Math 461 F
Spring 2011
Homework 4 Solutions
Drew Armstrong
A.1.
Euclid’s Lemma.
Suppose that
a
divides
bc
for
a, b, c
∈
Z
with
a
and
b
coprime
(i.e. they have no common factor except
±
1).
Prove
that
a
must divide
c
. (Hint: Since
a
and
b
are coprime, you may assume — without
proof — that there exist
x, y
∈
Z
such that
ax
+
by
= 1.)
Proof.
Since
a
and
b
are coprime they have greatest common divisor 1. You
may have seen in another class the fact that the greatest common divisor of
(
a, b
) is always an integer linear combination of
a
and
b
. That is, there exist
x, y
∈
Z
such that
ax
+
by
= 1. Now multiply both sides of this equation by
c
to get
axc
+ (
bc
)
y
=
c.
By assumption we have
bc
=
ak
for some
k
∈
Z
, hence
axc
+ (
bc
)
y
=
axc
+
aky
=
a
(
xc
+
ky
) =
c.
In other words,
a
divides
c
.
A.2. Prove
that
3
√
2 is not rational.
Proof.
Suppose for contradiction that
3
√
2 is rational.
Then we can write
3
√
2 =
a/b
as a fraction in lowst terms (i.e.
a, b
∈
Z
with
a, b
coprime).
Cubing both sides of this equation gives 2 =
a
3
/b
3
, or
a
3
= 2
b
3
. Since
a
3
is even we may conclude that
a
is even (for if
a
were odd then
a
3
would be
odd), and we write
a
= 2
k
for some
k
∈
Z
. But then we have 2
b
3
=
a
3
= 8
a
3
,
or
b
3
= 4
a
3
, which implies that
b
3
, and hence
b
, is even.
We have found
that
a
and
b
are both even, which contradicts the assumption that they are
coprime. Hence
3
√
2 is not rational.
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 Spring '11
 Armstrong
 Math, Algebra, Cos, Greatest common divisor, Drew Armstrong

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