Greatest common divisor

Template:Use American English Template:Short description In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers Template:Math, Template:Math, the greatest common divisor of Template:Math and Template:Math is denoted <math>\gcd (x,y)</math>. For example, the GCD of 8 and 12 is 4, that is, Template:Math.<ref name="Long 1972 33">Template:Harvtxt</ref><ref name="Pettofrezzo 1970 34">Template:Harvtxt</ref>

In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc.<ref> Template:Cite book.</ref><ref>Template:Cite book.</ref><ref name="Hardy&Wright 1979 20" /><ref>Some authors treat Template:Vanchor as synonymous with greatest common divisor. This contradicts the common meaning of the words that are used, as denominator refers to fractions, and two fractions do not have any greatest common denominator (if two fractions have the same denominator, one obtains a greater common denominator by multiplying all numerators and denominators by the same integer).</ref> Historically, other names for the same concept have included greatest common measure.<ref> Template:Cite book.</ref>

This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see Template:Section link below).

Overview

Definition

The greatest common divisor (GCD) of integers Template:Mvar and Template:Mvar, at least one of which is nonzero, is the greatest positive integer Template:Mvar such that Template:Mvar is a divisor of both Template:Mvar and Template:Mvar; that is, there are integers Template:Mvar and Template:Mvar such that Template:Math and Template:Math, and Template:Mvar is the largest such integer. The GCD of Template:Mvar and Template:Mvar is generally denoted Template:Math.Template:Refn

When one of Template:Math and Template:Math is zero, the GCD is the absolute value of the nonzero integer: Template:Math. This case is important as the terminating step of the Euclidean algorithm.

The above definition is unsuitable for defining Template:Math, since there is no greatest integer Template:Math such that Template:Math. However, zero is its own greatest divisor if greatest is understood in the context of the divisibility relation, so Template:Math is commonly defined as Template:Math. This preserves the usual identities for GCD, and in particular Bézout's identity, namely that Template:Math generates the same ideal as Template:Math.<ref>Thomas H. Cormen, et al., Introduction to Algorithms (2nd edition, 2001) Template:Isbn, p. 852</ref><ref>Bernard L. Johnston, Fred Richman, Numbers and Symmetry: An Introduction to Algebra Template:Isbn, p. 38</ref><ref>Martyn R. Dixon, et al., An Introduction to Essential Algebraic Structures Template:Isbn, p. 59</ref> This convention is followed by many computer algebra systems.<ref>e.g., Wolfram Alpha calculation and Maxima</ref> Nonetheless, some authors leave Template:Math undefined.<ref>Jonathan Katz, Yehuda Lindell, Introduction to Modern Cryptography Template:Isbn, 2020, section 9.1.1, p. 45</ref>

The GCD of Template:Mvar and Template:Mvar is their greatest positive common divisor in the preorder relation of divisibility. This means that the common divisors of Template:Mvar and Template:Mvar 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.

Example

The number 54 can be expressed as a product of two integers in several different ways:

<math> 54 \times 1 = 27 \times 2 = 18 \times 3 = 9 \times 6.</math>

Thus the complete list of divisors of 54 is 1, 2, 3, 6, 9, 18, 27, 54. Similarly, the divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. The numbers that these two lists have in common are the common divisors of 54 and 24, that is,

<math> 1, 2, 3, 6. </math>

Of these, the greatest is 6, so it is the greatest common divisor:

<math> \gcd(54,24) = 6. </math>

Computing all divisors of the two numbers in this way is usually not efficient, especially for large numbers that have many divisors. Much more efficient methods are described in Template:Slink.

Coprime numbers

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Two numbers are called relatively prime, or coprime, if their greatest common divisor equals Template:Math.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> For example, 9 and 28 are coprime.

A geometric view

File:24x60.svg

A 24-by-60 rectangle is covered with ten 12-by-12 square tiles, where 12 is the GCD of 24 and 60. More generally, an a-by-b rectangle can be covered with square tiles of side length c only if c is a common divisor of a and b.

For example, a 24-by-60 rectangular area can be divided into a grid of: 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, 6-by-6 squares or 12-by-12 squares. Therefore, 12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can thus be divided into a grid of 12-by-12 squares, with two squares along one edge (Template:Math) and five squares along the other (Template:Math).

