Modulo

• Given the integers a, b and n, the expression $a\backslash equiv\; b\backslash pmod\{n\}$ (pronounced "a is congruent to b modulo n") means that a and b have same remainder when divided by n, or equivalently, that a−b is a multiple of n. For more details, see modular arithmetic.

• In computing, given two integers, a and n, a mod n is the remainder of integer division of a by n. See modulo operation.

• Two members of a ring or an algebra are congruent modulo an ideal if the difference between them is in the ideal.

• Two members a and b of a group are congruent modulo a normal subgroup iff ab⁻¹ is a member of the normal subgroup. See quotient group and isomorphism theorem.

• Two subsets of an infinite set are equal modulo finite sets precisely if their symmetric difference is finite, that is, you can take a finite piece from the first infinite set, then add a finite piece to it, and get as result the second infinite set.

• In the mathematical community, the word modulo is also used informally, in many imprecise ways. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C". See modulo (jargon).

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