LCM & GCD - Revision history

Khanh: /* Calculation */

2021-12-11T17:47:39Z

Calculation

Khanh: /* Coprime numbers */

2021-12-11T17:42:08Z

Coprime numbers

Khanh: /* Definition */

2021-12-11T17:41:27Z

Definition

Khanh: /* In commutative rings */

2021-12-11T17:39:32Z

In commutative rings

Khanh: /* In commutative rings */

2021-12-11T17:37:44Z

In commutative rings

Khanh at 17:34, 11 December 2021

2021-12-11T17:34:38Z

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Khanh: Created page with "= Least common multiple = File:Symmetrical_5-set_Venn_diagram_LCM_2_3_4_5_7.svg|thumb|250px|A Venn diagram showing the least common multiples of combinations of 2, 3, 4, 5 a..."

2021-12-11T07:51:09Z

Created page with "= Least common multiple = File:Symmetrical_5-set_Venn_diagram_LCM_2_3_4_5_7.svg|thumb|250px|A Venn diagram showing the least common multiples of combinations of 2, 3, 4, 5 a..."

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= Least common multiple =
[[File:Symmetrical_5-set_Venn_diagram_LCM_2_3_4_5_7.svg|thumb|250px|A Venn diagram showing the least common multiples of combinations of 2, 3, 4, 5 and 7 (6 is skipped as it is 2 × 3, both of which are already represented). For example, a card game which requires its cards to be divided equally among up to 5 players requires at least 60 cards, the number at the intersection of the 2, 3, 4, and 5 sets, but not the 7 set.]]

In arithmetic and number theory, the '''least common multiple''', '''lowest common multiple''', or '''smallest common multiple''' of two integers ''a'' and ''b'', usually denoted by '''lcm(''a'', ''b'')''', is the smallest positive integer that is divisible by both ''a'' and ''b''. Since division of integers by zero is undefined, this definition has meaning only if ''a'' and ''b'' are both different from zero. However, some authors define lcm(''a'',0) as 0 for all ''a'', which is the result of taking the lcm to be the least upper bound in the lattice of divisibility.

The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The lcm of more than two integers is also well-defined: it is the smallest positive integer that is divisible by each of them.

== Overview ==
A multiple of a number is the product of that number and an integer. For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.

=== Notation ===
The least common multiple of two integers ''a'' and ''b'' is denoted as lcm(''a'', ''b''). Some older textbooks use [''a'', ''b''].

=== Example ===
:<math>\operatorname{lcm}(4, 6)</math>

Multiples of 4 are:

:<math> 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, ...</math>

Multiples of 6 are:

:<math> 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, ...</math>

''Common multiples'' of 4 and 6 are the numbers that are in both lists:

:<math> 12, 24, 36, 48, 60, 72, ...</math>

In this list, the smallest number is 12. Hence, the ''least common multiple'' is 12.

== Applications ==
When adding, subtracting, or comparing simple fractions, the least common multiple of the denominators (often called the lowest common denominator) is used, because each of the fractions can be expressed as a fraction with this denominator. For example,

:<math>{2\over21}+{1\over6}={4\over42}+{7\over42}={11\over42}</math>

where the denominator 42 was used, because it is the least common multiple of 21 and 6.

=== Gears problem ===
Suppose there are two meshing gears in a machine, having ''m'' and ''n'' teeth, respectively, and the gears are marked by a line segment drawn from the center of the first gear to the center of the second gear. When the gears begin rotating, the number of rotations the first gear must complete to realign the line segment can be calculated by using <math>\operatorname{lcm}(m, n)</math>. The first gear must complete <math>\operatorname{lcm}(m, n)\over m</math> rotations for the realignment. By that time, the second gear will have made <math>\operatorname{lcm}(m, n)\over n</math> rotations.

=== Planetary alignment ===
Suppose there are three planets revolving around a star which take ''l'', ''m'' and ''n'' units of time, respectively, to complete their orbits. Assume that ''l'', ''m'' and ''n'' are integers. Assuming the planets started moving around the star after an initial linear alignment, all the planets attain a linear alignment again after <math>\operatorname{lcm}(l, m, n)</math> units of time. At this time, the first, second and third planet will have completed <math>\operatorname{lcm}(l, m, n)\over l</math>, <math>\operatorname{lcm}(l, m, n)\over m</math> and <math>\operatorname{lcm}(l, m, n)\over n</math> orbits, respectively, around the star.

