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Math 3307 Practice Exams David Andrew Smith Abstract. Elementary number theory presents problems involving divisibility, prime numbers, Euclidean Algorithm, linear Diophantine equations, congruence problems, multiplicative functions, reciprocity laws, and various types of congruence applications. We also present several topics about elementary number theory including many reviews, practice exams, quizzes, and homework exercises. We briefly describe some of the history of number theory in terms of some of the main mathematicians that contributed to the theory of numbers. Early contributors were Archimedes and the Pythagoreans (including Euclid), and then later on Fermat, Euler and Gauss. Number theory may very well be one of the oldest subjects of mathematics. Roughly, num- ber theory is the mathematical treatment of ques- tions related to the integers; that is, the numbers ,- 1 , 1 ,- 2 , 2 ,- 3 , 3 ... and people has been manipu- lating them for thousands of years. The Ancient Greeks, in particular, Euclid (third century) and the Pythagoreans (sixth century B.C.) spent a considerable amount of attention to them; as well as Archimedes (third century). Indeed, one of the most important sets of numbers, the prime numbers; hold a key position in number theory since they are the building blocks of the integers; and per- haps the first question that comes to mind is whether there are infinitely many prime numbers. A proof of this amazing fact can be found in Euclids famous book: The Elements. Pierre de Fermat and Leonard Euler rekindled interest in number theory in the seventeenth and eighteenth centuries by using new (among others, calculus-related) techniques to arrive at important new results. Each new theorem of course, has led to many new questions and conjectures; and one of the fascinating aspects of number theory is that many unresolved questions can be understood with only a minor background in the subject. Even today there are many open problems; and some have a substan- tial reward for a solution! After Fermat and Euler, Carl Friedrich Gauss, one of the greatest mathematicians of all time, gave the first modern treatment of number theory. He defined the notion of congruence, and distinguished its importance; in fact, its his notation and ap- proach of number theory that we use today. Gausss many achievements in number theory are well docu- mented; and its Gauss who coined the phrase num- ber theory is the queen of the sciences. Interestingly enough, even in an elementary course of number the- ory other fields of mathematics come into play, such as the complex numbers, geometry, and abstract al- gebra. Various topics in elementary number theory including divisibility, congruence, quadratic reci- procity, and multiplicative functions. Upon comple- tion of Number Theory students should be able to: (1) Prove statements and solve problems in- volving divisibility, prime numbers and the Euclidean Algorithm; (2) Solve linear Diophantine equations and var-... View Full Document
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