Applications

Reducing fractions

Template:Further The greatest common divisor is useful for reducing fractions to the lowest terms.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> For example, Template:Math, therefore,

<math>\frac{42}{56}=\frac{3 \cdot 14 }{ 4 \cdot 14}=\frac{3 }{ 4}.</math>

Least common multiple

Template:Further The least common multiple of two integers that are not both zero can be computed from their greatest common divisor, by using the relation

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

Calculation

Using prime factorizations

Greatest common divisors can be computed by determining the prime factorizations of the two numbers and comparing factors. For example, to compute Template:Math, we find the prime factorizations 48 = 2⁴ · 3¹ and 180 = 2² · 3² · 5¹; the GCD is then 2^min(4,2) · 3^min(1,2) · 5^min(0,1) = 2² · 3¹ · 5⁰ = 12 The corresponding LCM is then 2^max(4,2) · 3^max(1,2) · 5^max(0,1) = 2⁴ · 3² · 5¹ = 720.

In practice, this method is only feasible for small numbers, as computing prime factorizations takes too long.

Euclid's algorithm

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The method introduced by Euclid for computing greatest common divisors is based on the fact that, given two positive integers Template:Mvar and Template:Mvar such that Template:Math, the common divisors of Template:Mvar and Template:Mvar are the same as the common divisors of Template:Math and Template:Mvar.

So, Euclid's method for computing the greatest common divisor of two positive integers consists of replacing the larger number with the difference of the numbers, and repeating this until the two numbers are equal: that is their greatest common divisor.

For example, to compute Template:Math, one proceeds as follows:

<math>\begin{align}\gcd(48,18)\quad&\to\quad \gcd(48-18, 18)= \gcd(30,18)\\

&\to \quad \gcd(30-18, 18)= \gcd(12,18)\\ &\to \quad \gcd(12,18-12)= \gcd(12,6)\\ &\to \quad \gcd(12-6,6)= \gcd(6,6).\end{align}</math> So Template:Math.

This method can be very slow if one number is much larger than the other. So, the variant that follows is generally preferred.

Euclidean algorithm

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:The Great Common Divisor of 62 and 36 is 2.ogv

Animation showing an application of the Euclidean algorithm to find the greatest common divisor of 62 and 36, which is 2.

A more efficient method is the Euclidean algorithm, a variant in which the difference of the two numbers Template:Mvar and Template:Mvar is replaced by the remainder of the Euclidean division (also called division with remainder) of Template:Mvar by Template:Mvar.

Denoting this remainder as Template:Math, the algorithm replaces Template:Math with Template:Math repeatedly until the pair is Template:Math, where Template:Mvar is the greatest common divisor.

For example, to compute gcd(48,18), the computation is as follows:

<math>\begin{align}\gcd(48,18)\quad&\to\quad \gcd(18, 48\bmod 18)= \gcd(18, 12)\\

&\to \quad \gcd(12, 18\bmod 12)= \gcd(12,6)\\ &\to \quad \gcd(6,12\bmod 6)= \gcd(6,0).\end{align}</math> This again gives Template:Math.

Binary GCD algorithm

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

The binary GCD algorithm is a variant of Euclid's algorithm that is specially adapted to the binary representation of the numbers, which is used in most computers.

The binary GCD algorithm differs from Euclid's algorithm essentially by dividing by two every even number that is encountered during the computation. Its efficiency results from the fact that, in binary representation, testing parity consists of testing the right-most digit, and dividing by two consists of removing the right-most digit.

The method is as follows, starting with Template:Math and Template:Math that are the two positive integers whose GCD is sought.

1. If Template:Math and Template:Math are both even, then divide both by two until at least one of them becomes odd; let Template:Mvar be the number of these paired divisions.
2. If Template:Math is even, then divide it by two until it becomes odd.
3. If Template:Math is even, then divide it by two until it becomes odd.

