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Modulo ○◂｜Definition｜1st｜20251119205401-00-⌔
Modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation.
Given two positive numbers a and n, a modulo n (often abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.1
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 a and n both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n is 0 to n − 1. a mod 1 is always 0.
When exactly one of a or n is negative, the basic definition breaks down, and programming languages differ in how these values are defined.
Printed 2026-06-28.
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Link to original Footnotes
Weisstein, Eric W. “Congruence”. Wolfram MathWorld. Retrieved 2020-08-27. ↩
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