The word modulus has several different meanings in mathematics with respect to complex numbers, congruences, elliptic integrals, quadratic invariants, sets, etc.
The modulus of a congruence 
is the number 
.
It is the "base" with respect to which a congruence
is computed (i.e., 
gives the number of multiples of 
that are "thrown out"). For example, when computing
the time of day using a 12-hour clock obtained by adding four hours to 9:00, the
answer, 1:00, is obtained by taking 
(i.e., adding the hours with modulus 12).
In many computer languages (such as FORTRAN or the Wolfram Language), the common residue of 
(mod 
) is written mod(b,m)
(FORTRAN) or Mod[b,m] (Wolfram
Language).
