Modulo operation

In computing, the modulo operation finds the remainder after division of one number by another (sometimes called modulus).

Given two positive numbers, $a$ (the dividend) and $n$ (the divisor), $a$ modulo $n$ (abbreviated as $a mod n$ ) is the remainder of the Euclidean division of $a$ by $n$ . For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Note that doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)

Although typically performed with $a$ and $n$ both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of $n$ is 0 to $n - 1$ . ( $a$ mod 1 is always 0; $a mod 0$ is undefined, possibly resulting in a division by zero error in programming languages.) See modular arithmetic for an older and related convention applied in number theory.

When either $a$ or $n$ is negative, the naive definition breaks down and programming languages differ in how these values are defined.

In mathematics, the result of the modulo operation is the remainder of the Euclidean division. However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware.

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