Modulo operation

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Quotient (red) and remainder (green) functions using different algorithms.

In computing, the modulo operation finds the remainder of division of one number by another.

Given two positive numbers, $a$ (the dividend) and $n$ (the divisor), a modulo n (abbreviated as a mod n) can be thought of as the remainder, on division of a by n. For instance, the ｅxpression "5 mod 4" would evaluate to 1 because 5 divided by 4 leaves a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Notice that doing the division with a calculator won't show you the result referred to here by this operation, the quotient will be expressed as a decimal.) When either $a$ or $n$ is negative, this naive definition breaks down and programming languages differ in how these values are defined. 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. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages) See modular arithmetic for an older and related convention applied in number theory.