== Calculation ==

=== Using the greatest common divisor ===
The following formula reduces the problem of computing the least common multiple to the problem of computing the greatest common divisor (gcd), also known as the greatest common factor:

:<math>\operatorname{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)}.</math>

This formula is also valid when exactly one of ''a'' and ''b'' is 0, since gcd(''a'', 0) = |''a''|. However, if both ''a'' and ''b'' are 0, this formula would cause division by zero; lcm(0, 0) = 0 is a special case.

There are fast algorithms for computing the gcd that do not require the numbers to be factored, such as the Euclidean algorithm. To return to the example above,

:<math>\operatorname{lcm}(21,6)
={21\cdot6\over\gcd(21,6)}
={21\cdot6\over\gcd(3,6)}
={21\cdot 6\over 3}= \frac{126}{3} = 42.</math>

Because gcd(''a'', ''b'') is a divisor of both ''a'' and ''b'', it is more efficient to compute the lcm by dividing ''before'' multiplying:

:<math>\operatorname{lcm}(a,b)=\left({|a|\over\gcd(a,b)}\right)\cdot |b|=\left({|b|\over\gcd(a,b)}\right)\cdot |a|.</math>

This reduces the size of one input for both the division and the multiplication, and reduces the required storage needed for intermediate results (that is, overflow in the ''a''×''b'' computation). Because gcd(''a'', ''b'') is a divisor of both ''a'' and ''b'', the division is guaranteed to yield an integer, so the intermediate result can be stored in an integer. Implemented this way, the previous example becomes:

:<math>\operatorname{lcm}(21,6)={21\over\gcd(21,6)}\cdot6={21\over\gcd(3,6)}\cdot6={21\over3}\cdot6=7\cdot6=42.</math>

=== Using prime factorization ===
The unique factorization theorem indicates that every positive integer greater than 1 can be written in only one way as a product of prime numbers. The prime numbers can be considered as the atomic elements which, when combined, make up a composite number.

For example:

:<math>90 = 2^1 \cdot 3^2 \cdot 5^1 = 2 \cdot 3 \cdot 3 \cdot 5. </math>

Here, the composite number 90 is made up of one atom of the prime number 2, two atoms of the prime number 3, and one atom of the prime number 5.

This fact can be used to find the lcm of a set of numbers.

Example: lcm(8,9,21)

Factor each number and express it as a product of prime number powers.

: <math>
\begin{align}
8 & = 2^3 \\
9 & = 3^2 \\
21 & = 3^1 \cdot 7^1
\end{align}
</math>

The lcm will be the product of multiplying the highest power of each prime number together. The highest power of the three prime numbers 2, 3, and 7 is 23, 32, and 71, respectively. Thus,

:<math>\operatorname{lcm}(8,9,21) = 2^3 \cdot 3^2 \cdot 7^1 = 8 \cdot 9 \cdot 7 = 504. </math>

This method is not as efficient as reducing to the greatest common divisor, since there is no known general efficient algorithm for integer factorization.

The same method can also be illustrated with a Venn diagram as follows, with the prime factorization of each of the two numbers demonstrated in each circle and ''all'' factors they share in common in the intersection. The lcm then can be found by multiplying all of the prime numbers in the diagram.

Here is an example:

: 48 = 2 × 2 × 2 × 2 × 3,
: 180 = 2 × 2 × 3 × 3 × 5,

sharing two "2"s and a "3" in common:

:[[Image:least common multiple.svg|400px]]

: Least common multiple = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720
: Greatest common divisor = 2 × 2 × 3 = 12

This also works for the greatest common divisor (gcd), except that instead of multiplying all of the numbers in the Venn diagram, one multiplies only the prime factors that are in the intersection. Thus the gcd of 48 and 180 is 2 × 2 × 3 = 12.

=== Using a simple algorithm ===
This method works easily for finding the lcm of several integers.

