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The Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), is the largest integer that divides two or more numbers exactly. Used in simplifying fractions, finding common denominators, and solving Diophantine equations.

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1. 1Prime factorization method: factor each number, take common prime factors with lowest powers
2. 2Euclidean algorithm: GCF(a,b) = GCF(b, a mod b), repeat until remainder is 0
3. 3GCF(a,b) × LCM(a,b) = a × b
4. 4GCF of two primes is always 1 (they share no common factors)

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