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Prime factorization involves breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number that can only be divided by 1 and itself without resulting in a fraction. For instance, the prime numbers up to 10 are 2, 3, 5, and 7. When we perform prime factorization, we express the given number as a product of these prime factors.
For example, in the exercise, we found the prime factorization of 48 and 56.
48 can be written as \[48 = 2^4 \times 3\]This means when 48 is divided by 2 four times, we are left with 3, which is also a prime number.
Similarly, 56 can be written as \[56 = 2^3 \times 7\]This means 56 can be broken down by dividing by 2 three times, leaving us with 7.

Once we have the prime factors of the given numbers, the next step is to identify the common prime factors. This means looking for the same prime numbers in the factorization of each number. In our example:
For 48: \[48 = 2^4 \times 3\]
For 56: \[56 = 2^3 \times 7\]
We see that the common prime factor is 2 because it's a factor in both numbers. This part is crucial because these common factors help in determining the greatest common factor (GCF).
It's worth noting that if there were no common prime factors, the GCF would be 1, as 1 is the only universal factor for all numbers.

The final step in finding the greatest common factor (GCF) involves using the lowest power of the common prime factors identified. The 'power' of a prime factor is simply the exponent it has in the factorization.
In our example, we have identified 2 as the common prime factor: \[48 = 2^4 \times 3\]
\[56 = 2^3 \times 7\]
The powers of 2 are 4 in 48 and 3 in 56. We take the lowest of these two, which is 3. This gives us \[2^3 = 8\]
Therefore, the GCF is \[8\]
This process ensures that the GCF is the largest number that can divide both original numbers without any remainder.