Modulo

The mod operator is used to give the remainder when one number is divided by another.

For example, 11 mod 2 is 1, because 2 fits into 11 five times, with a remainder of 1; 11 mod 4 is 3, because dividing 11 by 4 leaves a remainder of 3.

Disco> 11 mod 2
1
Disco> 11 mod 4
3
Disco> 6 mod 2
0
Disco> 6 mod 7
6
Disco> (-7) mod 2
1

Formally, the result of mod is defined in terms of the “Division Algorithm”: given a number \(n\) and a positive divisor \(d\), the remainder n mod d is the unique number \(r\) such that \(n = qd + r\), where \(0 \leq r < d\) and \(q\) is the quotient. (For negative divisors, we instead require \(d < r \leq 0\).)