The Greatest Common Factor (GCF)
The Greatest Common Factor (also known as the Greatest Common Divisor) is the largest positive integer that perfectly divides two or more numbers without a remainder. It is heavily used to simplify fractions.
The Euclidean Algorithm
While prime factorization works, the ancient Greek Euclidean Algorithm is faster for large numbers. Divide the larger number by the smaller one. Then divide the previous divisor by the remainder. Repeat until the remainder is zero. The last non-zero remainder is the GCF.
Simplifying Fractions
To reduce a fraction to its lowest terms, simply divide both the numerator and denominator by their GCF. For instance, in the fraction 24/36, the GCF is 12. Dividing both by 12 yields 2/3.