Let there be a finite sequence of positive integers ''X'' = (''x''1, ''x''2, ..., ''x''''n''), ''n'' > 1. The algorithm proceeds in steps as follows: on each step ''m'' it examines and updates the sequence ''X''(''m'') = (''x''1(''m''), ''x''2(''m''), ..., ''x''''n''(''m'')), ''X''(1) = ''X'', where ''X''(''m'') is the ''m''th iteration of ''X'', that is, ''X'' at step ''m'' of the algorithm, etc. The purpose of the examination is to pick the least (perhaps, one of many) element of the sequence ''X''(''m''). Assuming ''x''''k''0(''m'') is the selected element, the sequence ''X''(''m''+1) is defined as

: ''x''''k''(''m''+1) = ''x''''k''(''m''), ''k'' ≠ ''k''0
: ''x''''k''0(''m''+1) = ''x''''k''0(''m'') + ''x''''k''0(1).

In other words, the least element is increased by the corresponding ''x'' whereas the rest of the elements pass from ''X''(''m'') to ''X''(''m''+1) unchanged.

The algorithm stops when all elements in sequence ''X''(''m'') are equal. Their common value ''L'' is exactly lcm(''X'').

For example, if ''X'' = ''X''(1) = (3, 4, 6), the steps in the algorithm produce:
:''X''(2) = (6, 4, 6)
:''X''(3) = (6, 8, 6)
:''X''(4) = (6, 8, 12) - by choosing the second 6
:''X''(5) = (9, 8, 12)
:''X''(6) = (9, 12, 12)
:''X''(7) = (12, 12, 12) so lcm = 12.

=== Using the table-method ===
This method works for any number of numbers. One begins by listing all of the numbers vertically in a table (in this example 4, 7, 12, 21, and 42):
<div style="font-family: monospace;">
: 4
: 7
: 12
: 21
: 42
</div>
The process begins by dividing all of the numbers by 2. If 2 divides any of them evenly, write 2 in a new column at the top of the table, and the result of division by 2 of each number in the space to the right in this new column. If a number is not evenly divisible, just rewrite the number again. If 2 does not divide evenly into any of the numbers, repeat this procedure with the next largest prime number, 3 (see below).

{| class="wikitable" style="text-align:right"
|-
! ×
! 2
|-
| 4
| '''2'''
|-
| 7
| 7
|-
| 12
| '''6'''
|-
| 21
| 21
|-
| 42
| '''21'''
|}

Now, assuming that 2 did divide at least one number (as in this example), check if 2 divides again:

{| class="wikitable" style="text-align:right"
|-
! ×
! 2
! 2
|-
| 4
| '''2'''
| '''1'''
|-
| 7
| 7
| 7
|-
| 12
| '''6'''
| '''3'''
|-
| 21
| 21
| 21
|-
| 42
| '''21'''
| 21
|}

Once 2 no longer divides any number in the current column, repeat the procedure by dividing by the next larger prime, 3. Once 3 no longer divides, try the next larger primes, 5 then 7, etc. The process ends when all of the numbers have been reduced to 1 (the column under the last prime divisor consists only of 1's).

{| class="wikitable" style="text-align:right"
|-
! ×
! 2
! 2
! 3
! 7
|-
| 4
| '''2'''
| '''1'''
| 1
| 1
|-
| 7
| 7
| 7
| 7
| '''1'''
|-
| 12
| '''6'''
| '''3'''
| '''1'''
| 1
|-
| 21
| 21
| 21
| '''7'''
| '''1'''
|-
| 42
| '''21'''
| 21
| '''7'''
| '''1'''
|}

Now, multiply the numbers in the top row to obtain the lcm. In this case, it is 2 × 2 × 3 × 7 = 84.

As a general computational algorithm, the above is quite inefficient. One would never want to implement it in software: it takes too many steps and requires too much storage space. A far more efficient numerical algorithm can be obtained by using Euclid's algorithm to compute the gcd first, and then obtaining the lcm by division.

