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Topic: Greatest common divisor

	Greatest common divisor - Wikipedia, the free encyclopedia
		In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers.
		The greatest common divisor of a and b is written as gcd(a, b), or sometimes simply as (a, b).
		The greatest common divisor is useful for reducing vulgar fractions to be in lowest terms.
	en.wikipedia.org /wiki/Greatest_common_factor (1078 words)

	Greatest common divisor
		The GCD of a and b is often written as gcd(a,b).
		The GCD of 0 and 0 is usually defined to be 0.
		The greatest common divisor of a and b (not both zero) may be defined alternatively and equivalently as follows: it is the smallest positive integer d which can be written in the form ap+bq where p and q are integers.
	www.wordlookup.net /gr/greatest-common-divisor.html (655 words)

	Euclidean algorithm - Wikipedia, the free encyclopedia
		The Euclidean algorithm (also called Euclid's algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers.
		An alternative algorithm, the binary GCD algorithm, exploits the binary representation used by computers to avoid divisions and thereby increase efficiency, although it too is O(n²); it merely shrinks the constant hidden by the big-O notation on many real machines.
		The most general context in which the algorithm terminates with the greatest common divisor is the so-called Euclidean domain.
	en.wikipedia.org /wiki/Euclidean_algorithm (1243 words)

	PlanetMath: greatest common divisor
		Once the gcd is found, it can be written as a linear combination of the two integers.
		The idea of the gcd of two integers can be generalized in another direction: to the gcd of a set of integers.
		This is version 12 of greatest common divisor, born on 2001-10-16, modified 2006-07-16.
	planetmath.org /encyclopedia/GreatestCommonDivisor.html (170 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-06)
		The GCD of two numbers is the same as the GCD of the smaller of them and their difference.
		For instance, the GCD of 200 and 196, or GCD(200, 196), is the same as GCD(196, 4), where 4 = 200 - 196.
		GCD 60 and 48 is 12, since 12 goes into both of them evenly, but nothing larger than 12 does.] So, GCD(2046, 1122) = GCD(1122, 2046 - 1122) = GCD(1122, 924) [that is a little easier already.
	mathforum.org /library/drmath/view/52436.html (732 words)

	Least common multiple - Wikipedia, the free encyclopedia
		In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b.
		When adding or subtracting vulgar fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator.
		If a and b are not both zero, the least common multiple can be computed by using the greatest common divisor (gcd) of a and b:
	en.wikipedia.org /wiki/Least_common_multiple (468 words)

	calculate greatest common divisor
		The biggest common divisor of two numbers is generally called the greatest common divisor of the two numbers.
		and 25 is the dividend, 7 the divisor, 3 the quotient and 4 the remainder.
		Since the greatest common divisor of 9 and 25 is 1 we know that there will be an answer.
	www.softmath.com /tutorials2/calculate-greatest-common-divisor.html (2308 words)

	Cryptography Tutorial - The Euclidean Algorithm finds the Greatest Common Divisor of two Integers (Site not responding. Last check: 2007-11-06)
		We encountered that some ciphers require the knowledge of the greatest common divisor of two integers, others require the usage of two integers with a 1 as a common divisor.
		The greatest common divisor of integers a and b, denoted by gcd(a,b), is the largest integer that divides (without remainder) both a and b.
		We repeatedly divide the divisor by the remainder until the remainder is 0.
	www.antilles.k12.vi.us /math/cryptotut/euclidean_algorithm.htm (561 words)

	Great Common Divisor (Site not responding. Last check: 2007-11-06)
		of two positive integers a and b, sometimes written (a, b), is the largest divisor common to a and b.
		The greatest common divisor of a and b is implemented in
		However, even for Euclidean rings, the notion of GCD of two elements of a ring is not the same as the GCD of two ideals of a ring.
	www.cs.kent.edu /~yizhou/great_common_divisor.htm (330 words)

	Two properties of Greatest Common Divisor
		Greatest Common Divisor is one of the best known arithmetic notions.
		The following statement (which is implied by the first one) asserts the central role played by gcd among linear combinations as + bt.
		Note that as + bt = gcd(a,b) implies that any common divisor of a and b also divide gcd(a,b).
	www.cut-the-knot.org /Generalization/gcd.shtml (330 words)

