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| [[File:Euclidean division example.svg|thumb|17 is divided into 3 groups of 5 with 2 left over. Here the dividend is 17, the divisor is 5, the quotient is 3, and the remainder is 2.<br/>17 = 5 × 3 + 2]]
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| In [[arithmetic]], the '''Euclidean division''' is the conventional process of [[division (mathematics)|division]] of two [[integer]]s producing a [[quotient]] and a [[remainder]]. There is a [[theorem]] stating that the quotient and remainder exist, are unique, and have certain properties. Integer [[division algorithm]]s compute the quotient and remainder given two integers, the most well-known such algorithm being [[long division]]. The integer division algorithm is an important ingredient for other algorithms, such as the [[Euclidean algorithm]] for finding the [[greatest common divisor]] of two integers.
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| [[File:Pie division.svg|thumb|The pie has 9 slices, so each of the 4 people receive 2 slices and 1 is left over.]]
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| ==Intuitive example==
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| Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.
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| This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in all. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.
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| In general, if the number of slices is denoted ''a'' and the number of people is ''b'', one can divide the pie evenly among the people such that each person receives ''q'' slices (the quotient) and some number of slices ''r'' < ''b'' are left over (the remainder). Regardless, the equation ''a'' = ''bq'' + ''r'' holds.
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| If 9 slices were divided among 3 people instead of 4, each would receive 3 and no slices would be left over. In this case the remainder is zero, and it is said that 3 ''evenly divides'' 9, or that 3 ''[[divides]]'' 9.
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| Euclidean division can also be extended to negative integers using the same formula; for example −9 = 4 × (−3) + 3, so −9 divided by 4 is −3 with remainder 3. The remainder is the only one of the four numbers that can never be negative.
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| ==Statement of the theorem==
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| Given two integers ''a'' and ''b'', with ''b'' ≠ 0, there exist [[Uniqueness quantification|unique]] integers ''q'' and ''r'' such that ''a'' = ''bq'' + ''r'' and 0 ≤ ''r'' < |''b''|, where |''b''| denotes the [[absolute value]] of ''b''.<ref>{{cite book |title=Elementary Number Theory |last=Burton |first=David M. |year=2010 |publisher=McGraw-Hill |isbn=978-0-07-338314-9 |pages=17–19}}</ref>
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| The four integers that appear in this theorem have been given a name: ''a'' is called the '''dividend''', ''b'' is called the '''divisor''', ''q'' is called the '''quotient''' and ''r'' is called the '''remainder'''.
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| The computation of the quotient and the remainder from the dividend and the divisor is called '''division''' or, in case of ambiguity, '''Euclidean division'''. The theorem is frequently referred to as the ''division algorithm'', although it is a theorem and not an algorithm, because its proof as given below also provides a simple division algorithm for computing ''q'' and ''r''.
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| Division is not defined in the case where ''b'' = 0; see [[division by zero]].
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| == Examples ==
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| *If ''a'' = 7 and ''b'' = 3, then ''q'' = 2 and ''r'' = 1, since 7 = 3 × 2 + 1.
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| *If ''a'' = 7 and ''b'' = −3, then ''q'' = −2 and ''r'' = 1, since 7 = −3 × (−2) + 1.
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| *If ''a'' = −7 and ''b'' = 3, then ''q'' = −3 and ''r'' = 2, since −7 = 3 × (−3) + 2.
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| *If ''a'' = −7 and ''b'' = −3, then ''q'' = 3 and ''r'' = 2, since −7 = −3 × 3 + 2.
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| ==Proof==
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| The proof consists of two parts — first, the proof of the existence of ''q'' and ''r'', and second, the proof of the uniqueness of ''q'' and ''r''.
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| ===Existence===
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| Consider first the case ''b'' < 0. Setting ''b' ''= −''b'' and ''q' ''= −''q'', the equation ''a'' = ''bq'' + ''r'' may be rewritten ''a'' = ''b'q' ''+ ''r'' and the inequality 0 ≤ r < |''b''| may be rewritten 0 ≤ r < |''b'''|. This reduces the existence for the case ''b'' < 0 to that of the case ''b'' > 0.
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| Similarly, if ''a'' < 0 and ''b'' > 0, setting ''a' ''= −''a'', ''q' ''= −''q'' − 1 and ''r' ''= ''b'' − ''r'', the equation ''a'' = ''bq'' + ''r'' may be rewritten ''a' ''= ''bq' ''+ ''r' '' and the inequality 0 ≤ ''r'' < ''b'' may be rewritten 0 ≤ ''r' '' < ''b''. Thus the proof of the existence is reduced to the case ''a'' ≥ 0 and ''b'' > 0 and we consider only this case in the remainder of the proof.
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| Let ''q''<sub>1</sub> and ''r''<sub>1</sub>, both nonnegative, such that ''a'' = ''bq''<sub>1</sub> + ''r''<sub>1</sub>, for example ''q''<sub>1</sub> = 0 and ''r''<sub>1</sub> = ''a''. If ''r''<sub>1</sub> < ''b'', we are done. Otherwise ''q''<sub>2</sub> = ''q''<sub>1</sub> + 1 and ''r''<sub>2</sub> = ''r''<sub>1</sub> − ''b'' satisfy ''a'' = ''bq''<sub>2</sub> + ''r''<sub>2</sub> and 0 ≤ ''r''<sub>2</sub> < ''r''<sub>1</sub>. Repeating this process one gets eventually ''q'' = ''q''<sub>''k''</sub> and ''r'' = ''r''<sub>''k''</sub> such that ''a'' = ''bq'' + ''r'' and 0 ≤ ''r'' < ''b''.
