# LCM

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

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## Basic Examples(2)

Find the least common multiple of a set of numbers:

Plot the least common multiple for a number with :

## Scope(11)

### Numerical Manipulation(7)

LCM works over integers:

Gaussian integers:

Real rational numbers:

Complex rational numbers:

The one-argument form is identity for positive integers:

Compute for large integers:

### Symbolic Manipulation(4)

Reduce inequalities:

Solve equations:

Simplify expressions:

## Applications(9)

### Basic Applications(4)

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

Visualize the LCMs of two integers:

Fibonacci numbers:

The LCM of the first 100 integers:

Compute LCM for positive integers:

Compare with:

### Number Theory(5)

Cumulative LCMs:

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

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

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

LCMs of binomial coefficients:

Compare with:

Simplify expressions containing LCM:

## Properties & Relations(7)

Every divisor of a and b is a divisor of :

Use GCD to compute LCM:

Compare with:

The LCM of coprime numbers is equal to their product:

LCM for prime numbers is their product:

LCM for prime power representation :

LCM is commutative :

LCM is associative :

LCM is distributive :

Use LCM to compute MangoldtLambda:

Compare with:

## Possible Issues(3)

The arguments must be explicit integers: LCM sorts its arguments:

## Interactive Examples(1)

Visualize the LCMs of three numbers:

## Neat Examples(4)

Visualize the LCMs of Fibonacci numbers:

Plot the arguments of the Fourier transform of the LCM:

Plot the Ulam spiral of the LCM:

Form the LCMs of with rational numbers: