further questions or need clarification on any steps, feel free to ask!