The integer Euclidean remainder : irem remain smod mods mod %

### 2.6.11 The integer Euclidean remainder : `irem remain smod mods mod %`

`irem` (or `remain`) returns the integer remainder r from the Euclidean division of two integers a and b given as arguments (a=b*q+r avec 0≤ r< b).
For Gaussian integers, we choose q so that b*q is as near by a as possible and it can be proved that r may be choosen so that |r|2 ≤ |b|2/2.
Input :

`irem(148,5)`

Output :

`3`

`irem` works with long integers or with Gaussian integers.
Example :

`irem(factorial(148),factorial(45)+2 )`

Output :

`111615339728229933018338917803008301992120942047239639312`

Another example

`irem(25+12*i,5+7*i)`

Output :

`-4+i`

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

`smod` or `mods` is a prefixed function and has two integers a and b as arguments.
`smod` or `mods` returns the symetric remainder s of the Euclidean division of the arguments a and b (a=b*q+s with −b/2<s ≤ b/2).
Input :

`smod(148,5)`

Output :

`-2`

`mod` (or `%`) is an infixed function and has two integers a and b as arguments.
`mod` (or `%`) returns r% b of Z/bZ where r is the remainder of the Euclidean division of the arguments a and b.
Input :

`148 mod 5`

or

`148 % 5`

Output :

`3 % 5`

Note that the answer `3 % 5` is not an integer (3) but an element of Z/5Z (see 2.31 to have the possible operations in Z/5Z).