Math 230 E
Fall 2011
Homework 4
Drew Armstrong
Problem 1.
(a) Consider
a,b,d
∈
Z
with gcd(
a,d
) = 1.
Prove:
If
d

ab
then
d

b
.
(b) Consider
a,b,m
∈
Z
with gcd(
a,b
) = 1.
Prove:
If
a

m
and
b

m
then
ab

m
.
(c) Consider
a,b,c,n
∈
Z
with gcd(
c,n
) = 1.
Prove:
If
ca
≡
cb
mod
n
then
a
≡
b
mod
n
.
Proof.
First we prove (a). Consider
a,b,d
∈
Z
with
a,d
coprime, and suppose that
d

ab
. We
will show that
d

b
. Since
d

ab
, there exists
k
∈
Z
such that
dk
=
ab
. Then since
a,d
are
coprime, B´
ezout’s Lemma implies that there exist
x,y
∈
Z
such that
ax
+
dy
= 1. Finally, we
multiply both sides of this equality by
b
to get
b
(
ax
+
dy
) =
b
·
1
,
bax
+
bdy
=
b,
dkx
+
dby
=
b,
d
(
kx
+
by
) =
b.
We conclude that
d

b
. To show (b), consider
a,b,m
∈
Z
with
a,b
coprime. Suppose further
that
a

m
and
b

m
. We will show that
ab

m
. There are two possible solutions.
Solution 1.
(From Scratch)
There exist
k,‘
∈
Z
such that
ak
=
m
and
b‘
=
m
. Then since
a,b
are
coprime, B´
ezout’s Lemma gives us
x,y
∈
Z
such that
ax
+
by
= 1. Multiply both sides by
m
to get
m
(
ax
+
by
) =
m
·
1
,
max
+
mby
=
m,
b‘ax
+
akby
=
m,
ab
(
‘x
+
ky
) =
m.
Hence
ab

m
.
Solution 2. (Quote Part (a))
There exist
k,‘
∈
Z
such that
ak
=
m
=
b‘
,
hence
ak
=
b‘
. Since
b

ak
with
a,b
coprime, part (a) implies that
b

k
. That is, there exists
x
∈
Z
such that
k
=
bx
. Substitute this into
ak
=
m
to get
abx
=
m
. We conclude that
ab

m
. Finally, we will prove (c). Consider
a,b,c,n
∈
Z
with
c,n
coprime and
ca
≡
cb
mod
n
.
We will show that