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LCM
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LCM

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LCM[n1,n2,…]

gives the least common multiple of the ni.

Details

  • LCM is also known as smallest common multiple.
  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • LCM[n1,n2,…] is the smallest positive integer that is a multiple of each of the integers n1,n2,….
  • For rational numbers ri, LCM[r1,r2,…] gives the least rational number r for which all the r/ri are integers.
  • LCM works over Gaussian integers.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the least common multiple of a set of numbers:

In[1]:=1
✖
https://wolfram.com/xid/02cx6
Out[1]=1

Plot the least common multiple for a number with :

In[1]:=1
✖
https://wolfram.com/xid/02cx6-ysdzv
Out[1]=1

Scope  (11)Survey of the scope of standard use cases

Numerical Evaluation  (7)

LCM works over integers:

In[1]:=1
✖
https://wolfram.com/xid/02cx6-fgai5c
Out[1]=1

Gaussian integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-cmctw6
Out[1]=1

Real rational numbers:

In[1]:=1
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https://wolfram.com/xid/02cx6-hka5t4
Out[1]=1

Complex rational numbers:

In[1]:=1
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https://wolfram.com/xid/02cx6-hbja6m
Out[1]=1

The one-argument form is identity for positive integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-e8sr93
Out[1]=1

Compute for large integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-me32zh
Out[1]=1

LCM threads elementwise over lists:

In[1]:=1
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https://wolfram.com/xid/02cx6-g7y05f
Out[1]=1

Symbolic Manipulation  (4)

TraditionalForm formatting:

In[1]:=1
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https://wolfram.com/xid/02cx6-dey62h

Reduce inequalities:

In[1]:=1
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https://wolfram.com/xid/02cx6-kgta6s
Out[1]=1

Solve equations:

In[1]:=1
✖
https://wolfram.com/xid/02cx6-qg8p4k
Out[1]=1

Simplify expressions:

In[1]:=1
✖
https://wolfram.com/xid/02cx6-xrggl4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/02cx6-y7h28z
Out[2]=2

Applications  (9)Sample problems that can be solved with this function

Basic Applications  (4)

Table of the LCMs of the first 100 pairs of integers:

In[2]:=2
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https://wolfram.com/xid/02cx6-kzlod6
Out[2]=2

Visualize the LCMs of two integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-wipe5
Out[1]=1

Fibonacci numbers:

In[2]:=2
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https://wolfram.com/xid/02cx6-nbtmnk
Out[2]=2

The LCM of the first 100 integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-hcxlz
Out[1]=1

Compute LCM for positive integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-f5sa2v
In[2]:=2
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https://wolfram.com/xid/02cx6-hc12m
Out[2]=2

Compare with:

In[3]:=3
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https://wolfram.com/xid/02cx6-h1anbe
Out[3]=3

Number Theory  (5)

Cumulative LCMs:

In[1]:=1
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https://wolfram.com/xid/02cx6-zf9asi
Out[1]=1

Plot the logarithm of the data (if Riemann's hypothesis holds, this grows linearly):

In[2]:=2
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https://wolfram.com/xid/02cx6-lidfix
Out[2]=2

The sum of MangoldtLambda of the first n integers is equal to the natural log of the LCM of the first n integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-89poab
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/02cx6-6orsme
Out[2]=2

Maximal order of group elements from the symmetric group of order n (Landau's function):

In[1]:=1
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https://wolfram.com/xid/02cx6
Out[1]=1

LCMs of binomial coefficients:

In[1]:=1
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https://wolfram.com/xid/02cx6
Out[1]=1

Compare with:

In[2]:=2
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https://wolfram.com/xid/02cx6
Out[2]=2

Simplify expressions containing LCM:

In[1]:=1
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https://wolfram.com/xid/02cx6-bxhmfw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/02cx6-5elzc4
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/02cx6-7j66kw
Out[3]=3

Properties & Relations  (7)Properties of the function, and connections to other functions

Every divisor of a and b is a divisor of :

In[1]:=1
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https://wolfram.com/xid/02cx6-hu64el
Out[1]=1

Use GCD to compute LCM:

In[1]:=1
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https://wolfram.com/xid/02cx6-f2xcc7
Out[1]=1

Compare with:

In[2]:=2
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https://wolfram.com/xid/02cx6-emhzf3
Out[2]=2

