amount left over after dividing one integer by another

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In arithmetic, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder—an amount "left over" after the division—is also accepted.

The remainder for natural numbers


Given a natural number   and a non-zero natural number  , it can be shown that there exist unique integers   and  , such that   and  . The number   is called the quotient, while   is called the remainder.[1][2]

The case of general integers


If   and   are integers with   being non-zero, then a remainder is an integer   such that   for some integer  , and with  .

When defined this way, there are two possible remainders. For example, the division of -42 by -5 can be expressed as either




So the remainder is then either 3 or -2.

This ambiguity in the value of the remainder is not very serious; in the case above, the negative remainder is obtained from the positive one just by subtracting 5, which is  . This holds in general. When dividing by  , if the positive remainder is  , and the negative one is  , then


The remainder for real numbers


When   and   are real numbers, with   being non-zero, then   can be divided by   without remainder, with the quotient being another real number. If the quotient is constrained to being an integer, however, the concept of remainder is still necessary. It can be proven that there exists a unique integer quotient   and a unique real remainder   such that   with  . Similar to the case of division of integers, the remainder could be required to be negative, that is,  .

Extending the definition of remainder for real numbers, as described above, is not of theoretical importance in mathematics; however, many programming languages implement this definition—see modulation for more.



  1. "The Definitive Higher Math Guide to Long Division and Its Variants — for Integers". Math Vault. 2019-02-24. Retrieved 2020-08-27.
  2. "The quotient remainder theorem (article) | Khan Academy". Khan Academy. Retrieved 2020-08-27.