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]](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FLeastCommonMultiple%2FInline6.svg){kind=small}
,
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
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 
.
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) |
 |
(19) |
and satisfies the absorption law
 |
(20) |
It is also true that
See also
Chebyshev Functions, Greatest Common Divisor, Least Common Denominator, Mangoldt Function, Multiple, Relatively Prime Explore this topic in the MathWorld classroom
Related Wolfram sites
http://functions.wolfram.com/IntegerFunctions/LCM/
Explore with Wolfram|Alpha
References
Andrews, G. E. Number Theory. New York: Dover, 1994.Bressoud, D. M. and Wagon,
S. A
Course in Computational Number Theory. London: Springer-Verlag, 2000.Bronshtein,
I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook
of Mathematics, 5th ed. Berlin: Springer, 2007.D'Angelo, J. P.
and West, D. B. Mathematical
Thinking: Problem-Solving and Proofs, 2nd ed. Upper Saddle River, NJ: Prentice-Hall,
2000.Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H.; and
Künstner, H. (Eds.). VNR
Concise Encyclopedia of Mathematics, 2nd ed. New York: Van Nostrand Reinhold,
1989.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science. Reading, MA: Addison-Wesley,
1990.Guy, R. K. "Density of a Sequence with l.c.m. of Each
Pair Less than 
."
§E2 in Unsolved
Problems in Number Theory, 3rd ed. New York: Springer-Verlag, pp. 312-313,
2004.Jones, G. A. and Jones, J. M. "Least Common Multiples."
§1.3 in Elementary
Number Theory. Berlin: Springer-Verlag, pp. 12-13, 1998.Nagell,
T. "Least Common Multiple and Greatest Common Divisor." §5 in Introduction
to Number Theory. New York: Wiley, pp. 16-19, 1951.Nair,
M. "A New Method in Elementary Prime Number Theory." J. London Math.
Soc. 25, 385-391, 1982a.Nair, M. "On Chebyshev-Type
Inequalities for Primes." Amer. Math. Monthly 89, 126-129, 1982b.Råde,
L. and Westergren, B. Mathematics
Handbook for Science and Engineering. Berlin: Springer, 2004.Selmer,
E. S. "On the Number of Prime Divisors of a Binomial Coefficient."
Math. Scand. 39, 271-281, 1976.Sloane, N. J. A.
Sequence A003418/M1590 in "The On-Line
Encyclopedia of Integer Sequences."Tenenbaum, G. Introduction
à la théorie analytique et probabiliste des nombres. Publications
de l'Institut Cartan, pp. 12-13, 1990.Trott, M. The
Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.Yan,
S. Y. Number
Theory for Computing, 2nd ed. Berlin: Springer, 2002.Zwillinger,
D. (Ed.). "Least Common Multiple." §2.3.6 in CRC
Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press,
p. 91, 1996.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Least Common Multiple." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LeastCommonMultiple.html