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| This proves the existence and also gives a simple [[division algorithm]] to compute the quotient and the remainder. However this algorithm needs ''q'' steps and is thus not efficient.
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| ===Uniqueness===
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| Suppose there exists ''q'', ''q' '', ''r'', ''r' '' with 0 ≤ ''r'', ''r' '' < ''|b|'' such that ''a'' = ''bq'' + ''r'' and ''a'' = ''bq' '' + ''r' ''. Adding the two inequalities 0 ≤ ''r'' < |''b''| and −|''b''| < −''r' '' ≤ 0 yields −|''b''| < ''r'' − ''r' '' < |''b''|, that is |''r'' − ''r' ''| < |''b''|.
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| Subtracting the two equations yields: ''b''(''q' '' − ''q'') = (''r'' − ''r' ''). Thus |''b''| divides |''r'' − ''r' ''|. If |''r'' − ''r' ''| ≠ 0 this implies |''b''| < |''r'' − ''r' ''|, contradicting previous inequality. Thus, ''r'' = ''r' '' and ''b''(''q' '' − ''q'') = 0. As ''b'' ≠ 0, this implies ''q'' = ''q' '', proving uniqueness.
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| ===Other proofs===
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| Some proofs of the algorithm rely on the [[Well-ordering principle]].<ref>{{cite book|last=Durbin|first=John R.|title=Modern Algebra : an Introduction|year=1992|publisher=Wiley|location=New York|isbn=0-471-51001-7|pages=63|url=http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000258.html|edition=3rd ed.}}</ref>
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| == Effectiveness ==
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| Usually, an existence proof does not provide an algorithm to compute the existing object, but the above proof provides immediately an algorithm (see [[Division algorithm#Division_by_repeated_subtraction]]). However this is not a very efficient method, as it requires as many steps as the size of the quotient. This is related to the fact that it only uses addition, subtraction and comparison of the integers, without involving multiplication, nor any particular representation of the integers, such as decimal notation.
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| In terms of decimal notation, [[long division]] provides a much more efficient division algorithm. Its generalization to binary notation allows to use it in a computer. However, for large inputs, algorithms that reduce division to multiplication, like [[Division algorithm#Newton–Raphson division|Newton–Raphson]] one, are usually preferred, because they need a time which is proportional to the time of the multiplication needed to verify the result, independently of the multiplication algorithm which is used.
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| == Generalizations ==
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| === In domains other than integers ===
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| [[Euclidean domain]]s are defined as [[integral domain]]s which support the following generalization of Euclidean division:
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| Given an element ''a'' and a non-zero element ''b'' in a Euclidean domain ''R'' equipped with a '''Euclidean function''' ''d'', there exist ''q'' and ''r'' in ''R'' such that {{nowrap|''a'' {{=}} ''bq'' + ''r''}} and either {{nowrap|''r'' {{=}} 0}} or {{nowrap|''d''(''r'') < ''d''(''b'')}}. Unlike in the integer case, ''q'' and ''r'' need not be unique.
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| Examples of Euclidean domains include [[field (mathematics)|field]]s, [[polynomial ring]]s in one variable over a field, and the [[Gaussian integers]]. The Euclidean division of polynomials has been the object of specific developments. See [[Polynomial long division]], [[Polynomial greatest common divisor#Euclidean division]] and [[Polynomial greatest common divisor#Pseudo-remainder sequences]].
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| === Generalized division algorithms ===
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| The 1st generalized division algorithm: Given integers <math>m</math>, <math>a</math>, <math>d</math> with <math>m>0</math>, there exist unique integers <math>q</math> and <math>r</math> with <math>d \le r < m+d </math> such that <math>a = mq+r</math>.
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| Especially, if <math> d=- \left\lfloor \frac{m}{2} \right\rfloor </math> then <math> - \left\lfloor \frac{m}{2} \right\rfloor \le r < m-\left\lfloor \frac{m}{2} \right\rfloor </math> . In this case, <math>r</math> is called the least absolute remainder. As an application of this generalization, the original Euclidean algorithm for integers can be slightly sped up.
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| The 2nd generalized division algorithm: Given integers <math>m</math>, <math>a</math> with <math>m>0</math>, and
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| let <math>R^{-1} \in \mathbb Z_m ^*</math> be the multiplicative inverse of <math>R \in \mathbb Z_m ^*</math>. Then
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| there exist unique integers <math>q</math> and <math>r</math> with <math>0 \le r < m </math> such that <math> a = mq+R^{-1} \cdot r </math>.
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| This result generalizes Hensel's odd division (1900), and its proof can be found in.<ref>Haining Fan, Ming Gu, Jiaguang Sun and Kwok-Yan Lam: "Obtaining More Karatsuba-Like Formulae over the Binary Field", IET Information security, vol. 6, no. 1, pp. 14-19, 2012.</ref>
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| The value <math>r</math> in the 2nd generalization corresponds to the ''N''-residue defined in [[Montgomery reduction]].
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Division Algorithm}}
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| [[Category:Arithmetic]]
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| [[Category:Theorems in number theory]]
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| [[Category:Articles containing proofs]]
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| [[Category:Division]]
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