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.