How do you find the lowest common multiple of prime factors?

To find the Lowest Common Multiple (LCM) using prime factorization, follow these steps:

1. Find the Prime Factorization: Determine the prime factorization of each number you want to find the LCM for. This means breaking down each number into a product of its prime factors.

2. List Prime Factors with Highest Powers: For each prime factor that appears in any of the factorizations, list it with the highest power to which it appears in any of the individual prime factorizations.

3. Multiply: Multiply all the prime factors (with their highest powers) listed in the previous step. The result is the LCM.

Here's an example to illustrate the process:

Example: Find the LCM 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. List Prime Factors with Highest Powers:

   • The prime factors are 2 and 3.
   • The highest power of 2 is 2².
   • The highest power of 3 is 3².

3. Multiply:

   • LCM = 2² x 3² = 4 x 9 = 36

Therefore, the LCM of 12 and 18 is 36.

In essence: Prime factorization allows you to identify all the necessary prime components to construct a multiple common to all the original numbers, ensuring you choose the lowest such multiple by considering only the highest required power of each prime.