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
$$\mathrm{lcm}(a,b)=\prod _{i=1}^{m}{p}_{i}^{\mathrm{max}\{{\alpha}_{i},{\beta}_{i}\}}.$$ This can be generalized for lcm of several numbers.

2.
Because the greatest common divisor^{} has the expression $\mathrm{gcd}(a,b)={\prod}_{i=1}^{m}{p}_{i}^{\mathrm{min}\{{\alpha}_{i},{\beta}_{i}\}}$, we see that
$$\mathrm{gcd}(a,b)\cdot \mathrm{lcm}(a,b)=ab.$$ This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,
$$\mathrm{gcd}(a,b,c)\cdot \mathrm{lcm}(a,b,c)\ne abc.$$ 
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 nonzero unity. $R$ is a Prüfer ring iff Jensen’s formula
$$(\U0001d51e+\U0001d51f)(\U0001d51e\cap \U0001d51f)=\U0001d51e\U0001d51f$$ is true for all ideals $\U0001d51e$ and $\U0001d51f$ of $R$, with at least one of them having nonzerodivisors (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  20150506 19:07:25 
Last modified on  20150506 19:07:25 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  32 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 1100 
Synonym  least common dividend 
Synonym  lcm 
Related topic  Divisibility 
Related topic  PruferRing 
Related topic  SumOfIdeals 
Related topic  IdealOfElementsWithFiniteOrder 