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; Mathematica), l.c.m.
(Andrews 1994, p. 22; Guy 2004, pp. 312-313), or ![[a,b]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_sZLdJRApHNeVgfdfFvlS7MXl4bZEhis4pMkhUsY9jigpJOl5KVTA2N6JA57KprXefSR1C6h-0t6Vsaa-KSnB25ZSEXdCbCeLkRXU3uDm-DZGE-rDPZkZp7B3ejIBM5U14Dvc3OqhviuLY_zy9qJhxj=s0-d)
, is the smallest positive number 
for which there exist positive integers 
and 
such that
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; Mathematica), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that |
(1)
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The least common multiple 
of more than two numbers is similarly defined.
of more than two numbers is similarly defined.
The least common multiple of 
, 
, ... is implemented in Mathematica as LCM[a, b, ...].
, , ... is implemented in Mathematica as LCM[a, b, ...].
The least common multiple of two numbers 
and 
can be obtained by finding the prime factorization of each
and can be obtained by finding the prime factorization of each |  |  |
(2)
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 |  |  |
(3)
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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
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)
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For example, consider 
.
. |  |  |
(5)
|
 |  |  |
(6)
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so
 |
(7)
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The plot above shows 
for rational 
, which is equivalent to the numerator of the reduced form of 
.
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 
.
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, ... (Sloane's A003418; Selmer 1976), which is related to the Chebyshev function 
. For 
, 
(Nair 1982ab, Tenenbaum 1990). The prime number theorem implies that
positive integers for , 2, ... are 1, 2, 6, 12, 60, 60, 420, 840, ... (Sloane's A003418; Selmer 1976), which is related to the Chebyshev function . For , (Nair 1982ab, Tenenbaum 1990). The prime number theorem implies that |
(8)
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as 
, in other words,
, in other words, |
(9)
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as 
.
.
Let 
be a common multiple of 
and 
so that
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
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)
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 |
(12)
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The smallest 
is given by 
,
is given by , |
(13)
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so
 |
(14)
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The LCM is idempotent
 |
(15)
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commutative
 |
(16)
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associative
 |  |  |
(17)
|
 |  |  |
(18)
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distributive
 |
(19)
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and satisfies the absorption law
 |
(20)
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It is also true that
 |  |  |
(21)
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 |  |  |
(22)
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 |  |  |
(23)
Ref: mathworld.wolfram.com
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