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# Integer division

(Redirected from Least absolute remainder)

Integer division (or Euclidean division) is an operation on integers defined as

${\displaystyle m\backslash n:=\left\lfloor {\frac {m}{n}}\right\rfloor ,\quad m\in \mathbb {Z} ,n\in \mathbb {Z} ^{*},}$
where
 ⌊ ·⌋
is the floor function. In the mapping
${\displaystyle (m,n)\mapsto (q,r):=\left(m\backslash n,\,m-(m\backslash n)\,n\right)=\left(\left\lfloor {\frac {m}{n}}\right\rfloor ,m-\left\lfloor {\frac {m}{n}}\right\rfloor n\right),\quad m\in \mathbb {Z} ,n\in \mathbb {Z} ^{*},}$
given by the division algorithm, the integer quotient[1]
 q ∈ ℤ
, with
 0   ≤   r < | n |
as the remainder.[2]

## Least absolute remainder

The least absolute remainder integer division (or nearest integer division) is an operation on integers defined as

${\displaystyle \left\lfloor {\frac {m}{n}}+{\rm {sgn}}(n)\,{\frac {1}{2}}\right\rfloor ,\quad m\in \mathbb {Z} ,n\in \mathbb {Z} ^{*},}$
where
 sgn (n)
is the sign function. In the mapping
${\displaystyle (m,n)\mapsto (q,r):=\left(\left\lfloor {\frac {m}{n}}+{\rm {sgn}}(n)\,{\frac {1}{2}}\right\rfloor ,m-\left\lfloor {\frac {m}{n}}+{\rm {sgn}}(n)\,{\frac {1}{2}}\right\rfloor n\right),\quad m\in \mathbb {Z} ,n\in \mathbb {Z} ^{*},}$
given by the [least absolute remainder] division algorithm, the nearest integer quotient
 q ∈ ℤ
, with
 −  ⌊  | n |/2⌋   ≤   r < | n |  −  ⌊  | n |/2⌋
as the least absolute remainder.

### Examples

The allowed least absolute remainders
 mod ±2
are
 { − 1, 0}
:
•  (7, 2)
maps to  (4, 7  −  4  ×  2) = (4,  − 1)
, i.e.  7 = 4  ×  2 + ( − 1)
;
•  ( − 7, 2)
maps to  ( − 3,  − 7  −  ( − 3)  ×  2) = ( − 3,  − 1)
, i.e.  − 7 = ( − 3)  ×  2 + ( − 1)
;
•  (7,  − 2)
maps to  ( − 4, 7  −  ( − 4)  ×  ( − 2)) = ( − 4,  − 1)
, i.e.  7 = ( − 4)  ×  ( − 2) + ( − 1)
;
•  ( − 7,  − 2)
maps to  (3,  − 7  −  3  ×  ( − 2) = (3,  − 1)
, i.e.  − 7 = 3  ×  ( − 2) + ( − 1)
.
The allowed least absolute remainders
 mod ±256
are
 { − 128, ...,  − 1, 0, 1, ..., 127}
:
•  (128, 256)
maps to  (1, 128  −  1  ×  256) = (1,  − 128)
, i.e.  128 = 1  ×  256 + ( − 128)
;
•  ( − 128, 256)
maps to  0,  − 128  −  0  ×  256) = (0,  − 128)
, i.e.  − 128 = 0  ×  256 + ( − 128)
;
•  (128,  − 256)
maps to  ( − 1, 128  −  ( − 1)  ×  ( − 256)) = ( − 1,  − 128)
, i.e.  128 = ( − 1)  ×  ( − 256) + ( − 128)
;
•  ( − 128,  − 256)
maps to  (0,  − 128  −  0  ×  ( − 256) = (0,  − 128)
, i.e.  − 128 = 0  ×  ( − 256) + ( − 128)
.