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| {{DISPLAYTITLE:Multiplicative group of integers modulo ''n''}}
| | I am Eddy and աas born on 9 August 1971. Мy hobbies are Element collecting and Drawing.<br><br>my homepage: free trips around the world - [http://www.picnicbasketcity.com/sleep-sound-on-the-road-with-this-hotel-advice/ www.picnicbasketcity.com], |
| {{Distinguish|Integers modulo n}}
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| In [[modular arithmetic]] the set of [[Modular_arithmetic#Congruence_class|congruence classes]] [[relatively prime]] to the modulus number, say ''n'', form a [[group (mathematics)|group]] under multiplication called the '''multiplicative group of integers modulo ''n'''''. It is also called the group of '''primitive residue classes modulo ''n'''''. In the [[ring (algebra)|theory of rings]], a branch of [[abstract algebra]], it is described as the group of [[Unit (ring theory)|units]] of the ring of integers modulo ''n''. (Units refers to elements with a [[Modular multiplicative inverse|multiplicative inverse]].)
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| This group is fundamental in [[number theory]]. It has found applications in [[cryptography]], [[integer factorization]], and [[primality test]]ing. For example, by finding the order of this group, one can determine whether ''n'' is prime: ''n'' is prime [[if and only if]] the order is {{nowrap|''n'' − 1}}.
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| ==Group axioms==
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| It is a straightforward exercise to show that, under multiplication, the set of [[congruence class]]es modulo ''n'' which are relatively prime to ''n'' satisfy the axioms for an [[abelian group]].
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| Because {{nowrap|''a'' ≡ ''b'' (mod ''n'')}} implies that {{nowrap|1=gcd(''a'', ''n'') = gcd(''b'', ''n'')}}, the notion of congruence classes modulo ''n'' which are relatively prime to ''n'' is well-defined.
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| Since {{nowrap|1=gcd(''a'', ''n'') = 1}} and {{nowrap|1=gcd(''b'', ''n'') = 1}} implies {{nowrap|1=gcd(''ab'', ''n'') = 1}} the set of classes relatively prime to ''n'' is closed under multiplication.
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| The natural mapping from the integers to the congruence classes modulo ''n'' that takes an integer to its congruence class modulo ''n'' respects products. This implies that the class containing 1 is the unique multiplicative identity, and also the associative and commutative laws hold. In fact it is a [[ring homomorphism]].
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| Given ''a'', {{nowrap|1=gcd(''a'', ''n'') = 1}}, finding ''x'' satisfying {{nowrap|''ax'' ≡ 1 (mod ''n'')}} is the same as solving {{nowrap|1=''ax'' + ''ny'' = 1}}, which can be done by [[Bézout's lemma]]. The ''x'' found will have the property that {{nowrap|1=gcd(''x'', ''n'') = 1}}.
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| ==Notation==
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| The [[Factor_ring#Examples|(quotient) ring]] of integers modulo ''n'' is denoted <math>\mathbb{Z}/n\mathbb{Z}</math> or <math>\mathbb{Z}/(n)</math> (i.e., the ring of integers modulo the [[Ideal (ring theory)|ideal]] <!-- <math>n \mathbb{Z} = (n)</math> --> <math>n\mathbb{Z} = (n)</math> consisting of the multiples of ''n'') or by <math>\mathbb{Z}_n</math> (though the latter can be confused with the [[p-adic number|{{math|<var>p</var>}}-adic integers]] in the case <math>n=p</math>). Depending on the author, its group of units may be written <math>(\mathbb{Z}/n\mathbb{Z})^*,</math> <math>(\mathbb{Z}/n\mathbb{Z})^\times,</math> <math>U(\mathbb{Z}/n\mathbb{Z}),</math> <math>E(\mathbb{Z}/n\mathbb{Z})</math> (for German ''Einheit'' = unit) or similar notations. This article uses <math>(\mathbb{Z}/n\mathbb{Z})^\times.</math>
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| The notation <math>\mathrm{C}_n</math> refers to the [[cyclic group]] of order ''n''.
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| ==Structure==
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| ===''n'' = 1===
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| Modulo 1 any two integers are congruent, i.e. there is only one congruence class. Every integer is relatively prime to 1. Therefore the single congruence class modulo 1 is relatively prime to the modulus, so <math>(\mathbb{Z}/1\,\mathbb{Z})^\times \cong \mathrm{C}_1</math> is trivial. This implies that {{nowrap|1=φ(1) = 1}}. Since the first power of any integer is congruent to 1 modulo 1, λ(1) is also 1.
