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LCM






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Integer Functions > LCM[n1,n2,...,nm] > Introduction to the GCD and LCM (greatest common divisor and least common multiple)





The best-known properties and formulas of the GCD and LCM

Specific values for specialized variables

The functions GCD and LCM , , and have the following values for specialized values:

The first values of the greatest common divisor (gcd(m, n)) of the integers and for and are described in the following table:

The first values of the extended greatest common divisor () of the integers and for and are described in the following table:

The first values of the least common multiple () of the integers and for and are described in the following table:

Analyticity

The functions and are nonanalytical functions defined over with values in . The function is a vector‐valued nonanalytical function defined over .

Periodicity

All three functions , , and do not have periodicity.

Parity and symmetry

The functions and are even functions:

The functions and have permutation symmetry:

Series representations

The function has the following sum representations:

where is the floor function and is the Kronecker delta function.

Product representations

The functions and have the following product representations:

Generating functions

The function can be represented as the coefficients of the series expansion of corresponding generating functions, which includes a sum of the Euler totient function:

Transformations with multiple arguments

The GCD and LCM functions , , and satisfy special relations including multiple arguments, for example:

Identities

The GCD and LCM functions satisfy some parallel identities that can be presented in the forms shown in the following table:

Summation

There are many finite and infinite sums containing GCD and LCM functions, for example:

Limit operation

The following two related limits include the function . The third limit includes lcm():

Inequalities

The functions and satisfy various inequalities, for example:





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