Find GCD from Prime Factorizations: Reducing Fractions

[Holocene]

Is there any way to derive the greatest common divisor from the prime factorizations of the numerator and denominator?


For instance:

$$\displaystyle{\frac{48}{150} = \frac{ 2 * 2 * 2 * 2 * 3}{2 * 3 * 5 * 5}}$$

The GCD = 6 in this example, but is there any way to determine that from the prime factorizations alone?

 

[CRGreathouse]

Is there any way to derive the greatest common divisor from the prime factorizations of the numerator and denominator?

Yes, that's the easiest (if not fastest) way. Just choose pairs of identical prime factors until none are left that match, then multiply the primes together.