chapter  10
The Integers
Pages 8

In this chapter we begin to study the most basic, and also perhaps the mostfascinating, number system of all — the integers. Our first aim will be toinvestigate factorization properties of integers. We know already that every in-teger greater than 1 has a prime factorization (Proposition 8.1). This was quiteeasy to prove using Strong Induction. A somewhat more delicate questionis whether the prime factorization of an integer is always unique — in otherwords, whether, given an integer n, one can write it as a product of primes inonly one way. The answer is yes; and this is such an important result that ithas acquired the grandiose title of “The Fundamental Theorem of Arithmetic.”We shall prove it in the next chapter and try there to show why it is such animportant result by giving some examples of its use. In this chapter we lay thegroundwork for this.We begin with a familiar definition. DEFINITION Let a,b ∈ Z. We say a divides b (or a is a factor ofb) if b = ac for some integer c. When a divides b, we write a|b.