This preview shows pages 1–7. Sign up to view the full content.
Introduction
Fundamental Theorem of
Arithmetic
Discrete Mathematics
Andrei Bulatov
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document 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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Discrete Mathematics – Fundamental Theorem of Arithmetic
326
Least Common Multiple
A positive integer
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/14/2011 for the course MACM 101 taught by Professor Pearce during the Spring '08 term at Simon Fraser.
 Spring '08
 PEARCE

Click to edit the document details