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| Because of its trivial nature, the case of congruences modulo 1 is generally ignored. For example, the theorem "<math>(\mathbb{Z}/n\mathbb{Z})^\times</math> is cyclic if and only if {{nowrap|1=φ(''n'') = λ(''n'')}}" implicitly includes the case {{nowrap|1=''n'' = 1}}, whereas the usual statement of Gauss's theorem "<math>(\mathbb{Z}/n\mathbb{Z})^\times</math> is cyclic if and only if ''n'' = 2, 4, any power of an odd prime or twice any power of an odd prime," explicitly excludes 1.
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| ===Powers of 2===
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| Modulo 2 there is only one relatively prime congruence class, 1, so <math>(\mathbb{Z}/2\mathbb{Z})^\times \cong \mathrm{C}_1</math> is the [[trivial group]].
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| Modulo 4 there are two relatively prime congruence classes, 1 and 3, so <math>(\mathbb{Z}/4\mathbb{Z})^\times \cong \mathrm{C}_2,</math> the cyclic group with two elements.
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| Modulo 8 there are four relatively prime classes, 1, 3, 5 and 7. The square of each of these is 1, so <math>(\mathbb{Z}/8\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_2,</math> the [[Klein four-group]].
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| Modulo 16 there are eight relatively prime classes 1, 3, 5, 7, 9, 11, 13 and 15. <math>\{\pm 1, \pm 7\}\cong \mathrm{C}_2 \times \mathrm{C}_2,</math> is the 2-torsion subgroup (i.e. the square of each element is 1), so <math>(\mathbb{Z}/16\mathbb{Z})^\times</math> is not cyclic. The powers of 3, <math>\{1, 3, 9, 11\}</math> are a subgroup of order 4, as are the powers of 5, <math>\{1, 5, 9, 13\}.</math> Thus <math>(\mathbb{Z}/16\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_4.</math>
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| The pattern shown by 8 and 16 holds<ref>Gauss, DA, arts. 90–91</ref> for higher powers 2<sup>''k''</sup>, {{nowrap|''k'' > 2}}: <math>\{\pm 1, 2^{k-1} \pm 1\}\cong \mathrm{C}_2 \times \mathrm{C}_2,</math> is the 2-torsion subgroup (so <math>(\mathbb{Z}/2^k\mathbb{Z})^\times </math> is not cyclic) and the powers of 3 are a subgroup of order 2<sup>''k'' − 2</sup>, so <math>(\mathbb{Z}/2^k\mathbb{Z})^\times \cong \mathrm{C}_2 \times \mathrm{C}_{2^{k-2}}.</math>
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| ===Powers of odd primes===
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| For powers of odd primes ''p''<sup>''k''</sup> the group is cyclic:<ref>Gauss, DA, arts. 52–56, 82–89</ref>
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| :<math> (\mathbb{Z}/p^k\mathbb{Z})^\times \cong \mathrm{C}_{p^{k-1}(p-1)} \cong \mathrm{C}_{\varphi(p^k)} .</math>
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| ===General composite numbers===
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| The [[Chinese remainder theorem]]<ref>Riesel covers all of this. pp. 267–275</ref> says that if <math>\;\;n=p_1^{k_1}p_2^{k_2}p_3^{k_3}\dots, \;</math> then the ring <math>\mathbb{Z}/n\mathbb{Z}</math> is the [[Product of rings|direct product]] of the rings corresponding to each of its prime power factors:
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| :<math>\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/{p_1^{k_1}}\mathbb{Z}\; \times \;\mathbb{Z}/{p_2^{k_2}}\mathbb{Z} \;\times\; \mathbb{Z}/{p_3^{k_3}}\mathbb{Z}\dots\;\;</math>
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| Similarly, the group of units <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> is the direct product of the groups corresponding to each of the prime power factors:
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| :<math>(\mathbb{Z}/n\mathbb{Z})^\times\cong (\mathbb{Z}/{p_1^{k_1}}\mathbb{Z})^\times \times (\mathbb{Z}/{p_2^{k_2}}\mathbb{Z})^\times \times (\mathbb{Z}/{p_3^{k_3}}\mathbb{Z})^\times \dots\;.</math>
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| ====Subgroup of false witnesses====
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| If ''n'' is composite, there exists a subgroup of the multiplicative group, called the "group of false witnesses", in which the elements, when raised to the power {{nowrap|''n'' − 1}}, are congruent to 1 modulo ''n'' (since the residue 1, to any power, is congruent to 1 modulo ''n'', the set of such elements is nonempty).<ref>{{cite journal | zbl=0586.10003 | last1=Erdős | first1=Paul | author1-link=Paul Erdős | last2=Pomerance | first2=Carl | author2-link=Carl Pomerance | title=On the number of false witnesses for a composite number | journal=Math. Comput. | volume=46 | pages=259–279 | year=1986 }}</ref> One could say, because of [[Fermat's Little Theorem]], that such residues are "false positives" or "false witnesses" for the primality of ''n''. 2 is the residue most often used in this basic primality check, hence {{nowrap|1=341 = 11 × 31}} is famous since 2<sup>340</sup> is congruent to 1 modulo 341, and 341 is the smallest such composite number (with respect to 2). For 341, the false witnesses subgroup contains 100 residues and so is of index 3 inside the 300 element multiplicative group mod 341.
