least common multiple
If and are two positive integers, then their least common multiple, denoted by
is the positive integer satisfying the conditions
-
•
and ,
-
•
if and , then .
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 and are the prime factor of the positive integers and (, ), then
This can be generalized for lcm of several numbers.
-
2.
Because the greatest common divisor has the expression , we see that
This formula is sensible only for two integers; it can not be generalized for several numbers, i.e., for example,
-
3.
The preceding formula may be presented in of ideals of ; we may replace the integers with the corresponding principal ideals. The formula acquires the form
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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 be a commutative ring with non-zero unity. is a Prüfer ring iff Jensen’s formula
is true for all ideals and of , 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 |