== Formulas ==

=== Fundamental theorem of arithmetic ===
According to the fundamental theorem of arithmetic, a positive integer is the product of prime numbers, and this representation is unique up to the ordering of prime numbers:

:<math>n = 2^{n_2} 3^{n_3} 5^{n_5} 7^{n_7} \cdots = \prod_p p^{n_p},</math>

where the exponents ''n''2, ''n''3, ... are non-negative integers; for example, 84 = 22 31 50 71 110 130 ...

Given two positive integers <math display="inline">a = \prod_p p^{a_p}</math> and <math display="inline">b = \prod_p p^{b_p}</math>, their least common multiple and greatest common divisor are given by the formulas
:<math>\gcd(a,b) = \prod_p p^{\min(a_p, b_p)}</math>

and
:<math>\operatorname{lcm}(a,b) = \prod_p p^{\max(a_p, b_p)}.</math>

Since
:<math>\min(x,y) + \max(x,y) = x + y,</math>
this gives
:<math>\gcd(a,b) \operatorname{lcm}(a,b) = ab.</math>

In fact, every rational number can be written uniquely as the product of primes, if negative exponents are allowed. When this is done, the above formulas remain valid. For example:
:<math>\begin{align}
4 &= 2^2 3^0, & 6 &= 2^1 3^1, & \gcd(4, 6) &= 2^1 3^0 = 2, & \operatorname{lcm}(4,6) &= 2^2 3^1 = 12. \\[8pt]
\tfrac{1}{3} &= 2^0 3^{-1} 5^0, & \tfrac{2}{5} &= 2^1 3^0 5^{-1}, & \gcd\left(\tfrac13, \tfrac{2}{5}\right) &= 2^0 3^{-1} 5^{-1} = \tfrac{1}{15}, & \operatorname{lcm}\left(\tfrac{1}{3}, \tfrac{2}{5}\right) &= 2^1 3^0 5^0 = 2, \\[8pt]
\tfrac{1}{6} &= 2^{-1} 3^{-1}, & \tfrac{3}{4} &= 2^{-2} 3^1, & \gcd\left(\tfrac{1}{6}, \tfrac{3}{4}\right) &= 2^{-2} 3^{-1} = \tfrac{1}{12}, & \operatorname{lcm}\left(\tfrac{1}{6}, \tfrac{3}{4}\right) &= 2^{-1} 3^1 = \tfrac{3}{2}.
\end{align}</math>

=== Lattice-theoretic ===
The positive integers may be partially ordered by divisibility: if ''a'' divides ''b'' (that is, if ''b'' is an integer multiple of ''a'') write ''a'' ≤ ''b'' (or equivalently, ''b'' ≥ ''a''). (Note that the usual magnitude-based definition of ≤ is not used here.)

Under this ordering, the positive integers become a lattice, with meet given by the gcd and join given by the lcm. The proof is straightforward, if a bit tedious; it amounts to checking that lcm and gcd satisfy the axioms for meet and join. Putting the lcm and gcd into this more general context establishes a duality between them:

:''If a formula involving integer variables, gcd, lcm, ≤ and ≥ is true, then the formula obtained by switching gcd with lcm and switching ≥ with ≤ is also true.'' (Remember ≤ is defined as divides).

The following pairs of dual formulas are special cases of general lattice-theoretic identities.

{| style="margin:0;" cellpadding="0" border="0" cellspacing="0"
|
;Commutative laws
:<math>\operatorname{lcm}(a, b) = \operatorname{lcm}(b, a),</math>
:<math>\gcd(a, b) =\gcd( b, a).</math>
|     
|
;Associative laws
:<math>\operatorname{lcm}(a,\operatorname{lcm}(b, c)) = \operatorname{lcm}(\operatorname{lcm}(a , b),c),</math>
:<math>\gcd(a, \gcd(b, c)) = \gcd(\gcd(a,b), c).</math>
|     
|
;Absorption laws:
:<math>\operatorname{lcm}(a, \gcd(a,b)) = a,</math>
:<math>\gcd(a, \operatorname{lcm}(a, b)) = a.</math>
|}

{| style="margin:0;" cellpadding="0" border="0" cellspacing="0"
|
;Idempotent laws
:<math>\operatorname{lcm}(a, a) = a,</math>
:<math>\gcd(a, a) = a.</math>
|     
|
;Define divides in terms of lcm and gcd
:<math>a \ge b \iff a = \operatorname{lcm}(a,b),</math>
:<math>a \le b \iff a = \gcd(a,b).</math>
|}

