What is the least common multiple?

How it works

The greatest common divisor of some whole numbers is the largest number that divides every one of them without a remainder: the GCD of 24 and 36 is 12. The least common multiple is the smallest positive number that every one of them divides: the LCM of 4 and 6 is 12. The GCD is what you divide by to simplify a fraction, and the LCM is the common denominator for adding fractions.

The Euclidean algorithm finds the GCD by repeated division: divide the larger number by the smaller, then the smaller by the remainder, and so on until the remainder is 0. The last divisor is the GCD. For 48 and 18: 48 = 2 × 18 + 12, then 18 = 1 × 12 + 6, then 12 = 2 × 6 + 0, so the GCD is 6. The LCM then follows from lcm(a, b) = a ÷ gcd(a, b) × b, which gives 48 ÷ 6 × 18 = 144.

For more than two numbers the answer is built pair by pair: gcd(a, b, c) = gcd(gcd(a, b), c), and the same for the LCM. The signs of numbers are ignored. The GCD of 0 and a number is that number, gcd(0, 0) is defined as 0, and the LCM is 0 whenever one of the numbers is 0. Calculations are exact, for numbers of up to 40 digits.