	Greatest Common Divisors
		The greatest common divisor of f and g in a multivariate polynomial ring P. If either of the inputs is zero, then the result is the other input (and if the inputs are both zero then the result is zero).
		The content of f, that is, the greatest common divisor of the coefficients of f as an element of the coefficient ring.
		The content (the greatest common divisor of the coefficients) of f, as an element of the coefficient ring, as well as the primitive part (f divided by the content) of f.
	www.umich.edu /~gpcc/scs/magma/text595.htm (494 words)

	[No title]
		This is connected by a simple formula with the greatest common divisor of the two integers, a familiar topic from modern algebra and number theory.
		The purpose of this paper is to present a proof for the connection between least common multiple and greatest common divisor.
		Because c is a common multiple of a and b, we know both a and b are divisors of c.
	www.american.edu /academic.depts/cas/mathstat/People/kalman/pdffiles/assessment/sample.doc (701 words)

	Greatest Common Divisors
		We say that integers a and b have a common divisor c if c is a divisor if both a and b.
		c is a common divisor of a and b; and
		We have already met an algorithm to compute the greatest common divisor of two integers in Section 2.5.3, namely the Euclidean algorithm.
	www.maths.abdn.ac.uk /~igc/tch/mx3015/notes/node150.html (429 words)

	Math 511, Greatest Common Divisors
		One is that we have defined a greatest common divisor, not the greatest common divisor.
		The usual definition of a = gcd(e,f) is that a is a common divisor of e and f such that if b is any other common divisor of e and f, then ba.
		It is traditional to write the greatest common divisor of a pair of polynomials with a leading constant of 1.
	www.math.ksu.edu /math511/notes/1109.html (676 words)

	[Tutor] Help Needed [greatest common divisor] (Site not responding. Last check: 2007-11-06)
		Here's an example of such a fraction: 42 -- 8 A common divisor is some number D that we can use to divide a group of numbers.
		For example, if we have 42 and 8, a common divisor of those two numbers could be 2, since 2 divides 42 and it also divides 8.
		The "Greatest Common Divisor" is just the largest number we can use that will divide two numbers neatly.
	mail.python.org /pipermail/tutor/2002-April/013845.html (308 words)

	Some Number Theory
		The greatest common divisor and the Euclidean algorithm
		We have already met the greatest common divisor, or gcd, which is the largest integer which divides both of a pair of numbers.
		Then since d is a divisor of a and a divisor of b, it is also a divisor of a - bk.
	www.math.sunysb.edu /~scott/Book331/Some_Number_Theory.html (1440 words)

	Greatest Common Divisors (Site not responding. Last check: 2007-11-06)
		The greatest common divisor of two integers is the largest integer that is a divisor of both.
		Multiplying the primes that are common to both numbers gives the gcd.
		The primes common to both 396 and 120 are 2, 2, 3.
	sierra.nmsu.edu /morandi/CourseMaterials/GCD.html (366 words)

	Amby's Math Resources - Greatest Common Factors
		The Greatest Common Factor (G.C.F.) of two numbers is the largest number that is a divisor of both.
		Don't let the "greatest" in the name fool you - the GCF is no larger than the smallest of the numbers.
		Sometimes only the number '1' will work as a common divisor; for example: 21 and 16 have no common factor other than 1.
	amby.com /educate/math/2-1_GCF.html (440 words)

	Greatest Common Divisor (Site not responding. Last check: 2007-11-06)
		greatest common divisor definition of greatest common divisor in computing dicti...
		GCD - The Greatest Common Divisor of two integers...
		Greatest Common Divisor, Least Common Multiple, Decimal To Fraction for the HP-6...
	www.scienceoxygen.com /math/10.html (127 words)

	2 Greatest common divisor (Site not responding. Last check: 2007-11-06)
		This algorithm is perhaps the most common operation that is used in computing with large integers after the basic operations.
		It is also the first an algorithm was written (since this algorithm was written before the base 10 arithmetic operations that we saw earlier).
		The fundamental question is to find the greatest common divisor or highest common factor d of a pair a, b of integers.
	www.imsc.res.in /~kapil/crypto/notes/node8.html (133 words)

