Least common multiple

The smallest positive number among the common multiples of a finite set of integers or, in particular, of natural numbers, . The least common multiple of the numbers exists if . It is usually denoted by .

Properties of the least common multiple are:

1) the least common multiple of is a divisor of any other common multiple;

2) ;

3) if the integers are expressed as

where are distinct primes, , , and , , then

4) if , then , where is the greatest common divisor of and .

Thanks to the last property, the least common multiple of two numbers can be found with the aid of the Euclidean algorithm. The concept of the least common multiple can be defined for elements of an integral domain, and also for ideals of a commutative ring.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	A.A. Bukhshtab, "Number theory" , Moscow (1966) (In Russian)
[3]	R. Faure, A. Kaufman, M. Denis-Papin, "Mathématique nouvelles" , 1–2 , Dunod (1964)

Comments

Other frequently used notations for the least common multiple are: , , , etc. In a unique factorization domain least common multiples exist and are unique (up to units).