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Module 5: Basic Number Theory
Theme 1: Division
Given two integers, say
and
,thequo
t
ien
t
may or may not be an integer (e.g.,
but
). Number theory concerns the former case, and discovers criteria upon which one can
decide about divisibility of two integers.
More formally, for
we say that
divides
if there is another integer
such that
and we write
.Inshor
t
:
if and only if
This simple deFnition leads to many properties of divisibility. ±or example, let us establish the
following lemma.
Lemma 1
If
and
,then
.
Proof
.W
eg
i
v
ead
i
r
e
c
tp
roo
f
.±
romth
ed
eFn
i
t
iono
fd
i
v
i
s
ib
i
l
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View Full Document Theme 2: Primes
Primes numbers occupy very prominent role in number theory. A
prime
number
is an integer
greater than
that is divisible
only
by
and itself. A number that is not prime is called
composite
.
Example 1
:Thepr
imeslessthan
are:
How many primes are there? We Frst prove that there are inFnite number of primes.
Theorem 1
.
There are infnite number oF primes.
Proof
.W
ep
r
o
v
i
d
eap
r
o
o
fb
yc
o
n
t
r
a
d
i
c
t
i
o
n
.A
c
t
u
a
l
l
y
,i
ti
sd
u
et
oE
u
c
l
i
da
n
di
where
are exponents of
(i.e., the number of times
occurs in the factorization of
)
.
Proof
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This note was uploaded on 03/26/2012 for the course STAT 350 taught by Professor Staff during the Spring '08 term at Purdue University.
 Spring '08
 Staff
 Statistics

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