McCombs Math 381
Greatest Common Divisor
1
Formal Definition:
Given
,
ab
∈
Z
, with
0
a
≠
or
0
b
≠
. The
greatest common divisor
of
a
a
n
d
b
, written
( )
gcd
,
, is the largest integer that divides both
a
a
n
d
b
. In other words,
( )
gcd
,
d
=
means that

da
and

db
, and
i
f

ca
and

cb
, then
cd
<
.
Special Case:
If
()
gcd
,
1
=
, we say that
a
and
b
are
relatively prime (coprime)
.
Other Important Results:
1.
Given
,,
abw
∈
Z
, the equation
ax
by
d
+
=
has integer solutions
,
x y
if and only if
gcd
,

ab w
.
2.
Given
abc
positive integers such that
( )
gcd
,
1
=
.
If

c
, then

ac
.
3.
Given
123
,
,
,...,
n
aa a
a
∈
Z
and any prime number
p
.
If

...
n
p
a
⋅⋅⋅
⋅
, then

k
p a
for some 1
kn
≤
≤
.
4.
Given integers
3
12
3
...
a
aa
a
n
n
ap p p
p
=⋅
⋅
⋅
and
3
3
...
b
bb
b
n
n
bp p p
p
⋅
⋅
,
where each exponent is a nonnegative integer, and
,
,
,...,
n
p
pp
p
are prime.
(i)
(
)
(
)
(
)
,
,
min
min
min
min
34
11
2 2
3
gcd
,
...
a b
nn
n
p
p
p
p
⋅
⋅
(ii)
(
)
(
)
(
)
,
,
max
max
max
max
3
lcm
,
...
n
p
p
p
p
⋅
⋅
(iii)
( ) ( )
lcm
,
gcd
,
⋅=
⋅
Crucial Theorem:
If
( )
gcd
,
d
=
, there exist integers
,
x y
such that
ax
by
d
+=
.
Integers
a
and
b
are relatively prime if and only if there
exist integers
,
xy
such that
1
ax
by
+
=
.