The least common multiple of two numbers 
and 
, variously denoted 
(this work; Zwillinger 1996, p. 91; Råde
and Westergren 2004, p. 54), 
(Gellert et al. 1989, p. 25; Graham et
al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West
2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325;
Wolfram Language), l.c.m.
(Andrews 1994, p. 22; Guy 2004, pp. 312-313),
or ![[a,b]](/images/equations/LeastCommonMultiple/Inline6.svg)
,
is the smallest positive number (multiple) 
for which there exist positive integers 
and 
such that
 |
(1)
|
The least common multiple 
of more than two numbers is similarly defined.
The least common multiple of 
, 
, ... is implemented in the Wolfram
Language as LCM[a,
b, ...].
The least common multiple of two numbers 
and 
can be obtained by finding the prime
factorization of each
 |  |  |
(2)
|
 |  |  |
(3)
|
where the 
s
are all prime factors of 
and 
, and if 
does not occur in one factorization, then the corresponding
exponent is taken as 0. The least common multiple is then given by
 |
(4)
|
For example, consider 
.
 |  |  |
(5)
|
 |  |  |
(6)
|
so
 |
(7)
|
The plot above shows 
for rational 
, which is equivalent to the numerator
of the reduced form of 
.
The above plots show a number of visualizations of 
in the 
-plane. The figure on the left is simply 
, the figure in the middle is the absolute values of
the two-dimensional discrete Fourier transform
of 
(Trott 2004, pp. 25-26), and the figure at right is the absolute value of the
transform of 
.
The least common multiples of the first 
positive integers for 
, 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, ... (OEIS A003418; Selmer 1976), which is related to the
Chebyshev function 
. For 
, 
(Nair 1982ab, Tenenbaum 1990). The prime number theorem implies that
 |
(8)
|
as 
,
in other words,
 |
(9)
|
as 
.
Let 
be a common multiple of 
and 
so that
 |
(10)
|
Write 
and 
,
where 
and 
are relatively prime by definition of the greatest common divisor 
. Then 
, and from the division
lemma (given that 
is divisible by 
and 
), we have 
is divisible by 
, so
 |
(11)
|
 |
(12)
|
The smallest 
is given by 
,
 |
(13)
|
so
 |
(14)
|
The LCM is idempotent
 |
(15)
|
 |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
 |
(19)
|
and satisfies the absorption law
 |
(20)
|
It is also true that
 |  |  |
(21)
|
 |  |  |
(22)
|
 |  |  |
(23)
|
Polynomial least common multiples are computed in the Wolfram Language using PolynomialLCM.
."
§E2 in