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Prime factorization is the process of breaking down a composite number into its prime factors. Prime factors are the prime numbers that multiply together to get the original number. For instance, in the exercise, the first step is to find the prime factors for 48 and 96.

For 48:
We start by dividing by the smallest prime number, 2, repeatedly until we get to the prime factor 3:
48 = 2 × 24
24 = 2 × 12
12 = 2 × 6
6 = 2 × 3
So, 48 can be expressed as 48 = 2^4 × 3.

For 96:
We again start by dividing by 2 until we get to the prime factor 3:
96 = 2 × 48
48 = 2 × 24
24 = 2 × 12
12 = 2 × 6
6 = 2 × 3
So, 96 can be expressed as 96 = 2^5 × 3.

Now we have completed the prime factorization for both numbers!

When finding the Least Common Multiple (LCM) using prime factorization, the next step is to identify the highest powers of all prime factors from the factorizations. This ensures we include each prime factor the necessary number of times to cover both numbers.

In our example, the prime factors of 48 and 96 are 2 and 3. We look at the exponents of these prime factors.

For factor 2:

• From 48, we have 2^4.
• From 96, we have 2^5.

The highest power of 2 between these two is 2^5.

For factor 3:

• Both 48 and 96 contribute the factor 3 to the power of 1.

So, 3^1 is the highest power for the prime factor 3.

By selecting the highest power for each prime factor, we ensure that the LCM will be able to divide both original numbers without a remainder.

Once we have the highest powers of all involved prime factors, the final step is to multiply them together to find the LCM.

From our identified highest powers:

• The highest power of 2 is 2^5.
• The highest power of 3 is 3.

Now, we multiply these together to get the LCM:

\[ \text{LCM} = 2^5 \times 3 \ = 32 \times 3 \ = 96 \ \text{Thus, the LCM of 48 and 96 is } 96.\]

This means 96 is the smallest number that is a multiple of both 48 and 96. Each step—prime factorization, identifying the highest powers, and multiplying them—ensures that the LCM calculation is both accurate and easy to understand.