McCombs Math 81
Working with gcd(
a
,
b
)
1
Important Theorems:
Theorem:
Given
,
a b
!
Z
, with
a
and
b
not both 0. If
(
)
gcd
,
d
a b
=
, then there
exist
,
x y
!
Z
such that
d
ax
by
=
+
.
Theorem:
Given
,
a b
!
Z
, with
a
and
b
not both 0. If
(
)
gcd
,
d
a b
=
, then
d
is the
smallest positive integer that can be expressed as the linear combination
d
ax
by
=
+
, where
,
x y
!
Z
.
Theorem:
Integers
a
and
b
are relatively prime if and only if there exist
,
x y
!
Z
such that
1
ax
by
+
=
.
Proof Examples:
1.
Given
, ,
a b w
!
Z
. The equation
ax
by
w
+
=
has integervalued solutions
,
x y
if and
only if
(
)
gcd
,
a b
w
.
Part (i):
Assume the equation
ax
by
w
+
=
has integervalued solutions
,
x y
.
Let
(
)
gcd
,
d
a b
=
. We know that
a
kd
=
and
b
md
=
for some
integers
m
and
n
.
(
)
(
)
ax
by
w
kd
x
md
y
w
+
=
!
+
=
So
(
)
w
d kx
my
=
+
. Since
k
,
x
,
m
, and
y
are all integers,
d w
.
Thus,
(
)
gcd
,
a b
w
.
Part (ii):
Assume
(
)
gcd
,
a b
w
.
Let
(
)
gcd
,
d
a b
=
. We know that
a
kd
=
and
b
md
=
for some
integers
m
and
n
. We also know that there exist integers
m
and
n
such
that
ma
nb
d
+
=
.
(
)
gcd
,
a b
w
w
dk
!
=
for some integer
k
.
So we have
(
)
w
dk
ma
nb k
kma
knb
=
=
+
=
+
.
Thus,
x
mk
=
and
y
nk
=
are integervalued solutions for the
equation
ax
by
w
+
=
.
Therefore, the equation
ax
by
w
+
=
has integervalued solutions
,
x y
if and
only if
(
)
gcd
,
a b
w
.
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McCombs Math 81
Working with gcd(
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 Summer '11
 NA
 Math, Natural number, Prime number, Euclidean algorithm, gcd

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