7.1.9  Integer Euclidean remainder

The irem or remain command finds the remainder of two integers (see Section 7.1.8).

• irem takes two arguments: a and b, integers.
• irem(a,b) returns the remainder r of a divided by b.

Examples

Here r=a−b q=−4+i and |−4+i|²=17<|5+7 i|²/2=74/2=37.

The smod or mods command finds the symmetric remainder of two (ordinary) integers.

• smod takes two arguments: a and b, integers.
• smod(a,b) returns the symmetric remainder s of the Euclidean division of a and b; namely, the value s with a=b q+s and −b/2<s ≤ b/2.

Example

The mod or % operator is an infixed operator which takes an integer to a modular integer.

• mod has two operands: a and b, ordinary integers.
• a mod b returns r % b in ℤ/bℤ, where r is the remainder of the Euclidean division of the arguments a and b.

Example

or:

Note that the result -2 % 5 is not an integer (−2) but an element of Z/5Z (see Section 11.8 for the possible operations in Z/5Z).