Editing Modulo (section)

{{About|the binary operation ''mod({{mvar|a,n}})''|the ''(mod {{mvar|n}})'' notation|Modular arithmetic|other uses}}
{{short description|Computational operation}}
In [[computing]] and [[mathematics]], the '''modulo operation''' returns the [[remainder]] or signed remainder of a [[Division (mathematics)|division]], after one number is divided by another, the latter being called the ''[[modular arithmetic|modulus]]'' of the operation.

Given two positive numbers {{math|''a''}} and {{math|''n''}}, {{math|''a''}} modulo {{math|''n''}} (often abbreviated as {{math|''a'' mod ''n''}}) is the remainder of the [[Euclidean division]] of {{math|''a''}} by {{math|''n''}}, where {{math|''a''}} is the [[Division (mathematics)|dividend]] and {{math|''n''}} is the [[divisor]].<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Congruence|url=https://mathworld.wolfram.com/Congruence.html|access-date=2020-08-27|website=Wolfram MathWorld |language=en}}</ref>

For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a [[quotient]] of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0.

Although typically performed with {{math|''a''}} and {{math|''n''}} both being [[integer]]s, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of {{math|''n''}} is 0 to {{math|''n'' − 1}}. {{math|''a''}} mod 1 is always 0.

When exactly one of {{math|''a''}} or {{math|''n''}} is negative, the basic definition breaks down, and [[programming languages]] differ in how these values are defined.