least common multiple

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

\mathrm{lcm}\!(a,\,b),

is the positive integer $f$ satisfying the conditions

•

$a\mid f$ and $b\mid f$ ,
•

if $a\mid c$ and $b\mid c$ , then $f\mid c$ .

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

Properties

1.

If $a=\prod_{i=1}^{m}p_{i}^{\alpha_{i}}$ and $b=\prod_{i=1}^{m}p_{i}^{\beta_{i}}$ are the prime factor of the positive integers $a$ and $b$ ( $\alpha_{i}\geqq 0$ , $\beta_{i}\geqq 0$ $\forall i$ ), then

$\mathrm{lcm}\!(a,\,b)=\prod_{i=1}^{m}p_{i}^{\max\{\alpha_{i},\,\beta_{i}\}}.$

This can be generalized for lcm of several numbers.
2.

Because the greatest common divisor has the expression $\gcd(a,\,b)=\prod_{i=1}^{m}p_{i}^{\min\{\alpha_{i},\,\beta_{i}\}}$ , we see that

$\gcd(a,\,b)\cdot\mathrm{lcm}\!(a,\,b)=ab.$

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

$\gcd(a,\,b,\,c)\cdot\mathrm{lcm}(a,\,b,\,c)\neq abc.$
3.

The preceding formula may be presented in of ideals of $\mathbb{Z}$ ; we may replace the integers with the corresponding principal ideals. The formula acquires the form

$((a)+(b))((a)\cap(b))=(a)(b).$
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

$(\mathfrak{a}+\mathfrak{b})(\mathfrak{a}\cap\mathfrak{b})=\mathfrak{ab}$

is true for all ideals $\mathfrak{a}$ and $\mathfrak{b}$ 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