# 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. 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. 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. 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. 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