How to Find the HCF of a Number Using the Prime Factorization Method?

The prime factorization method to find the Highest Common Factor (HCF) involves breaking down each number into its prime factors and then identifying the common prime factors raised to the lowest power they appear in either number. The HCF is the product of these common prime factors.

Here's a step-by-step guide:

1. Find the Prime Factorization of Each Number: Express each number as a product of its prime factors. You can use a factor tree or division method for this.

2. Identify Common Prime Factors: Compare the prime factorizations of all the numbers. List the prime factors that are common to all the numbers.

3. Determine the Lowest Power of Each Common Prime Factor: For each common prime factor, identify the lowest power (exponent) to which it appears in any of the prime factorizations.

4. Calculate the HCF: Multiply the common prime factors, each raised to its lowest power. The result is the HCF of the numbers.

Example:

Let's find the HCF of 36 and 48 using the prime factorization method.

1. Prime Factorization:

   • 36 = 2 x 2 x 3 x 3 = 2² x 3²
   • 48 = 2 x 2 x 2 x 2 x 3 = 2⁴ x 3¹

2. Common Prime Factors: The common prime factors are 2 and 3.

3. Lowest Power:

   • The lowest power of 2 is 2² (since 2 appears as 2² in 36 and 2⁴ in 48).
   • The lowest power of 3 is 3¹ (since 3 appears as 3² in 36 and 3¹ in 48).

4. HCF Calculation: HCF (36, 48) = 2² x 3¹ = 4 x 3 = 12

Therefore, the HCF of 36 and 48 is 12.

In summary: Decompose the numbers into their prime factors, identify the common prime factors, find the lowest power of each common factor, and multiply those lowest powers together to get the HCF.