Unique Factorization in $\mathbb {Z}$

Abstract

This chapter presents the concept of induction, in two equivalent forms: complete induction, and the well-ordering principle. One or the other is used to prove the existence and uniqueness of factorization of any number > 1 into a product of prime numbers (the so-called Fundamental Theorem of Arithmetic), to prove the Division Theorem, and to prove the existence of Bezout’s Identity. Uniqueness of factorization permits an alternative description of when one number divides another. It also provides alternate descriptions of the greatest common divisor and the least common multiple of two numbers.