GCF: the biggest shared divisor
The greatest common factor (also called HCF or GCD) of a set of numbers is the largest whole number that divides every one of them. This calculator finds it with the Euclidean algorithm, applied pairwise across your list: gcd(a, b) = gcd(b, a mod b) until the remainder hits zero.
GCF(a, b, c) = GCF(GCF(a, b), c) · LCM(a, b) = a ÷ GCF(a, b) × b
Worked example: 12, 18, 30
Prime factorizations: 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5. The GCF takes each prime to the lowest power appearing in all three: 2 × 3 = 6. The LCM takes each prime to the highest power anywhere: 2² × 3² × 5 = 180. Pairwise, the Euclidean route gives gcd(12, 18) = 6, then gcd(6, 30) = 6; and lcm(12, 18) = 12 ÷ 6 × 18 = 36, then lcm(36, 30) = 180.
Why compute LCM via the GCF?
Multiplying first (a × b ÷ GCF) can overflow for large inputs, so the calculator divides first: a ÷ GCF × b. Inputs are capped at 10¹², and if a set's LCM grows beyond what integers can represent exactly (about 9 × 10¹⁵), the calculator says so rather than showing a silently rounded value.
Where you will actually use these
The GCF is the number you divide by to reduce a fraction or ratio to lowest terms — it is why 12/18 simplifies to 2/3. The LCM is the common denominator when adding fractions (1/12 + 1/18 = 3/36 + 2/36), the answer to scheduling questions ("gym every 12 days, pool every 18 — both on day 36"), and the cycle length of repeating patterns. The prime factorization table under the result shows exactly where both numbers come from.