Common Factor Calculator

A factor of an integer n is any whole number that divides n evenly with no remainder. The common factors of a set of numbers are the factors shared by every member of the set. The greatest common factor (GCF), also called the greatest common divisor (GCD), is the largest such shared factor. The least common multiple (LCM) is the smallest positive integer divisible by every number in the set. Together, GCF and LCM solve fraction simplification, ratio reduction, synchronized repeating events, and gear-mesh problems.

For 12 and 18, list factors: 12 has 1, 2, 3, 4, 6, 12; 18 has 1, 2, 3, 6, 9, 18. Common factors are 1, 2, 3, 6, so GCF = 6. LCM = 36 because 36 is the smallest number divisible by both 12 and 36. Check via prime factorization: 12 = 2² × 3 and 18 = 2 × 3². GCF takes the minimum exponent of each shared prime: 2¹ × 3¹ = 6. LCM takes the maximum exponent: 2² × 3² = 36. The identity GCF(a,b) × LCM(a,b) = a × b holds for two positive integers and is a fast sanity check.

With three or more numbers, extend the same prime-exponent logic. For 24, 36, and 48: 24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. Shared primes are 2 and 3. GCF uses minimum exponents 2² × 3 = 12. LCM uses maximum exponents 2⁴ × 3² = 144. Common factors of all three are 1, 2, 3, 4, 6, 12. The Euclidean algorithm efficiently computes GCF for large pairs without full factor lists — repeatedly replace the larger number with the remainder until zero remains.

GCF simplifies fractions: 24/36 divided by GCF 12 yields 2/3. LCM gives common denominators when adding unlike fractions — denominators 12 and 18 need LCM 36. In music, LCM counts when two rhythmic patterns realign. In algebra, factoring polynomials often starts with numeric GCF extraction. Link to the dedicated GCF calculator and LCM calculator for focused two-number workflows, the prime factorization calculator for building blocks, and the factor calculator to list all divisors of a single integer.

Enter two or more positive integers separated by commas. Negative inputs are treated by absolute value for factor purposes; zero is excluded because every nonzero integer divides zero, making GCF/LCM undefined in the usual sense. Results include the full common factor list, GCF, LCM, individual prime factorizations, and shared prime exponents. Use these outputs to reduce ratios with the ratio calculator, simplify fractions, or verify divisibility homework without manual factor trees.

When numbers are coprime (GCF = 1), their LCM equals their product. When one number divides another, the GCF is the smaller number and the LCM is the larger. Recognizing these shortcuts speeds mental checks before you rely on the full algorithmic output displayed below your inputs.