Introduction
Common Divisors
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics – Common Divisors
312
Previous Lecture
Representation of numbers
Prime and composite numbers
Discrete Mathematics – Common Divisors
313
The Greatest Common Divisor
For integers
a
and
b,
a positive integer
c
is said to be a
common divisor of
a
and
b
if
c  a
and
c  b
Let
a, b
be integers such that
a
≠
0
or
b
≠
0.
Then a positive
integer
c
is called the
greatest common divisor of
a,
b
if
(a)
c  a
and
c  b
(that is
c
is a common divisor of
a, b)
(b)
for any common divisor
d
of
a
and
b,
we have
d  c
The greatest common divisor of
a
and
b
is denoted by
gcd(a,b)
Discrete Mathematics – Common Divisors
314
The Greatest Common Divisor
(cntd)
Theorem
For any
positive integers
a
and
b,
there is a unique positive integer
c
such that
c
is the greatest common divisor of a
and
b
First try:
Take the largest common divisor, in the sense of usual order
Does not work:
Why every other common divisor divides it?
4
1
2
3
5
11
7
10
6
9
12
8
a
b
gcd(a,b)
Discrete Mathematics – Common Divisors
315
The Greatest Common Divisor
(cntd)
Proof
.
Given
a, b,
let
S = { as + bt  s,t
∈
Z
,
as + bt > 0 }.
Since
S
≠ ∅
,
it has a least element
c.
We show that
c = gcd(a,b)
We have
c = ax + by
for some integers
x
and
y.
If
d  a
and
d  b,
then
d  ax + by = c.
If
c  a,
we can use the division algorithm to find
a = qc + r,
where
q,r are integers and
0 < r < c.
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 Spring '10
 AndreiBulatov
 Prime number, Greatest common divisor, common divisor, common divisors

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