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Prime factorization is the process of breaking down a number into its prime factors. Prime factors are the building blocks of all numbers, as they are numbers greater than 1 that can only be divided by 1 and themselves. For example, the prime factors of 8 are determined by repeatedly dividing by the smallest prime, which is 2. We can write:
\( 8 = 2 \times 2 \times 2 = 2^3 \).

Similarly, to find the prime factors of 48, we divide by the smallest prime until we can’t divide further:
- \( 48 = 2 \times 24 \)
- \( 24 = 2 \times 12 \)
- \( 12 = 2 \times 6 \)
- \( 6 = 2 \times 3 \)
As such, we write \( 48 = 2^4 \times 3^1 \).
This method helps in determining which prime numbers combine to create the original number.

When finding the Least Common Multiple (LCM), identifying the highest power of each prime factor is crucial. This means looking at the prime factorizations of each number involved and choosing the highest exponent for each prime number.
For the example of 8 and 48:
- For the prime factor 2, the highest power is \( 2^4 \) from 48.
- For the prime factor 3, the highest power is \( 3^1 \) from 48.
By selecting the highest power of each prime factor, we ensure that all multiples of the original numbers are covered in the calculated LCM.

The final step in finding the LCM involves multiplying the highest powers of all the prime factors together. This ensures that the LCM is a multiple of all the original numbers. Continuing with our example:

• 2^4 = 16
• 3^1 = 3

By multiplying these together, we get the LCM: \( 2^4 \times 3 = 16 \times 3 = 48 \).
This final multiplication step consolidates the prime factors to give us the smallest number that is divisible by both 8 and 48.