Flow chart of an algorithm ( Euclid's algorithm ) for calculating the greatest common divisor ( g. c. d.

Euclid poses the problem: " Given two numbers not prime to one another, to find their greatest common measure ".

Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers ' common measure is in fact the greatest.

* The greatest common divisor and least common multiple functions act associatively.

Two whole numbers m and n are called coprime if their greatest common divisor is 1 ; i. e., if there is no prime number that divides both of them.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Bézout's identity ( also called Bezout's lemma ) is a theorem in the elementary theory of numbers: let a and b be integers, not both zero, and let d be their greatest common divisor.

That is, if R is a PID, and a and b are elements of R, and d is a greatest common divisor of a and b,

This is the same thing as their greatest common divisor being 1.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

Though nearly all modern mathematicians consider nonconstructive methods just as sound as constructive ones, Euclid's constructive proofs often supplanted fallacious nonconstructive ones — e. g., some of the Pythagoreans ' proofs that involved irrational numbers, which usually required a statement such as " Find the greatest common measure of ..."

This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.

In particular, the greatest common divisor of any two elements exists and can be written as a linear combination

An arbitrary PID has much the same " structural properties " of a Euclidean domain ( or, indeed, even of the ring of integers ), but knowing an explicit algorithm for Euclidean division, and thus also for greatest common divisor computation, gives a concreteness which is useful for algorithmic applications.

Euclid's method for finding the greatest common divisor ( GCD ) of two starting lengths BA and DC, both defined to be multiples of a common " unit " length.

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor ( GCD ) of two integers, also known as the greatest common factor ( GCF ) or highest common factor ( HCF ).

That number then is the greatest common divisor of the original pair.

The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the smaller number is subtracted from the larger number.

That k may also represent the greatest common divisor is proven below.

Suppose it is desired to find the greatest common divisor of 1989 and 867.

If we reduce the larger number by subtracting the smaller one from it, the greatest common divisor does not change.

Since the two numbers are equal, we have found the answer: the greatest common divisor of 1989 and 867 must be 51.

The dweller at p is last to hear about a new cure, the slowest to announce to his neighbors his urgent distresses, the one who goes the farthest to trade, and the one with the greatest difficulty of all in putting over an idea or getting people to join him in a cooperative effort.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Carl says it is the greatest poem ever written to the guitar because he has never heard of any other poem to that subtle instrument.

In conformance with the maximization principle we affirm that Gentile-Jewish relations will be harmonious or inharmonious to the degree that one relation or the other is expected by the active participants to yield the greatest net advantage, taking all value outcomes and effects into consideration.

but the possibility of this effort is bound up with that development of historical thought which is the greatest achievement of our civilization in the last two centuries, and it is utterly impossible to people in whom this development has not taken place.

On the other hand, the bright vision of the future has been directly stated in science fiction concerned with projecting ideal societies -- science fiction, of course, is related, if sometimes distantly, to that utopian literature optimistic about science, literature whose period of greatest vigor in the late nineteenth and early twentieth centuries produced Edward Bellamy's Looking Backward and H. G. Wells's A Modern Utopia.

Even in its present form, however, the first part of Malraux's unrecoverable novel is among the greatest works of mid-twentieth century literature ; ;

But the secretary insists that the success of the American farmer is the `` greatest single source of strength '' in the struggle to insure freedom around the world.

Please do put more pictures and articles in about Liberace, as he is truly one of our greatest entertainers and a really wonderful person.

What I mean is, he was a Pole and the greatest soldier in the Ulanys.

What Sam Rayburn's life proves to us all is the magnificent lesson in political science that one can devotedly and with absolute dedication represent the seemingly provincial interests of one's own community, one's own district, one's own State, and by that help himself represent even better the sweep and scope of the problems of this the greatest nation of all time.

Sam Rayburn is one of the greatest American public figures in the history of our country and I consider that I have been singly honored in the privilege of knowing Sam Rayburn and sharing with him the rights and obligations of a Member of the House of Representatives in the Congress of the United States.

The need here is most clearly felt and our capacity to recruit and train qualified volunteers in a short period of time is greatest.

One of the greatest obstacles to the achievement of this goal is the lack of trained men and women with the skill to teach the young and assist in the operation of development projects -- men and women with the capacity to cope with the demands of swiftly evolving economics, and with the dedication to put that capacity to work in the villages, the mountains, the towns and the factories of dozens of struggling nations.

The cost of developing a major weapon system is now so enormous that the greatest care must be exercised in selecting new systems for development, in determining the most satisfactory rate of development, and in deciding the proper time at which either to place a system into production or to abandon it.

The Pushup done in this manner is the greatest pectoral-ribcage stretcher ever invented!!

This is true only if a very wide grip is used and only when the greatest possible stretch is achieved.

Artur Schnabel was one of the greatest Schubert-Beethoven-Mozart players of all time, and any commentary of his on this repertory is valuable.

A body of water is usually the center of interest at parks which attract the greatest picnic and camping use.

The latest and, significantly, greatest fruit of this theatrical vine is The, an adaptation of Basho's classic frog-haiku by Roger Entwhistle, a former University of Maryland chemistry instructor.