Least common multiple

The smallest positive number among the common multiples of a finite set of integers or, in particular, of natural numbers, $a_1,\ldots,a_n$. The least common multiple of the numbers $a_1,\ldots,a_n$ exists if $a_1 \cdots a_n \neq 0$. It is usually denoted by $[a_1,\ldots,a_n]$.

Properties of the least common multiple are:

1) the least common multiple of $a_1,\ldots,a_n$ is a divisor of any other common multiple;

2) $[a_1,\ldots,a_{n+1}] = [[a_1,\ldots,a_n],a_{n+1}]$;

3) if the integers $a_1,\ldots,a_n$ are expressed as $$ a_i = p_1^{e_{i1}} \cdots p_k^{e_{ik}} $$ where $p_1,\ldots,p_k$ are distinct primes, $e_{ij} \ge 0$, $i=1,\ldots,n$, $j=1,\ldots,k$, and $f_j = \max\{e_{1j},\ldots,e_{nj} \}$, $j=1,\ldots,k$, then $$ [a_1,\ldots,a_n] = p_1^{f_1} \cdots p_k^{f_k} \ ; $$

4) if $ab > 0$, then $ab = [a,b](a,b)$, where $(a,b)$ is the greatest common divisor of $a,b$.

Thanks to the last property, the least common multiple of two numbers (and so by (2), of any finite set) 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. In a unique factorization domain least common multiples exist and are unique (up to units).

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: $\text{lcm}(a_1,\ldots,a_n)$, $\text{LCM}(a_1,\ldots,a_n)$, $\text{l.c.m.}(a_1,\ldots,a_n)$, etc.