The bestknown 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:
