Mod[m, n] gives the remainder on division of m by n.
Mod[m, n, d] uses an offset d.
For integers and Mod[m, n] lies between 0 and .
Mod[m, n, 1] gives a result in the range to , suitable for use in functions such as Part.
Mod[m, n, d] gives a result such that and .
The sign of Mod[m, n] is always the same as the sign of n, at least so long as m and n are both real.
Mod[m, n] is equivalent to m - n Quotient[m, n].
Mod[m, n, d] is equivalent to m - n Quotient[m, n, d].
The arguments of Mod can be any numeric quantities, not necessarily integers.
Mod[x, 1] gives the fractional part of x.
For exact numeric quantities, Mod internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.
See The Mathematica Book: Section 1.1.3 and Section 3.2.4.
See also: PowerMod, Quotient, FractionalPart, MantissaExponent, PolynomialMod, PolynomialRemainder, Xor.