How can prime factorization be used to find a greatest common divisor?

To find the greatest common divisor (GCD) of a set of positive integers using prime factorization, follow these steps:

1. Factorize each number: Express each integer as a product of its prime factors, indicating the number of times each distinct prime occurs as an exponent (e.g., $3,000=2^3\times 3^1\times 5^3$).
2. Identify common primes: Determine which prime numbers appear in the factorization of every integer in the set.
3. Select the smallest power: For each of these common primes, identify the smallest exponent (the fewest number of times it occurs) among all the factorizations.
4. Calculate the product: Multiply these common primes together, each raised to its respective smallest exponent identified in the previous step. The resulting product is the GCD.1

For example, to find the GCD of $3,000$ ($2^3\times 3^1\times 5^3$) and $2,646$ ($2^1\times 3^3\times 7^2$), you take the common primes $2$ and $3$ at their lowest powers ($2^1$ and $3^1$), resulting in a GCD of $2\times 3=6$.1