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| =====Examples=====
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| ; ''n'' = 9
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| The smallest example with a nontrivial subgroup of false witnesses is {{nowrap|1=9 = 3 × 3}}. There are 6 residues relatively prime to 9: 1, 2, 4, 5, 7, 8. Since 8 is congruent to {{nowrap|−1 modulo 9}}, it follows that 8<sup>8</sup> is congruent to 1 modulo 9. So 1 and 8 are false positives for the "primality" of 9 (since 9 is not actually prime). These are in fact the only ones, so the subgroup {1,8} is the subgroup of false witnesses. The same argument shows that {{nowrap|''n'' − 1}} is a "false witness" for any odd composite ''n''.
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| ; ''n'' = 561
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| 561 is a [[Carmichael number]], thus ''n''<sup>560</sup> is congruent to 1 modulo 561 for any number ''n'' coprime to 561. Thus the subgroup of false witnesses is in this case not proper, it is the entire group of multiplicative units modulo 561, which consists of 320 residues.
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| ==Properties==
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| ===Order===
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| The order of the group is given by [[Euler's totient function]]: <math>| (\mathbb{Z}/n\mathbb{Z})^\times|=\varphi(n).</math> This is the product of the orders of the cyclic groups in the direct product.
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| ===Exponent===
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| The exponent is given by the [[Carmichael function]] <math>\lambda(n),</math> the [[least common multiple]] of the orders of the cyclic groups. This means that given ''n'', <math>a^{\lambda(n)} \equiv 1 \pmod n,</math> for any ''a'' relatively prime to ''n'', and <math>\lambda(n)</math> is the smallest such number.
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| ===Generators===
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| <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> is cyclic if and only if <math>\varphi(n)=\lambda(n).</math> This is the case when ''n'' is 2, 4, ''p''<sup>''k''</sup> or 2''p''<sup>''k''</sup>, where ''p'' is an odd prime and {{nowrap|''k'' > 0}}. For all other values of ''n'' (except 1) the group is not cyclic.<ref>{{MathWorld|title=Modulo Multiplication Group|urlname=ModuloMultiplicationGroup}}
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| </ref><ref>[http://www.encyclopediaofmath.org/index.php/Primitive_root Primitive root], [[Encyclopedia of Mathematics]]</ref> The single generator in the cyclic case is called a '''[[primitive root modulo n]]'''.
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| Since all the <math>(\mathbb{Z}/n\mathbb{Z})^\times,</math> {{nowrap|''n'' ≤ 7}} are cyclic, another way to state this is: If {{nowrap|''n'' < 8}} then <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> has a primitive root. If {{nowrap|''n'' ≥ 8}} then <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> has a primitive root unless ''n'' is divisible by 4 or by two distinct odd primes.
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| In the general case there is one generator for each cyclic direct factor.
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| ==Examples==
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| This table shows the cyclic decomposition of <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> and a [[Generating set of a group|generating set]] for small values of ''n''. The generating sets are not unique; e.g. modulo 16 both {{nowrap|{−1, 3} }} and {{nowrap|{−1, 5} }} will work. The generators are listed in the same order as the direct factors.
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| For example take {{nowrap|1=''n'' = 20}}. <math>\varphi(20)=8</math> means that the order of <math>(\mathbb{Z}/20\mathbb{Z})^\times</math> is 8 (i.e. there are 8 numbers less than 20 and coprime to it); <math>\lambda(20)=4</math> that the fourth power of any number relatively prime to 20 is {{nowrap|≡ 1 (mod 20)}}; and as for the generators, 19 has order 2, 3 has order 4, and every member of <math>(\mathbb{Z}/20\mathbb{Z})^\times</math> is of the form {{nowrap|19<sup>''a''</sup> × 3<sup>''b''</sup>}}, where ''a'' is 0 or 1 and ''b'' is 0, 1, 2, or 3.