The LCM of coprime numbers is equal to their product:

In[1]:=1
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https://wolfram.com/xid/02cx6-yea5ne
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/02cx6-bxw2rg
Out[2]=2

LCM for prime numbers is their product:

In[1]:=1
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https://wolfram.com/xid/02cx6-c11ruk
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/02cx6-b6ga13
Out[2]=2

LCM for prime power representation :

In[1]:=1
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https://wolfram.com/xid/02cx6-kbwtft
In[2]:=2
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https://wolfram.com/xid/02cx6-d2nf8
Out[2]=2

LCM is commutative :

In[1]:=1
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https://wolfram.com/xid/02cx6-1b13cp
Out[1]=1

LCM is associative :

In[2]:=2
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https://wolfram.com/xid/02cx6-4xyiz3
Out[2]=2

LCM is distributive :

In[3]:=3
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https://wolfram.com/xid/02cx6-j40va9
Out[3]=3

Use LCM to compute MangoldtLambda:

In[1]:=1
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https://wolfram.com/xid/02cx6-htptxl
Out[1]=1

Compare with:

In[2]:=2
✖
https://wolfram.com/xid/02cx6-u8c3yh
Out[2]=2

Possible Issues  (3)Common pitfalls and unexpected behavior

Signs are discarded:

In[1]:=1
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https://wolfram.com/xid/02cx6-kc9x8g
Out[1]=1

The arguments must be explicit integers:

In[1]:=1
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https://wolfram.com/xid/02cx6-bksbzi
Out[1]=1

LCM sorts its arguments:

In[1]:=1
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https://wolfram.com/xid/02cx6-0v6xu3
Out[1]=1

Interactive Examples  (1)Examples with interactive outputs

Visualize the LCMs of three numbers:

In[1]:=1
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https://wolfram.com/xid/02cx6-lorgr
Out[1]=1

Neat Examples  (4)Surprising or curious use cases

Visualize the LCMs of Fibonacci numbers:

In[10]:=10
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https://wolfram.com/xid/02cx6-e7s0yr
Out[10]=10

Plot the arguments of the Fourier transform of the LCM:

In[1]:=1
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https://wolfram.com/xid/02cx6-hw26ik
Out[1]=1

Plot the Ulam spiral of the LCM:

In[1]:=1
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https://wolfram.com/xid/02cx6-5mjdwo
In[2]:=2
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https://wolfram.com/xid/02cx6-hx9nbv
Out[2]=2

Form the LCMs of with rational numbers:

In[1]:=1
✖
https://wolfram.com/xid/02cx6
Out[1]=1
Wolfram Research (1988), LCM, Wolfram Language function, https://reference.wolfram.com/language/ref/LCM.html (updated 1999).
✖
Wolfram Research (1988), LCM, Wolfram Language function, https://reference.wolfram.com/language/ref/LCM.html (updated 1999).

Text

Wolfram Research (1988), LCM, Wolfram Language function, https://reference.wolfram.com/language/ref/LCM.html (updated 1999).

✖
Wolfram Research (1988), LCM, Wolfram Language function, https://reference.wolfram.com/language/ref/LCM.html (updated 1999).

CMS

Wolfram Language. 1988. "LCM." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1999. https://reference.wolfram.com/language/ref/LCM.html.

✖
Wolfram Language. 1988. "LCM." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1999. https://reference.wolfram.com/language/ref/LCM.html.

APA

Wolfram Language. (1988). LCM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LCM.html

✖
Wolfram Language. (1988). LCM. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LCM.html

BibTeX

@misc{reference.wolfram_2025_lcm, author="Wolfram Research", title="{LCM}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/LCM.html}", note=[Accessed: 23-March-2025 ]}

✖
@misc{reference.wolfram_2025_lcm, author="Wolfram Research", title="{LCM}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/LCM.html}", note=[Accessed: 23-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_lcm, organization={Wolfram Research}, title={LCM}, year={1999}, url={https://reference.wolfram.com/language/ref/LCM.html}, note=[Accessed: 23-March-2025 ]}

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@online{reference.wolfram_2025_lcm, organization={Wolfram Research}, title={LCM}, year={1999}, url={https://reference.wolfram.com/language/ref/LCM.html}, note=[Accessed: 23-March-2025 ]}