Elementary Number Theory Problems. Part VI

Problems in Elementary Number Theory

Introduction The heart of Mathematics is its problems. Paul Halmos Number Theory is a beautiful branch of Mathematics. The purpose of this book is to present a collection of interesting problems in elementary Number Theory. Many of the problems are mathematical competition problems from all over the world like IMO, APMO, APMC, Putnam and many others. The book has a supporting website at which h...

Part VI Example Problems

An object is cloaked if its presence cannot be detected by an observer using electromagnetic or other forms of imaging. In fact, the observer should not notice that cloaking is even occurring. Cloaking has a long history in science fiction, but recent developments have put the idea on a firmer mathematical and physical basis. Suppose that an observer seeks to image some bounded region Ω in spac...

Elementary Number Theory

Elementary number theory

The division relation plays a prominent role throughout this note, and so, we start by presenting some of its basic properties and their relation to addition and multiplication. First, it is re exive because multiplication has a unit (i.e., m= 1×m) and it is transitive, since multiplication is associative. It is also (almost) preserved by linear combination because multiplication distributes ov...

Elementary Number Theory

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natu...