How is the GCF found using prime factorization?

The Greatest Common Factor (GCF) is found using prime factorization by identifying the prime factors common to all numbers and then multiplying those common prime factors together.

Here's a breakdown of the process:

1. Find the Prime Factorization of Each Number: Break down each number into its prime factors. This is often done using a factor tree. A prime factor is a number that is only divisible by 1 and itself (e.g., 2, 3, 5, 7, 11, etc.).

2. Identify Common Prime Factors: Look for the prime factors that are common to all of the numbers you are considering.

3. Determine the Lowest Power of Each Common Prime Factor: If a prime factor appears more than once in the prime factorization of a number, identify the lowest power (or exponent) of that prime factor among all the numbers. For example, if one number has 2³ and another has 2², the lowest power is 2².

4. Multiply the Common Prime Factors: Multiply the common prime factors, each raised to its lowest power identified in step 3. The result is the GCF.

Example:

Let's find the GCF of 12 and 18.

1. Prime Factorization:

   • 12 = 2 x 2 x 3 = 2² x 3
   • 18 = 2 x 3 x 3 = 2 x 3²

2. Common Prime Factors:

   • Both 12 and 18 have the prime factors 2 and 3.

3. Lowest Power:

   • The lowest power of 2 is 2¹ (since 18 has 2¹ and 12 has 2²).
   • The lowest power of 3 is 3¹ (since 12 has 3¹ and 18 has 3²).

4. Multiply:

   • GCF = 2¹ x 3¹ = 2 x 3 = 6

Therefore, the GCF of 12 and 18 is 6.