Find GCD from Prime Factorizations: Reducing Fractions

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The greatest common divisor (GCD) can be derived from the prime factorizations of the numerator and denominator by identifying and multiplying the lowest powers of common prime factors. In the example of 48 and 150, the prime factorizations reveal shared factors of 2 and 3. By selecting pairs of identical prime factors, the GCD can be calculated efficiently. This method is straightforward and effective for determining the GCD from prime factorizations. Understanding this process simplifies reducing fractions.
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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?
 
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Holocene said:
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.
 
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