The word modulo is the Latin

Latin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

 ablative of modulus which itself means "a small measure."
It was introduced into mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 in the book Disquisitiones Arithmeticae

Disquisitiones Arithmeticae

The Disquisitiones Arithmeticae is a textbook of number theory written in Latin by Carl Friedrich Gauss in 1798 when Gauss was 21 and first published in 1801 when he was 24...

by Carl Friedrich Gauss

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

 in 1801. Ever since, however, "modulo" has gained many meanings, some exact and some imprecise.

Usage

• (This usage is from Gauss's book.) Given the integer
  Integer
  The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
  
  s a, b and n, the expression a ≡ b (mod n) means that a − b is a multiple of n, or equivalently, a and b both leave the same remainder when divided by n. For more details, see modular arithmetic
  Modular arithmetic
  In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
  
  .
• In computing
  Computing
  Computing is usually defined as the activity of using and improving computer hardware and software. It is the computer-specific part of information technology...
  
  , given two numbers (either integer or real), a and n, a modulo n is the remainder
  Remainder
  In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
  
   after numerical division
  Division (mathematics)
  right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
  
   of a by n, under certain constraints. See modulo operation
  Modulo operation
  In computing, the modulo operation finds the remainder of division of one number by another.Given two positive numbers, and , a modulo n can be thought of as the remainder, on division of a by n...
  
  .
• Two members of a ring
  Ring (mathematics)
  In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
  
   or an algebra are congruent modulo an ideal
  Ideal (ring theory)
  In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
  
   if the difference between them is in the ideal.
• Two members a and b of a group
  Group (mathematics)
  In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
  
   are congruent modulo a normal subgroup
  Normal subgroup
  In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
  
   if and only if
  If and only if
  In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
  
   ab⁻¹ is a member of the normal subgroup. See quotient group
  Quotient group
  In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
  
   and isomorphism theorem
  Isomorphism theorem
  In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
  
  .
• Two subsets of an infinite set are equal modulo finite sets precisely if their symmetric difference
  Symmetric difference
  In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....
  
   is finite, that is, you can remove a finite piece from the first subset, then add a finite piece to it, and get as result the second subset.
• The most general precise definition is simply in terms of an equivalence relation
  Equivalence relation
  In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
  
   R. We say that a is equivalent or congruent to b modulo R if aRb.

Example

Using Gauss's definintion

13 is congruent 63 modulo 10

to mean

13 and 63 differ by a multiple of 10

However, the word modulo has acquired several related definitions with time, many of which have become integrated into popular mathematical jargon

Mathematical jargon

The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal...

.

Generally, to say:

A is the same as B modulo C

means, "more-or-less", as in:

A and B are the same except for differences accounted for or explained by C.

Up to

The up to

Up to

In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

 concept is often talked about this way, using modulo as a term alerting the hearer. The use of the term in modular arithmetic is a special case of that usage, and that is how this more general usage evolved. The operation of "modding out by C" is that of identifying with each other any two things that are the same modulo C.

Here are several ways in which modulo is used.

• "http and https
  Https
  Hypertext Transfer Protocol Secure is a combination of the Hypertext Transfer Protocol with SSL/TLS protocol to provide encrypted communication and secure identification of a network web server...
  
   are the same, modulo encryption." - means "the only difference between http and https is the addition of encryption".

• "These two characters are equal." "You mean, equal modulo case." - indicates that the first speaker's words are true only for a relaxed sense of equality. In computing, letter case
  Letter case
  In orthography and typography, letter case is the distinction between the larger majuscule and smaller minuscule letters...
  
  is sometimes treated as significant, and sometimes not.

• "The two students performed equally well on the exam, modulo some minor computational mistakes." - means that the two students demonstrated an equal understanding of the material and its application, but at least one of them lost some points for minor computational mistakes which the other did not make.

• "This code is finished modulo testing" - means "this code is finished except for testing".