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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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 Fall '08
 EVERAGE
 Algebra, Division, Integers, Natural Numbers, Prime Numbers, Prime number

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