It can also be shown that this lattice is distributive; that is, lcm distributes over gcd and gcd distributes over lcm:

:<math>\operatorname{lcm}(a,\gcd(b,c)) = \gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c)),</math>
:<math>\gcd(a,\operatorname{lcm}(b,c)) = \operatorname{lcm}(\gcd(a,b),\gcd(a,c)).</math>

This identity is self-dual:
:<math>\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(b,c),\operatorname{lcm}(a,c))=\operatorname{lcm}(\gcd(a,b),\gcd(b,c),\gcd(a,c)).</math>

=== Other ===
* Let ''D'' be the product of ''ω''(''D'') distinct prime numbers (that is, ''D'' is squarefree).

Then<ref>Crandall & Pomerance, ex. 2.4, p. 101.</ref>

:<math>|\{(x,y) \;:\; \operatorname{lcm}(x,y) = D\}| = 3^{\omega(D)},</math>

where the absolute bars || denote the cardinality of a set.

* If none of <math>a_1, a_2, \ldots , a_r</math> is zero, then

:<math>\operatorname{lcm}(a_1, a_2, \ldots , a_r) = \operatorname{lcm}(\operatorname{lcm}(a_1, a_2, \ldots , a_{r-1}), a_r). </math>

== In commutative rings ==
The least common multiple can be defined generally over commutative rings as follows: Let ''a'' and ''b'' be elements of a commutative ring ''R''. A common multiple of ''a'' and ''b'' is an element ''m'' of ''R'' such that both ''a'' and ''b'' divide ''m'' (that is, there exist elements ''x'' and ''y'' of ''R'' such that ''ax'' = ''m'' and ''by'' = ''m''). A least common multiple of ''a'' and ''b'' is a common multiple that is minimal, in the sense that for any other common multiple ''n'' of ''a'' and ''b'', ''m'' divides ''n''.

In general, two elements in a commutative ring can have no least common multiple or more than one. However, any two least common multiples of the same pair of elements are associates. In a unique factorization domain, any two elements have a least common multiple. In a principal ideal domain, the least common multiple of ''a'' and ''b'' can be characterised as a generator of the intersection of the ideals generated by ''a'' and ''b'' (the intersection of a collection of ideals is always an ideal).

== Licensing ==
Content obtained and/or adapted from:
* [https://en.wikipedia.org/wiki/Least_common_multiple Least common multiple, Wikipedia] under a CC BY-SA license
* [https://en.wikipedia.org/wiki/Greatest_common_divisor Greatest common divisor, Wikipedia] under a CC BY-SA license