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| The powers of 19 are {±1} and the powers of 3 are {3, 9, 7, 1}. The latter and their negatives modulo 20, {17, 11, 13, 19} are all the numbers less than 20 and prime to it. The fact that the order of 19 is 2 and the order of 3 is 4 implies that the fourth power of every member of <math>\mathbb{Z}_{20}^\times</math> is ≡ 1 (mod 20).
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| {| class="wikitable" style="text-align:center" cellpadding="2"
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| |+ Group Structure of '''(Z/''n''Z)<sup>×</sup>'''
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| |-
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| ! <math>n\;</math> || <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> || <math>\varphi(n)</math> || <math>\lambda(n)\;</math> || generating set
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| | width="25" rowspan="33" |
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| ! <math>n\;</math> || <math>(\mathbb{Z}/n\mathbb{Z})^\times</math> || <math>\varphi(n)</math> || <math>\lambda(n)\;</math> || generating set
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| |-
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| ! 1
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| | C<sub>1</sub> || 1 || 1 || 1
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| ! 33
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| | C<sub>2</sub>×C<sub>10</sub> || 20 || 10 || 10, 2
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| |-
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| ! 2
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| | C<sub>1</sub> || 1 || 1 || 1
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| ! 34
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| | C<sub>16</sub> || 16 || 16 || 3
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| |-
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| ! 3
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| | C<sub>2</sub> || 2 || 2 || 2
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| ! 35
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| | C<sub>2</sub>×C<sub>12</sub> || 24 || 12 || 6, 2
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| |-
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| ! 4
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| | C<sub>2</sub> || 2 || 2 || 3
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| ! 36
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| | C<sub>2</sub>×C<sub>6</sub> || 12 || 6 || 19, 5
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| |-
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| ! 5
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| | C<sub>4</sub> || 4 || 4 || 2
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| ! 37
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| | C<sub>36</sub> || 36 || 36 || 2
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| |-
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| ! 6
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| | C<sub>2</sub> || 2 || 2 || 5
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| ! 38
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| | C<sub>18</sub> || 18 || 18 || 3
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| |-
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| ! 7
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| | C<sub>6</sub> || 6 || 6 || 3
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| ! 39
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| | C<sub>2</sub>×C<sub>12</sub> || 24 || 12 || 38, 2
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| |-
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| ! 8
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| | C<sub>2</sub>×C<sub>2</sub> || 4 || 2 || 7, 3
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| ! 40
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| | C<sub>2</sub>×C<sub>2</sub>×C<sub>4</sub> || 16 || 4 || 39, 11, 3
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| |-
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| ! 9
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| | C<sub>6</sub> || 6 || 6 || 2
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| ! 41
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| | C<sub>40</sub> || 40 || 40 || 6
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| |-
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| ! 10
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| | C<sub>4</sub> || 4 || 4 || 3
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| ! 42
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| | C<sub>2</sub>×C<sub>6</sub> || 12 || 6 || 13, 5
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| |-
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| ! 11
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| | C<sub>10</sub> || 10 || 10 || 2
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| ! 43
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| | C<sub>42</sub> || 42 || 42 || 3
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| |-
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| ! 12
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| | C<sub>2</sub>×C<sub>2</sub> || 4 || 2 || 5, 7
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| ! 44
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| | C<sub>2</sub>×C<sub>10</sub> || 20 || 10 || 43, 3
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| |-
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| ! 13
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| | C<sub>12</sub> || 12 || 12 || 2
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| ! 45
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| | C<sub>2</sub>×C<sub>12</sub> || 24 || 12 || 44, 2
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| |-
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| ! 14
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| | C<sub>6</sub> || 6 || 6 || 3
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| ! 46
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| | C<sub>22</sub> || 22 || 22 || 5
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| |-
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| ! 15
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| | C<sub>2</sub>×C<sub>4</sub> || 8 || 4 || 14, 2
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| ! 47
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| | C<sub>46</sub> || 46 || 46 || 5
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| |-
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| ! 16
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| | C<sub>2</sub>×C<sub>4</sub> || 8 || 4 || 15, 3
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| ! 48
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| | C<sub>2</sub>×C<sub>2</sub>×C<sub>4</sub> || 16 || 4 || 47, 7, 5
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| |-
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| ! 17
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| | C<sub>16</sub> || 16 || 16 || 3
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| ! 49
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| | C<sub>42</sub> || 42 || 42 || 3
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| |-
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| ! 18
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| | C<sub>6</sub> || 6 || 6 || 5
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| ! 50
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| | C<sub>20</sub> || 20 || 20 || 3
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| |-
