This preview shows pages 1–2. Sign up to view the full content.
Math
465, Exam I (100 pts)
February 25, 2008
Print Name:
Ae Ja Yee
This exam is closedbook, closednotes. Show all your work for full credit. Partial credit will
be given based on what is written. You have 50 minutes to ﬁnish the exam.
1. (15 pts) Use the Euclidean algorithm to compute (171
,
30) and express (171
,
30) in the
form 171
x
+ 30
y
.
Solution.
Applying the Euclidean algorithm, we obtain
171 = 30
·
5 + 21
30 = 21
·
1 + 9
21 = 9
·
2 + 3
9 = 3
·
3 + 0
Therefore, (171
,
30) = 3. Working backwards through the equations, we obtain
3 = 21

9
·
2
= 21

(30

21)
·
2
= 21
·
3

30
·
2
= (171

30
·
5)
·
3

30
·
2
= 171
·
3

30
·
17
Thus we have 3 = 171
·
3 + 30
·
(

17).
2. Let Let
a, b, c
be integers. Prove that
(a) (10 pts) if (
a, b
) = 1 and
b

ac
, then
b

c
;
(b) (10 pts) if (
a, b
) = 1, then (
a

b,
2
a
+
b
) = 1 or 3;
(c) (10 pts) if 2
p

1 is prime then
p
is prime.
Solution.
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.
 Spring '08
 YEE
 Math, Number Theory

Click to edit the document details