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 ,
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