This preview shows pages 1–2. Sign up to view the full content.
1
Math 417 Exam I: July 2, 2009; Solutions
1
.
If
a
and
b
are relatively prime and each of them divides an integer
n
, prove
that their product
ab
also divides
n
.
Here are two proofs (of course, either one suﬃces for full credit).
By hypothesis,
n
=
ak
=
b`
. Since
b

ak
, Corollary 1.40 [if (
b, a
) = 1 and
b

ak
, then
b

k
] gives
b

k
; that is,
k
=
bb
0
. Therefore,
n
=
ak
=
abb
0
, and so
ab

n
.
The second proof does not use Corollary 1.40.
There are integers
s
and
t
with 1 =
sa
+
tb
. Hence,
n
=
san
+
tbn
=
sab`
+
tbak
=
ab
(
s`
+
tk
)
.
Hence,
ab

n
.
2
.
Let
I
be a subset of
Z
such that
1. 0
∈
I
;
2. if
a, b
∈
I
, then
a

b
∈
I
;
3. if
a
∈
I
and
q
∈
Z
, then
qa
∈
I
.
Prove that there is a nonnegative
d
∈
I
such that
I
consists precisely of all the
multiples of
d
.
This is Corollary 1.37 in the text.
3
.
(
i
) (
5 points
) Write gcd(7
,
37) as a linear combination of 7 and 37.
The Euclidean algorithm ends very quickly here:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 Math, Algebra

Click to edit the document details