   Now, Template:Math and Template:Math are both odd and will remain odd until the end of the computation

4. While Template:Math do
   • If Template:Math, then replace Template:Mvar with Template:Math and divide the result by two until Template:Mvar becomes odd (as Template:Math and Template:Math are both odd, there is, at least, one division by 2).
   • If Template:Math, then replace Template:Mvar with Template:Math and divide the result by two until Template:Mvar becomes odd.
5. Now, Template:Math, and the greatest common divisor is <math>2^d a.</math>

Step 1 determines Template:Mvar as the highest power of Template:Math that divides Template:Math and Template:Math, and thus their greatest common divisor. None of the steps changes the set of the odd common divisors of Template:Math and Template:Math. This shows that when the algorithm stops, the result is correct. The algorithm stops eventually, since each steps divides at least one of the operands by at least Template:Math. Moreover, the number of divisions by Template:Math and thus the number of subtractions is at most the total number of digits.

Example: (a, b, d) = (48, 18, 0) → (24, 9, 1) → (12, 9, 1) → (6, 9, 1) → (3, 9, 1) → (3, 3, 1) ; the original GCD is thus the product 6 of Template:Math and Template:Math.

The binary GCD algorithm is particularly easy to implement and particularly efficient on binary computers. Its computational complexity is

<math>O((\log a + \log b)^2).</math>

The square in this complexity comes from the fact that division by Template:Math and subtraction take a time that is proportional to the number of bits of the input.

The computational complexity is usually given in terms of the length Template:Math of the input. Here, this length is Template:Math, and the complexity is thus

<math>O(n^2)</math>.

Lehmer's GCD algorithm

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Lehmer's algorithm is based on the observation that the initial quotients produced by Euclid's algorithm can be determined based on only the first few digits; this is useful for numbers that are larger than a computer word. In essence, one extracts initial digits, typically forming one or two computer words, and runs Euclid's algorithms on these smaller numbers, as long as it is guaranteed that the quotients are the same with those that would be obtained with the original numbers. The quotients are collected into a small 2-by-2 transformation matrix (a matrix of single-word integers) to reduce the original numbers. This process is repeated until numbers are small enough that the binary algorithm (see below) is more efficient.

This algorithm improves speed, because it reduces the number of operations on very large numbers, and can use hardware arithmetic for most operations. In fact, most of the quotients are very small, so a fair number of steps of the Euclidean algorithm can be collected in a 2-by-2 matrix of single-word integers. When Lehmer's algorithm encounters a quotient that is too large, it must fall back to one iteration of Euclidean algorithm, with a Euclidean division of large numbers.

Other methods

File:Greatest common divisor.png

Thomae's function

If Template:Math and Template:Math are both nonzero, the greatest common divisor of Template:Math and Template:Math can be computed by using least common multiple (LCM) of Template:Math and Template:Math:

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

but more commonly the LCM is computed from the GCD.

Using Thomae's function Template:Math,

<math>\gcd(a,b) = a f\left(\frac b a\right),</math>

which generalizes to Template:Math and Template:Math rational numbers or commensurable real numbers.

Keith Slavin has shown that for odd Template:Math:

<math>\gcd(a,b)=\log_2\prod_{k=0}^{a-1} (1+e^{-2i\pi k b/a})</math>

which is a function that can be evaluated for complex b.<ref>Template:Cite journal</ref> Wolfgang Schramm has shown that

<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_d(k) is Ramanujan's sum.<ref>Template:Cite journal</ref>

Complexity

The computational complexity of the computation of greatest common divisors has been widely studied.<ref>Template:Cite book</ref> If one uses the Euclidean algorithm and the elementary algorithms for multiplication and division, the computation of the greatest common divisor of two integers of at most Template:Mvar bits is Template:Math.Template:Cn This means that the computation of greatest common divisor has, up to a constant factor, the same complexity as the multiplication.

However, if a fast multiplication algorithm is used, one may modify the Euclidean algorithm for improving the complexity, but the computation of a greatest common divisor becomes slower than the multiplication. More precisely, if the multiplication of two integers of Template:Math bits takes a time of Template:Math, then the fastest known algorithm for greatest common divisor has a complexity Template:Math. This implies that the fastest known algorithm has a complexity of Template:Math.

Previous complexities are valid for the usual models of computation, specifically multitape Turing machines and random-access machines.

