The greatest common factor (GCF) — also called the greatest common divisor (GCD) — of two or more numbers is the largest number that divides evenly into all of them. The GCF of 12 and 18 is 6: it's the biggest number that divides both without a remainder, even though both numbers have other, smaller shared factors too (1, 2, 3, and 6 all divide both 12 and 18, but 6 is the greatest of that shared set).
The most reliable method for finding the GCF is prime factorization: break each number down into its prime factors, then multiply together the primes that appear in both factorizations, using the lower of the two exponents whenever a prime appears in both. For 12 = 2² × 3 and 18 = 2 × 3², the shared primes are 2 and 3; take the lower power of each (2¹ from 12's 2² vs. 18's 2¹, and 3¹ from 12's 3¹ vs. 18's 3²), giving 2¹ × 3¹ = 6.
Worked example one: GCF of 24 and 36. 24 = 2³ × 3, 36 = 2² × 3². Shared primes at the lower power: 2² × 3¹ = 12. Worked example two: GCF of 45 and 60. 45 = 3² × 5, 60 = 2² × 3 × 5. Shared primes: 3¹ × 5¹ = 15.
A faster method for smaller numbers, when the full prime factorization feels like overkill, is the Euclidean algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller number, until the remainder is 0 — the last nonzero remainder is the GCF. For 48 and 18: 48 ÷ 18 leaves remainder 12; 18 ÷ 12 leaves remainder 6; 12 ÷ 6 leaves remainder 0, so the GCF is 6. This method is faster for large numbers precisely because it avoids factoring them at all.
A common mistake is confusing GCF with LCM (least common multiple) — the two are related but opposite in a specific sense: GCF is the largest shared *divisor*, always less than or equal to the smaller of the two numbers, while LCM is the smallest shared *multiple*, always greater than or equal to the larger number. Mixing them up (using the LCM formula when a problem asks for GCF, or vice versa) is one of the most common errors in this topic area.
GCF extends naturally to three or more numbers by finding it pairwise and combining: the GCF of 12, 18, and 30 can be found by first taking GCF(12,18) = 6, then GCF(6,30) = 6 — the order of pairing doesn't affect the final answer. GCF also shows up outside pure arithmetic in genuinely practical splitting problems: if you have 36 apples and 24 oranges and want to make identical gift bags using all the fruit with nothing left over, the largest number of bags you can make is GCF(36,24) = 12, each containing 3 apples and 2 oranges.
GCF has a direct practical use in simplifying fractions — dividing a fraction's numerator and denominator by their GCF is exactly how you reduce it to lowest terms, which is what the Simplify Fractions Calculator does automatically. For two numbers' GCF and LCM together (they're related by the identity GCF × LCM = the product of the two numbers), see the LCM Calculator, Two numbers with a GCF of exactly 1 — meaning they share no prime factors at all — are called coprime (or relatively prime), even if neither number is itself prime: 8 and 15 are coprime (8 = 2³, 15 = 3×5, no shared primes), despite neither being a prime number individually. Coprimality shows up again in simplified-fraction form specifically: a fraction is fully reduced exactly when its numerator and denominator are coprime, tying this concept directly back to the Simplify Fractions Calculator above.
For the method taught with more worked examples, see What Is GCF, and How Do You Find It?