least common multiple

If a and b are two positive integers, then their least common multipleMathworldPlanetmath, denoted by


is the positive integer f satisfying the conditions

  • af and bf,

  • if ac and bc, then fc.

Note:   The definition can be generalized for several numbers.  The positivePlanetmathPlanetmath lcm of positive integers is uniquely determined. (Its negative satisfies the same two conditions.)


  1. 1.

    If  a=i=1mpiαi  and  b=i=1mpiβi  are the prime factorMathworldPlanetmathPlanetmath of the positive integers a and b (αi0,  βi0i), then


    This can be generalized for lcm of several numbers.

  2. 2.

    Because the greatest common divisorMathworldPlanetmathPlanetmath has the expression  gcd(a,b)=i=1mpimin{αi,βi}, we see that


    This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,

  3. 3.

    The preceding formula may be presented in of ideals of ; we may replace the integers with the corresponding principal idealsMathworldPlanetmathPlanetmathPlanetmath.  The formula acquires the form

  4. 4.

    The recent formula is valid also for other than principal ideals and even in so general systems as the Prüfer rings; in fact, it could be taken as defining property of these rings:   Let R be a commutative ring with non-zero unity.  R is a Prüfer ring iff Jensen’s formula


    is true for all ideals 𝔞 and 𝔟 of R, with at least one of them having non-zero-divisors (http://planetmath.org/ZeroDivisor).


  • 1 M. Larsen and P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).
Title least common multiple
Canonical name LeastCommonMultiple
Date of creation 2015-05-06 19:07:25
Last modified on 2015-05-06 19:07:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 32
Author pahio (2872)
Entry type Definition
Classification msc 11-00
Synonym least common dividend
Synonym lcm
Related topic Divisibility
Related topic PruferRing
Related topic SumOfIdeals
Related topic IdealOfElementsWithFiniteOrder