The computation of the greatest common divisors belongs thus to the class of problems solvable in quasilinear time. A fortiori, the corresponding decision problem belongs to the class P of problems solvable in polynomial time. The GCD problem is not known to be in NC, and so there is no known way to parallelize it efficiently; nor is it known to be P-complete, which would imply that it is unlikely to be possible to efficiently parallelize GCD computation. Shallcross et al. showed that a related problem (EUGCD, determining the remainder sequence arising during the Euclidean algorithm) is NC-equivalent to the problem of integer linear programming with two variables; if either problem is in NC or is P-complete, the other is as well.<ref>Template:Cite book</ref> Since NC contains NL, it is also unknown whether a space-efficient algorithm for computing the GCD exists, even for nondeterministic Turing machines.

Although the problem is not known to be in NC, parallel algorithms asymptotically faster than the Euclidean algorithm exist; the fastest known deterministic algorithm is by Chor and Goldreich, which (in the CRCW-PRAM model) can solve the problem in Template:Math time with Template:Math processors.<ref>Template:Cite journal</ref> Randomized algorithms can solve the problem in Template:Math time on <math>\exp\left(O\left(\sqrt{n \log n}\right)\right)</math> processors Template:Clarify (this is superpolynomial).<ref>Template:Cite conference</ref>

Properties

• For positive integers Template:Math, Template:Math.
• Every common divisor of Template:Math and Template:Math is a divisor of Template:Math.
• Template:Math, where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form Template:Math, where p and q are integers. This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm.
• Template:Math, for Template:Math, since any number is a divisor of 0, and the greatest divisor of a is Template:Math.<ref name="Pettofrezzo 1970 34" /><ref name="Hardy&Wright 1979 20">Template:Harvtxt</ref> This is usually used as the base case in the Euclidean algorithm.
• If a divides the product b⋅c, and Template:Math, then a/d divides c.
• If m is a positive integer, then Template:Math.
• If m is any integer, then Template:Math. Equivalently, Template:Math.
• If m is a positive common divisor of a and b, then Template:Math.
• If Template:Math, then Template:Math.
• The GCD is a commutative function: Template:Math.
• The GCD is an associative function: Template:Math. Thus Template:Math can be used to denote the GCD of multiple arguments.
• The GCD is a multiplicative function in the following sense: if a₁ and a₂ are relatively prime, then Template:Math.
• Template:Math is closely related to the least common multiple Template:Math: we have

  Template:Math.

This formula is often used to compute least common multiples: one first computes the GCD with Euclid's algorithm and then divides the product of the given numbers by their GCD.

• The following versions of distributivity hold true:

  Template:Math

  Template:Math.

• If we have the unique prime factorizations of Template:Math and Template:Math where Template:Math and Template:Math, then the GCD of a and b is

  Template:Math.

• It is sometimes useful to define Template:Math and Template:Math because then the natural numbers become a complete distributive lattice with GCD as meet and LCM as join operation.<ref>

Template:Cite book. Footnote 27, p. 9: "For example, the natural numbers with gcd (greatest common divisor) as meet and lcm (least common multiple) as join operation determine a (complete distributive) lattice." Including these definitions for 0 is necessary for this result: if one instead omits 0 from the set of natural numbers, the resulting lattice is not complete.</ref> This extension of the definition is also compatible with the generalization for commutative rings given below.

• In a Cartesian coordinate system, Template:Math can be interpreted as the number of segments between points with integral coordinates on the straight line segment joining the points Template:Math and Template:Math.
• For non-negative integers Template:Math and Template:Math, where Template:Math and Template:Math are not both zero, provable by considering the Euclidean algorithm in base n:<ref>Template:Cite book</ref>

  Template:Math.

• An identity involving Euler's totient function:

  <math> \gcd(a,b) = \sum_{k|a \text{ and }k|b} \varphi(k) .</math>

• GCD Summatory function (Pillai's arithmetical function):

<math>\sum_{k=1}^n \gcd(k,n) = \sum_{d|n} d \varphi \left( \frac n d \right) =n\sum_{d|n}\frac{\varphi(d)}{d} =n\prod_{p|n}\left(1+\nu_p(n)\left(1-\frac{1}{p}\right)\right)</math> where <math>\nu_p(n)</math> is the Template:Math-adic valuation. (sequence A018804 in the OEIS)

