Euclids Algorithm.Euclids Algorithm appears as the solution to the Proposition VII.Elements.Given two numbers not prime to one another, to find their greatest common measure.What Euclid called common measure is termed nowadays a common factor or a common divisor.Euclid VII.Not surprisingly, the algorithm bears Euclids name.The algorithm is based on the following two observations If ba then gcda, b b.Java Data Structures 2nd Edition End of the World Production, LLC.Paperback ISBN 9780989472104 Hardcover ISBN 9780989472111 This book is an introduction to the standard methods of proving mathematical theorems.GeorgiaStandards.Org GSO is a free, public website providing information and resources necessary to help meet the educational needs of students.This is indeed so because no number b, in particular may have a divisor greater than the number itself I am talking here of non negative integers.If a bt r, for integers t and r, then gcda, b gcdb, r.Indeed, every common divisor of a and b also divides r.Thus gcda, b divides r.But, of course, gcda, bb.Therefore, gcda, b is a common divisor of b and r and hence gcda, b gcdb, r.The reverse is also true because every divisor of b and r also divides a.Example.Let a 2.Therefore, gcd2.For any pair a and b, the algorithm is bound to terminate since every new step generates a similar problem that of finding gcd for a pair of smaller integers.Let Eulena, b denote the length of the Euclidean algorithm for a pair a, b.Eulen2.Eulen3. 0, 6 1. Ill use.Corollary. Sherlock Holmes And The Hound Of The Baskervilles Game more. For every pair of whole numbers a and b there are two integers s and t such that as bt gcda, b.Example.Proof. Let a b. The proof is by induction on Eulena, b.If Eulena, b 1, i.Hence, a 1 ub b gcda, b.We can take s 1 and t 1 u.Cd Burning Software For Win 95 Iso Download on this page.Assume the Corollary has been established for all pairs of numbers for which Eulen is less than n.Let Eulena, b n.Apply one step of the algorithm a bu r.Eulenb, r n 1.Problems And Solutions In Mathematics 2Nd Edition' title='Problems And Solutions In Mathematics 2Nd Edition' />By the inductive assumption, there exist x and y such that bx ry gcdb,r gcda,b.Express r as r a bu.Hence, ry ay buy bx ay buy gcda, b.Finally, bx uy ay gcda, b and we can take s x uy and t y.There is also a simple proof that employs the Pigeonhole Principle.Remark.Note that any linear combination as bt is divisible by any common factor of a and b.In particular, any common factor of a and b also divides gcda, b.In a reverse application, any linear combination as bt is divisible by gcda, b.From here it follows that gcda, b is the least positive integer representable in the form as bt.Euclids Algorithm appears as the solution to the Proposition VII.Elements Given two numbers not prime to one another, to find their greatest common measure.Problems And Solutions In Mathematics 2Nd Edition' title='Problems And Solutions In Mathematics 2Nd Edition' />All the rest are multiples of gcda, b.The generalization of the Corollary to what is known as Principal ideal domain is known as Bzouts identity or Bzouts Lemma after the French mathematician ttiene Bzout 1.Corollary is also often referred to as Bzouts identity or Bzouts Lemma.For coprime numbers we get existence of s and t such that as bt 1.This Corollary is a powerful tool.It appeared in the 3 Glass and Hour Glass problems.For example, lets.Euclids Proposition VII.If two numbers, multiplied by one another make some number, and any prime number measures.Let a prime p divide the product ab.Assume pa.Then gcda, p 1.By Corollary, ax py 1 for some x and y.Multiply by b abx pby b.Now, pab and ppb.Hence, pb.Actually, this proves a generalization of the Proposition VII.Problems And Solutions In Mathematics 2Nd Edition' title='Problems And Solutions In Mathematics 2Nd Edition' />I used several times on these pages.Let mab and gcda, m 1.Then mb.Proposition VII.Fundamental Theorem of Arithmetic although Euclid has never stated it explicitly.The first time it was formulated in 1.Gauss in his Disquisitiones arithmeticae.Fundamental Theorem of Arithmetic.Any integer N can be represented as a product of primes.Such a representation is unique up to the.Since, by definition, a number is composite if it has factors other than 1 and itself, and these factors are bound to be smaller than the number, we can keep extracting the factors until only prime factors remain.This shows existence of the representation N pqr., where all p, q, r.To prove.N pqr.We see that p divides uvw.By Corollary, it divides one of the factors u, v, w,.Cancel them out.We can go on chipping away on the factors left and.Representation of a number as the product of primes is called prime number decomposition or prime factorization.The Fundamental Theorem of Arithmetic asserts that each integer has a unique prime number decomposition.Note Euclids Algorithm is not the only way to determine the greatest common factor of two integers.If you can find the prime factorizations of the two numbers you can easily determine their gcd as the intersection of the multisets formed by their prime factors.Factor Trees offer a convenient bookkeeping for finding prime factorizations of integers.References.H. Davenport, The Higher Arithmetic, Harper Brothers, NY.R.Graham, D. Knuth, O.Patashnik, Concrete Mathematics, 2nd edition, Addison Wesley, 1.Oystein Ore, Number Theory and Its History, Dover Publications, 1.S.K. Stein, Mathematics The Man Made Universe, 3rd edition, Dover, 2.ContactFront pageContentsAlgebraopyright 1.Alexander Bogomolny.Book of Proof.BOOK OF PROOFSecond EditionRichard Hammack.Paperback ISBN 9.Hardcover ISBN 9.This book is an introduction to the standard methods of proving mathematical theorems.It has been approved by the American Institute of Mathematics Open Textbook Initiative.Also see the Mathematical Association of America Math DL review of the 1st edition, the Amazon reviews,and a brief news story on VCU In.Sight.
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