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MAS 3300  FINAL EXAM  FALL 1990
Wednesday December 19
Do ALL questions. All questions are of equal value. Write on ONE side of your paper.
Calculators are allowed. Time allowed – TWO hours.
1.
Show that if
a
∈
R
then
a
0 = 0.
2.
Use mathematical induction to prove that
1 + 2 +
···
+
n
=
n
(
n
+ 1)
2
,
for all integers
n
≥
1.
3.
Show that if
a
,
b
,
c
are integers and
a

b
and
b

c
then
a

c
.
4.
Use unique factorization to prove that
√
2 is irrational. You may NOT assume Theorem
(4.3).
5.
(a) Use the Euclidean algorithm to ﬁnd gcd(285
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Unformatted text preview: , 165). (b) Use prime factorizations to compute gcd(285 , 165). 6. Show that √ 2 6∈ Q [ √ 3]. 7. Show that if n is an integer, n ≥ 2, a ∈ Z n and a and n are relatively prime then a has a multiplicative inverse in Z n . 8. Find 131 in Z 23 . 9. For z 1 , z 2 ∈ C show that z 1 z 2 = z 1 z 2 . 10. Determine all solutions in C of z 8 = 1 . Write the solutions in rectangular form and simplify. 1...
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This note was uploaded on 06/03/2011 for the course MAS 3300 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
 Staff

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