2. INTEGERS AND ALGORITHMS
155
2. Integers and Algorithms
2.1. Euclidean Algorithm. Euclidean Algorithm.
Suppose
a
and
b
are in
tegers with
a
≥
b >
0.
(1) Apply the division algorithm:
a
=
bq
+
r
, 0
≤
r < b
.
(2) Rename
b
as
a
and
r
as
b
and repeat until
r
= 0.
The last nonzero remainder is the greatest common divisor of
a
and
b
.
The Euclidean Algorithm depends upon the following lemma.
Lemma
2.1.1
.
If
a
=
bq
+
r
, then
GCD(
a, b
) = GCD(
b, r
)
.
Proof.
We will show that if
a
=
bq
+
r
, then an integer
d
is a common divisor
of
a
and
b
if, and only if,
d
is a common divisor of
b
and
r
.
Let
d
be a common divisor of
a
and
b
. Then
d

a
and
d

b
. Thus
d

(
a

bq
), which
means
d

r
, since
r
=
a

bq
. Thus
d
is a common divisor of
b
and
r
.
Now suppose
d
is a common divisor of
b
and
r
. Then
d

b
and
d

r
. Thus
d

(
bq
+
r
),
so
d

a
. Therefore,
d
must be a common divisor of
a
and
b
.
Thus, the set of common divisors of
a
and
b
are the same as the set of common
divisors of
b
and
r
. It follows that
d
is the
greatest
common divisor of
a
and
b
if and
only if
d
is the greatest common divisor of
b
and
r
.
Discussion
The fact that the Euclidean algorithm actually gives the greatest common divi
sor of two integers follows from the division algorithm and the equality in Lemma
2.1.1. Applying the division algorithm repeatedly as indicated yields a sequence of
remainders
r
1
> r
2
>
· · ·
> r
n
>
0 =
r
n
+1
, where
r
1
< b
. Lemma 2.1.1 says that
GCD(
a, b
) = GCD(
b, r
1
) = GCD(
r
1
, r
2
) =
· · ·
= GCD(
r
n

1
, r
n
)
.
But, since
r
n
+1
= 0,
r
n
divides
r
n

1
, so that
GCD(
r
n

1
, r
n
) =
r
n
.
Thus, the last nonzero remainder is the greatest common divisor of
a
and
b
.
Example
2.1.1
.
Find
GCD
(1317, 56).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2. INTEGERS AND ALGORITHMS
156
1317
= 56(23) + 29
56
= 29(1) + 27
29
= 27(1) + 2
27
= 2(13) + 1
2
= 1(2) + 0
GCD
(1317,56)=1
Example 2.1.1 shows how to apply the Euclidean algorithm.
Notice that when
you proceed from one step to the next you make the new dividend the old divisor
(replace
a
with
b
) and the new divisor becomes the old remainder (replace
b
with
r
). Recall that you can find the quotient
q
by dividing
b
into
a
on your calculator
and rounding
down
to the nearest integer. (That is,
q
=
b
a/b
c
.) You can then solve
for
r
.
Alternatively, if your calculator has a
mod
operation, then
r
=
mod
(
a, b
)
and
q
= (
a

r
)
/b
. Since you only need to know the remainders to find the greatest
common divisor, you can proceed to find them recursively as follows:
Basis.
r
1
=
a
mod
b
,
r
2
=
b
mod
r
1
.
Recursion.
r
k
+1
=
r
k

1
mod
r
k
, for
k
≥
2. (Continue until
r
n
+1
= 0 for some
n
. )
2.2.
GCD
’s and Linear Combinations.
Theorem
2.2.1
.
If
d
= GCD(
a, b
)
, then there are integers
s
and
t
such that
d
=
as
+
bt.
Moreover,
d
is the smallest positive integer that can be expressed this way.
Discussion
Theorem 2.2.1 gives one of the most useful characterizations of the greatest com
mon divisor of two integers. Given integers
a
and
b
, the expression
as
+
bt
, where
s
and
t
are also integers, is called a
linear combination
of
a
and
b
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 PenelopeKirby
 Natural number, Prime number, Greatest common divisor, Euclidean algorithm, Fundamental theorem of arithmetic

Click to edit the document details