Probabilities and expected value

In 1972, James E. Nymann showed that Template:Math integers, chosen independently and uniformly from Template:Math, are coprime with probability Template:Math as Template:Math goes to infinity, where Template:Math refers to the Riemann zeta function.<ref name="nymann">Template:Cite journal</ref> (See coprime for a derivation.) This result was extended in 1987 to show that the probability that Template:Math random integers have greatest common divisor Template:Math is Template:Math.<ref name="chid">Template:Cite journal</ref>

Using this information, the expected value of the greatest common divisor function can be seen (informally) to not exist when Template:Math. In this case the probability that the GCD equals Template:Math is Template:Math, and since Template:Math we have

<math>\mathrm{E}( \mathrm{2} ) = \sum_{d=1}^\infty d \frac{6}{\pi^2 d^2} = \frac{6}{\pi^2} \sum_{d=1}^\infty \frac{1}{d}.</math>

This last summation is the harmonic series, which diverges. However, when Template:Math, the expected value is well-defined, and by the above argument, it is

<math> \mathrm{E}(k) = \sum_{d=1}^\infty d^{1-k} \zeta(k)^{-1} = \frac{\zeta(k-1)}{\zeta(k)}. </math>

For Template:Math, this is approximately equal to 1.3684. For Template:Math, it is approximately 1.1106.

In commutative rings

The notion of greatest common divisor can more generally be defined for elements of an arbitrary commutative ring, although in general there need not exist one for every pair of elements.<ref>Template:Cite book</ref>

• If Template:Mvar is a commutative ring, and Template:Mvar and Template:Mvar are in Template:Mvar, then an element Template:Mvar of Template:Mvar is called a common divisor of Template:Mvar and Template:Mvar if it divides both Template:Mvar and Template:Mvar (that is, if there are elements Template:Mvar and Template:Mvar in Template:Mvar such that d·x = a and d·y = b).
• If Template:Mvar is a common divisor of Template:Mvar and Template:Mvar, and every common divisor of Template:Mvar and Template:Mvar divides Template:Mvar, then Template:Mvar is called a greatest common divisor of Template:Mvar and b.

With this definition, two elements Template:Mvar and Template:Mvar may very well have several greatest common divisors, or none at all. If Template:Mvar is an integral domain, then any two GCDs of Template:Mvar and Template:Mvar must be associate elements, since by definition either one must divide the other. Indeed, if a GCD exists, any one of its associates is a GCD as well.

Existence of a GCD is not assured in arbitrary integral domains. However, if Template:Mvar is a unique factorization domain or any other GCD domain, then any two elements have a GCD. If Template:Mvar is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance when Template:Math where Template:Mvar is a field, or when Template:Mvar is the ring of Gaussian integers), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

The following is an example of an integral domain with two elements that do not have a GCD:

<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 Template:Math and Template:Math are two maximal common divisors (that is, any common divisor which is a multiple of Template:Math is associated to Template:Math, the same holds for Template:Math, but they are not associated, so there is no greatest common divisor of Template:Mvar and Template:Math.

Corresponding to the Bézout property we may, in any commutative ring, consider the collection of elements of the form Template:Math, where Template:Mvar and Template:Mvar range over the ring. This is the ideal generated by Template:Mvar and Template:Mvar, and is denoted simply Template:Math. 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 Template:Math; then this Template:Mvar is a greatest common divisor of Template:Mvar and Template:Math. But the ideal Template:Math can be useful even when there is no greatest common divisor of Template:Mvar and Template:Math. (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 Template:Mvar, whence the ring-theoretic term.)

See also

• Bézout domain
• Lowest common denominator
• Unitary divisor

References

• Template:Cite book
• Template:Cite book
• Template:Cite book
• Template:Cite book

Further reading

• Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. Template:ISBN. Section 4.5.2: The Greatest Common Divisor, pp. 333–356.
• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. Template:ISBN. Section 31.2: Greatest common divisor, pp. 856–862.
• Saunders Mac Lane and Garrett Birkhoff. A Survey of Modern Algebra, Fourth Edition. MacMillan Publishing Co., 1977. Template:ISBN. 1–7: "The Euclidean Algorithm."

External links

• gcd(x,y) = y function graph: https://www.desmos.com/calculator/6nizzenog5