26 1. ALGEBRAIC THEMES Although the multiplicative structure of the integers is subordinate to the additive structure, many of the most interesting properties of the in-tegers have to do with divisibility , factorization, and prime numbers. Of course, these concepts are already familiar to you from school mathemat-ics, so the emphasis in this section will be more on a systematic, logical development of the material, rather than on exploration of unknown terri-tory. The main goal will be to demonstrate that every natural numbers has a unique factorization as a product of prime numbers; this is trickier than one might expect, the uniqueness being the difﬁcult part. On the way, we will, of course, be practicing with logical argument, and we will have an introduction to computational issues: How do we actually compute some abstractly deﬁned quantity? Let’s begin with a deﬁnition of divisibility. We say that an integer a divides an integer b (or that b is divisible by a ) if there is an integer
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