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Introduction
Fundamental Theorem of
Arithmetic
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Fundamental Theorem of Arithmetic
322
Previous Lecture
Common divisors
The greatest common divisor
Euclidean algorithm
Discrete Mathematics – Fundamental Theorem of Arithmetic
323
More Primes
Prime numbers have some very special properties with respect to
division
Properties of primes
.
(1)
If
a,b are integers and
p
is prime such that
p  ab then
p  a
or p  b.
(2)
Let
be an integer for
1
≤
i
≤
n,
and
p
is prime and
then
for
some
1
≤
i
≤
n
i
a
n
2
1
a
a
a

p
K
i
a

p
Discrete Mathematics – Fundamental Theorem of Arithmetic
324
The Fundamental Theorem of Arithmetic
Theorem
.
Every integer
n > 1
can be represented as a product of primes
uniquely, up to the order of the primes.
Proof
.
Existence
By contradiction.
Suppose that there is an
n > 1 that cannot be
represented as a product of primes,
and let
m
be the smallest
such number.
m
is not prime, therefore
m = st for some
s
and t
But then
s
and
t
can be written as products of primes, because
s < m
and
t < m.
Therefore
m
is a product of primes
Discrete Mathematics – Fundamental Theorem of Arithmetic
325
Example
Find the prime factorization of
980,220
Discrete Mathematics – Fundamental Theorem of Arithmetic
326
Least Common Multiple
A positive integer
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This note was uploaded on 05/18/2010 for the course MACM MACM 101 taught by Professor Andreibulatov during the Spring '10 term at Simon Fraser.
 Spring '10
 AndreiBulatov

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