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| ! 19
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| | C<sub>18</sub> || 18 || 18 || 2
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| ! 51
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| | C<sub>2</sub>×C<sub>16</sub> || 32 || 16 || 50, 5
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| |-
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| ! 20
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| | C<sub>2</sub>×C<sub>4</sub> || 8 || 4 || 19, 3
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| ! 52
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| | C<sub>2</sub>×C<sub>12</sub> || 24 || 12 || 51, 7
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| |-
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| ! 21
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| | C<sub>2</sub>×C<sub>6</sub> || 12 || 6 || 20, 2
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| ! 53
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| | C<sub>52</sub> || 52 || 52 || 2
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| |-
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| ! 22
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| | C<sub>10</sub> || 10 || 10 || 7
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| ! 54
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| | C<sub>18</sub> || 18 || 18 || 5
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| |-
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| ! 23
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| | C<sub>22</sub> || 22 || 22 || 5
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| ! 55
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| | C<sub>2</sub>×C<sub>20</sub> || 40 || 20 || 21, 2
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| |-
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| ! 24
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| | C<sub>2</sub>×C<sub>2</sub>×C<sub>2</sub> || 8 || 2 || 5, 7, 13
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| ! 56
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| | C<sub>2</sub>×C<sub>2</sub>×C<sub>6</sub> || 24 || 6 || 13, 29, 3
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| |-
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| ! 25
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| | C<sub>20</sub> || 20 || 20 || 2
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| ! 57
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| | C<sub>2</sub>×C<sub>18</sub> || 36 || 18 || 20, 2
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| |-
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| ! 26
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| | C<sub>12</sub> || 12 || 12 || 7
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| ! 58
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| | C<sub>28</sub> || 28 || 28 || 3
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| |-
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| ! 27
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| | C<sub>18</sub> || 18 || 18 || 2
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| ! 59
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| | C<sub>58</sub> || 58 || 58 || 2
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| |-
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| ! 28
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| | C<sub>2</sub>×C<sub>6</sub> || 12 || 6 || 13, 3
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| ! 60
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| | C<sub>2</sub>×C<sub>2</sub>×C<sub>4</sub> || 16 || 4 || 11, 19, 7
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| |-
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| ! 29
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| | C<sub>28</sub> || 28 || 28 || 2
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| ! 61
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| | C<sub>60</sub> || 60 || 60 || 2
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| |-
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| ! 30
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| | C<sub>2</sub>×C<sub>4</sub> || 8 || 4 || 11, 7
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| ! 62
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| | C<sub>30</sub> || 30 || 30 || 3
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| |-
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| ! 31
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| | C<sub>30</sub> || 30 || 30 || 3
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| ! 63
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| | C<sub>6</sub>×C<sub>6</sub> || 36 || 6 || 2, 5
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| |-
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| ! 32
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| | C<sub>2</sub>×C<sub>8</sub> || 16 || 8 || 31, 3
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| ! 64
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| | C<sub>2</sub>×C<sub>16</sub> || 32 || 16 || 63, 3
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| |}
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| ==See also==
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| * [[Lenstra elliptic curve factorization]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Clarke | first2 = Arthur A. (translator into English)
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| | title = Disquisitiones Arithemeticae (Second, corrected edition)
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| | publisher = [[Springer Science+Business Media|Springer]]
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| | location = New York
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| | year = 1986
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| | isbn = 0-387-96254-9}}
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| *{{citation
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| | last1 = Gauss | first1 = Carl Friedrich
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| | last2 = Maser | first2 = H. (translator into German)
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| | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
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| | publisher = Chelsea
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| | location = New York
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| | year = 1965
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| | isbn = 0-8284-0191-8}}
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| *{{citation
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| | last1 = Riesel | first1 = Hans
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| | title = Prime Numbers and Computer Methods for Factorization (second edition)
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| | publisher = Birkhäuser
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| | location = Boston
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| | year = 1994
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| | isbn = 0-8176-3743-5}}
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| ==External links==
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| *[http://www.lombok.demon.co.uk/maths/MultiGrpModN.html Calculator] by Shing Hing Man
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| [[Category:Modular arithmetic]]
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| [[Category:Group theory]]
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| [[Category:Finite groups]]
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| [[Category:Multiplication]]
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