LCM Calculator

What It Does

This page answers one list of whole numbers in four directions. In its least-common-multiple mode it prints the smallest number every entry divides into; in its greatest-common-divisor mode, the largest number that divides every entry; in its both mode it prints the pair together with the identity lcm(a, b) × gcd(a, b) = a × b, written out for the first two numbers; and in its factorisation mode it builds the multiple from the highest power of each prime. Beside the result it always prints each entry’s prime factorisation and the divisions the Euclidean algorithm took. The list holds two to six numbers.

Real-World Usage

Use it to add or compare fractions, whose denominators have to be brought to a common multiple first; to work out when two repeating events fall together again, such as two machines on different cycles or two bodies in orbit; to split a quantity into equal groups with nothing left over, which is what the divisor answers; and to check a prime factorisation you have done by hand, since the page prints the primes it used.

Local Context

The least common multiple and the greatest common divisor are properties of the numbers and not of the country, so nothing here changes with your location and one page serves all nine countries. Two things are worth knowing about the numbers themselves. The names differ by language — LCM and GCD in English, MCM and MCD in Spanish, PPCM and PGCD in French, mcm and MCD in Italian, MMC and MDC in Portuguese, kgV and ggT in German — and this page uses the names of the language you are reading it in. And the figures are formatted in that language’s own convention, so the same answer reads 9,007 in German and 9,007 in English while a four-digit figure reads 1.800 in Spanish and 1,800 in English. The entries themselves are separated by the point in every language, exactly as in 9.5.

What is the difference between the least common multiple and the greatest common divisor?

They are opposite questions about the same list. The greatest common divisor is the largest number that divides every entry: for 9, 12 and 15 that is 3, because 3 divides all three and no larger number does. The least common multiple is the smallest number that every entry divides into: 180, because 180 is divisible by 9, by 12 and by 15, and nothing smaller is. One asks for the biggest shared divisor, the other for the smallest shared multiple.

Why does this page refuse a zero instead of showing 0?

Because the least common multiple is not defined for zero. Some calculators answer 0 whenever an entry is zero, which reads as if zero were a multiple of the other numbers the way 15 is a multiple of 5 — it is not. The convention gcd(0, x) = x does exist for the divisor, but this page holds to one rule for the whole list: every entry is a positive integer, and a zero is refused with its reason rather than answered with a figure that means something different from the rest.

Why is a decimal refused?

Because divisibility is a question only a whole number can answer. It is true that 2.5 divides 5, but "the largest number that divides every entry" would then be a question about real numbers, where the answer is not a divisor at all — every positive real divides every other one. The page therefore asks for whole numbers and says so, instead of silently cutting the decimal off and computing a different question.

Why print the factorisations and the Euclidean divisions at all?

Because the working is what lets you check the answer by hand. Every figure on this page comes from the same integer arithmetic: the primes printed beside the result are the primes the multiple is built from, and the divisions are the ones the greatest common divisor actually took — the last non-zero remainder is the answer. Nothing is recomputed for display, so the working and the answer cannot disagree.