least common multiple


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

lcm(a,b),

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.)

Properties

  1. 1.

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

    lcm(a,b)=i=1mpimax{αi,βi}.

    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

    gcd(a,b)lcm(a,b)=ab.

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

    gcd(a,b,c)lcm(a,b,c)abc.
  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

    ((a)+(b))((a)(b))=(a)(b).
  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

    (𝔞+𝔟)(𝔞𝔟)=𝔞𝔟

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

References

  • 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