@@ Line 384: / Line 384: @@
 === Using prime factorizations ===
-Greatest common divisors can be computed by determining the [[prime factorization]]s of the two numbers and comparing factors. For example, to compute gcd(48,&nbsp;180), we find the prime factorizations 48&nbsp;=&nbsp;2<sup>4</sup>&nbsp;·&nbsp;3<sup>1</sup> and 180&nbsp;=&nbsp;2<sup>2</sup>&nbsp;·&nbsp;3<sup>2</sup>&nbsp;·&nbsp;5<sup>1</sup>; the GCD is then 2<sup>min(4,2)</sup>&nbsp;·&nbsp;3<sup>min(1,2)</sup>&nbsp;·&nbsp;5<sup>min(0,1)</sup> = 2<sup>2</sup>&nbsp;·&nbsp;3<sup>1</sup>&nbsp;·&nbsp;5<sup>0</sup>  =&nbsp;12, as shown in the Venn diagram. The corresponding LCM is then
+Greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors. For example, to compute gcd(48,&nbsp;180), we find the prime factorizations 48&nbsp;=&nbsp;2<sup>4</sup>&nbsp;·&nbsp;3<sup>1</sup> and 180&nbsp;=&nbsp;2<sup>2</sup>&nbsp;·&nbsp;3<sup>2</sup>&nbsp;·&nbsp;5<sup>1</sup>; the GCD is then 2<sup>min(4,2)</sup>&nbsp;·&nbsp;3<sup>min(1,2)</sup>&nbsp;·&nbsp;5<sup>min(0,1)</sup> = 2<sup>2</sup>&nbsp;·&nbsp;3<sup>1</sup>&nbsp;·&nbsp;5<sup>0</sup>  =&nbsp;12, as shown in the Venn diagram. The corresponding LCM is then
 <sup>max(4,2)</sup>&nbsp;·&nbsp;3<sup>max(1,2)</sup>&nbsp;·&nbsp;5<sup>max(0,1)</sup> =
 <sup>4</sup>&nbsp;·&nbsp;3<sup>2</sup>&nbsp;·&nbsp;5<sup>1</sup>  =&nbsp;720.
@@ Line 404: / Line 404: @@
 ===Euclidean algorithm===
 A more efficient method is the ''Euclidean algorithm'', a variant in which the difference of the two numbers {{mvar|a}} and {{mvar|b}} is replaced by the ''remainder'' of the Euclidean division (also called ''division with remainder'') of {{mvar|a}} by {{mvar|b}}.
@@ Line 470: / Line 468: @@
 :<math>\gcd(a,b)=\sum\limits_{k=1}^a \exp (2\pi ikb/a) \cdot \sum\limits_{d\left| a\right.} \frac{c_d (k)}{d} </math>
-is an [[entire function]] in the variable ''b'' for all positive integers ''a'' where ''c''<sub>''d''</sub>(''k'') is Ramanujan's sum.
+is an entire function in the variable ''b'' for all positive integers ''a'' where ''c''<sub>''d''</sub>(''k'') is Ramanujan's sum.
 === Complexity===

@@ Line 363: / Line 363: @@
 === Coprime numbers ===
-Two numbers are called relatively prime, or [[coprime]], if their greatest common divisor equals 1. For example, 9 and 28 are Coprime.
+Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1. For example, 9 and 28 are Coprime.
 === A geometric view ===

@@ Line 341: / Line 341: @@
 The ''greatest common divisor'' (GCD) of two nonzero integers {{mvar|a}} and {{mvar|b}}  is the greatest positive integer {{mvar|d}} such that {{mvar|d}} is a divisor of both {{mvar|a}} and {{mvar|b}}; that is, there are integers {{mvar|e}} and {{mvar|f}} such that {{math|1=''a'' = ''de''}} and {{math|1=''b'' = ''df''}}, and {{mvar|d}} is the largest such integer. The GCD of {{mvar|a}} and {{mvar|b}} is generally denoted {{math|gcd(''a'', ''b'')}}.
-This definition also applies when one of {{mvar|a}} and {{mvar|b}} is zero. In this case, the GCD is the absolute value of the non zero integer: {{math|1=gcd(''a'', 0) = gcd(0, ''a'') = {{abs|''a''}}}}. This case is important as the terminating step of the Euclidean algorithm.
+This definition also applies when one of {{mvar|a}} and {{mvar|b}} is zero. In this case, the GCD is the absolute value of the non zero integer: gcd(''a'', 0) = gcd(0, ''a'') = |''a''|. This case is important as the terminating step of the Euclidean algorithm.
-The above definition cannot be used for defining {{math|gcd(0, 0)}}, since {{math|1=0 × ''n'' = 0}}, and zero thus has no greatest divisor. However, zero is its own greatest divisor if ''greatest'' is understood in the context of the divisibility relation, so {{math|gcd(0, 0)}} is commonly defined as 0. This preserves the usual identities for GCD, and in particular Bézout's identity, namely that {{math|gcd(''a'', ''b'')}} generates the same ideal as {{math|{{mset|''a'', ''b''}}}}. This convention is followed by many computer algebra systems. Nonetheless, some authors leave {{math|gcd(0, 0)}} undefined.
+The above definition cannot be used for defining {{math|gcd(0, 0)}}, since {{math|1=0 × ''n'' = 0}}, and zero thus has no greatest divisor. However, zero is its own greatest divisor if ''greatest'' is understood in the context of the divisibility relation, so {{math|gcd(0, 0)}} is commonly defined as 0. This preserves the usual identities for GCD, and in particular Bézout's identity, namely that {{math|gcd(''a'', ''b'')}} generates the same ideal as {''a'', ''b''}. This convention is followed by many computer algebra systems. Nonetheless, some authors leave {{math|gcd(0, 0)}} undefined.
 The GCD of {{mvar|a}} and {{mvar|b}} is their greatest positive common divisor in the preorder relation of divisibility. This means that the common divisors of {{mvar|a}} and {{mvar|b}} are exactly the divisors of their GCD. This is commonly proved by using either Euclid's lemma, the fundamental theorem of arithmetic, or the Euclidean algorithm. This is the meaning of "greatest" that is used for the generalizations of the concept of GCD.

