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

d
a
and

d b
, and
if

c a
and

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

a b
w
.
2.
Given
, ,
a b c
positive integers such that
(
)
gcd
,
1
a b
=
.
If
(
)

a
bc
, then

a c
.
3.
Given
1
2
3
,
,
,...,
n
a
a
a
a
∈
Z
and any prime number
p
.
If
(
)
1
2
3

...
n
p
a
a
a
a
⋅
⋅
⋅
⋅
, then

k
p a
for some
1
k
n
≤
≤
.
4.
Given integers
3
1
2
1
2
3
...
a
a
a
a
n
n
a
p
p
p
p
=
⋅
⋅
⋅
and
3
1
2
1
2
3
...
b
b
b
b
n
n
b
p
p
p
p
=
⋅
⋅
⋅
,
where each exponent is a nonnegative integer, and
1
2
3
,
,
,...,
n
p
p
p
p
are prime.
(i)
(
)
(
)
(
)
(
)
(
)
,
,
,
,
min
min
min
min
3
4
1 1
2
2
1
2
3
gcd
,
...
a
b
a b
a
b
a
b
n
n
n
a b
p
p
p
p
=
⋅
⋅
⋅
(ii)
(
)
(
)
(
)
(
)
(
)
,
,
,
,
max
max
max
max
3
4
1 1
2
2
1
2
3
lcm
,
...
a
b
a b
a
b
a
b
n
n
n
a b
p
p
p
p
=
⋅
⋅
⋅
(iii)
(
)
(
)
lcm
,
gcd
,
a b
a b
a b
⋅
=
⋅
Crucial Theorem:
If
(
)
gcd
,
a b
d
=
, there exist integers
,
x y
such that
ax
by
d
+
=
.
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 Summer '11
 NA
 Math, Prime number, Greatest common divisor, Euclidean algorithm, gcd

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