Modulus->n is an option that can be given in certain algebraic functions to specify that integers should be treated modulo n.
Equations for Modulus can be given in Solve and related functions.
Modulus appears as an option in Factor, PolynomialGCD and PolynomialLCM, as well as in linear algebra functions such as Inverse, LinearSolve and Det.
Arithmetic is usually done over the full ring of integers; setting the option Modulus specifies that arithmetic should instead be done in the finite ring .
The setting Modulus -> 0 specifies the full ring of integers.
Some functions require that Modulus be set to a prime, or a power of a prime. is a finite field when is prime.
See The Mathematica Book: Section 3.3.5 and Section 3.4.11.
See also: Extension.