	Greatest Common Divisor and Least Common Multiple
		The value of that member is the greatest common factor or least common multiple, respectively, of the template arguments.
		The program common_factor_test.cpp is a demonstration of the results from instantiating various examples of the run-time GCD and LCM function templates and the compile-time GCD and LCM class templates.
		The greatest common divisor and least common multiple functions are greatly used in some numeric contexts, including some of the other Boost libraries.
	www.boost.org /regression-logs/cs-win32_metacomm/doc/html/boost_math/greatest_common_divisor_and_least_common_multiple.html (697 words)

	Greatest Common Divisors (Site not responding. Last check: 2007-11-06)
		The greatest common divisor (GCD) of two integers a and b is defined to be the largest integer that divides both a and b with no remainder.
		For example, the GCD of 16 and 28 is 4.
		The idea of the algorithm is based on the observation that, if r is the remainder when a is divided by b, then the common divisors of a and b are precisely the same as the common divisors of b and r.
	mitpress.mit.edu /sicp/chapter1/node16.html (449 words)

	Greatest Common Divisor, Least Common Multiple, Decimal To Fraction for the HP-67
		This routine is independent of the first, although both share a common subroutine, LBL e.
		For greatest common divisor, go to step 3; for least common multiple, go to step 5; for decimal to fraction, go to step 6.
		Key in integers a and b and find their greatest common divisor.
	www.hpmuseum.org /software/67pacs/67gcdlcm.htm (959 words)

	greatest common divisor calculator
		We have an extensive database of resources on greatest common divisor calculator.
		#!/bin/bash # gcd.sh: greatest common divisor # Uses Euclid's algorithm # The "greatest common divisor" (gcd) of two integers #+ is the largest integer that will divide both, leaving no remainder.
		gcd $1 $2 echo; echo "GCD of $1 and $2 = $dividend"; echo # Exercise : # -------- # Check command-line arguments to make sure they are integers, #+ and exit the script with an appropriate error message if not.
	www.softmath.com /tutorials2/greatest-common-divisor-calculator.html (1275 words)

	calculate the greatest common divisor of two numbers - greatest common divisor
		calculate the greatest common divisor of two numbers - greatest common divisor
		This script calculates the greatest common divisor ot two numbers.
		The gcd of two numbers is the greatest common divisor of this numbers, the biggest number this first two numbers can be divided.
	www.mathepower.com /english/ggt.php (69 words)

	CWE - Crazy Maths - Number Analysis - Greatest Common Divisor (Site not responding. Last check: 2007-11-06)
		This page will calculate the greatest common divisor (or factor) of two numbers that you enter.
		Two numbers are considered coprime if their greatest common divisor is 1.
		The quickest way to find the greatest common divisor of two integers, a and b (written as gcd(a,b)), is the Euclidean Algorithm.
	www.coolwebeffects.net /cm/gcd.html (114 words)

	Find The Greatest Common Divisor of Two Integers Using Euclid's Algorithm
		The GCD is simply the largest positive integer that will perfectly divide any two integer values in common.
		The signs of the integers are ignored and the GCD is always expressed as a positive value.
		In integer fractional arithmetic, the GCD may be used to reduce a fraction to its lowest terms by dividing both the numerator and the denominator of the fraction by this GCD.
	www.freevbcode.com /ShowCode.asp?ID=1778 (222 words)

	Fraction A/B Reductions and Greatest Common Divisor (GCD) (Site not responding. Last check: 2007-11-06)
		Given two integers, (±A, ±B), the program attempts to find the greatest common divisor (GCD) which is the largest positive integer that will perfectly divide both values.
		In the case where the numbers are mutually prime, it means that they have no perfect divisors in common except for 1 (unity).
		In such a case, when interpreted as a fraction, it means that the fraction is already expressed in the lowest terms and can not be reduced any further.
	www.neoprogrammics.com /gcd (355 words)

	Greatest Common Divisor Algorithm (Site not responding. Last check: 2007-11-06)
		It writes the greatest common divisor of a and b as a linear combination of them.
		The greatest common divisor of a and b will be denoted by gcd(a,b).
		The least common multiple of a and b will be denoted by lcm(a,b).
	www.math.unl.edu /~ijeib/CSCE235_Spring_01/GCD_Form_Prog.html (77 words)

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