@@ Line 538: / Line 538: @@
 :<math>R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.</math>
-The elements 2 and 1&nbsp;+&nbsp;<math>\sqrt{−3}</math> are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1&nbsp;+&nbsp;<math>\sqrt{−3}</math>, but they are not associated, so there is no greatest common divisor of {{mvar|a}} and&nbsp;''b''.
+The elements 2 and 1&nbsp;+&nbsp;<math>\sqrt{-3}</math> are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + <math>\sqrt{-3}</math>, but they are not associated, so there is no greatest common divisor of {{mvar|a}} and&nbsp;''b''.
 Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form ''pa''&nbsp;+&nbsp;''qb'', where {{mvar|p}} and {{mvar|q}} range over the ring. This is the ideal generated by {{mvar|a}} and {{mvar|b}}, and is denoted simply (''a'',&nbsp;''b''). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element ''d''; then this {{mvar|d}} is a greatest common divisor of {{mvar|a}} and ''b''. But the ideal (''a'',&nbsp;''b'') can be useful even when there is no greatest common divisor of {{mvar|a}} and ''b''. (Indeed, Ernst Kummer used this ideal as a replacement for a GCD in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ''ideal'', ring element {{mvar|d}}, whence the ring-theoretic term.)

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 :<math>R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.</math>
-The elements 2 and 1&nbsp;+&nbsp;{{sqrt|−3}} are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1&nbsp;+&nbsp;{{sqrt|−3}}, but they are not associated, so there is no greatest common divisor of {{mvar|a}} and&nbsp;''b''.
+The elements 2 and 1&nbsp;+&nbsp;<math>\sqrt{−3}</math> are two maximal common divisors (that is, any common divisor which is a multiple of 2 is associated to 2, the same holds for 1&nbsp;+&nbsp;<math>\sqrt{−3}</math>, but they are not associated, so there is no greatest common divisor of {{mvar|a}} and&nbsp;''b''.
-Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form ''pa''&nbsp;+&nbsp;''qb'', where {{mvar|p}} and {{mvar|q}} range over the ring. This is the ideal generated by {{mvar|a}} and {{mvar|b}}, and is denoted simply (''a'',&nbsp;''b''). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element ''d''; then this {{mvar|d}} is a greatest common divisor of {{mvar|a}} and ''b''. But the ideal (''a'',&nbsp;''b'') can be useful even when there is no greatest common divisor of {{mvar|a}} and ''b''. (Indeed, Ernst Kummer used this ideal as a replacement for a GCD in his treatment of [[Fermat's Last Theorem]], although he envisioned it as the set of multiples of some hypothetical, or ''ideal'', ring element {{mvar|d}}, whence the ring-theoretic term.)
+Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form ''pa''&nbsp;+&nbsp;''qb'', where {{mvar|p}} and {{mvar|q}} range over the ring. This is the ideal generated by {{mvar|a}} and {{mvar|b}}, and is denoted simply (''a'',&nbsp;''b''). In a ring all of whose ideals are principal (a principal ideal domain or PID), this ideal will be identical with the set of multiples of some ring element ''d''; then this {{mvar|d}} is a greatest common divisor of {{mvar|a}} and ''b''. But the ideal (''a'',&nbsp;''b'') can be useful even when there is no greatest common divisor of {{mvar|a}} and ''b''. (Indeed, Ernst Kummer used this ideal as a replacement for a GCD in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ''ideal'', ring element {{mvar|d}}, whence the ring-theoretic term.)
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