Math 391 Practice Final Exam, Fall 2002
The real nal exam is 15:1517:05 W Dec 11, Deady 306. It will be similar in length
and topics to this practice exam! (You should have plenty of time to nish the
391 Homework 8 solutions
Exercises 2.3: 18. Show that Z[i] is an integral domain, describe its
eld of fractions and nd the units.
There are two ways to show it is an integral domain. The rst is
to ob
391 Homework 7 solutions
Exercises 2.1: 14. Show that 1 Z.
2/
Solution. This is rather a mean question the easiest things are
always the most confusing to prove! You have to prove it from the
way we
391 Homework 6
Exercises 1.4: 13, 15.
13 Let p be a prime. Show that (p 1)! 1 (mod p).
Note 1.(p 1) 1 (mod p). So we just need to show that
2.3. . . . .(p 2) 1 (mod p).
Notice that for 2 a p 2, we ha
391 Homework 5
Exercises 1.3: 20(c),(f), 21(a),(b),(c).
20(c) 243x + 17 101 (mod 725).
We need to solve 243x 84 (mod 725). So we need to nd
2431 in Z725 . Use the Euclidean algorithm.
725 = 2.243 + 2
391 Homework 4 solutions
Exercises 1.3: 7, 8, 9, 13, 14.
7. Use Proposition 3.3 to show that 65 is not a prime.
Proof. Suppose that 65 is prime. Then by 3.3, 265 2 (mod 65).
Now compute 265 . It is (
391 Homework 3 solutions
Exercises 1.1: 4(e), 18.
4(e) For n 3, n + 4 < 2n .
Proof. Proceed by induction on n = 3, 4, . . .
Base case. If n = 3, 3 + 4 = 7 < 8 = 23 .
Induction step: Assume true for n
391 Homework 2 solutions
Exercises 1.1: 4(d),(g).
4(d) For n 1, n3 n is divisible by 3.
Proof. Proceed by induction on n.
Base case: n = 1, 13 1 = 0 which is divisible by 3.
Induction step: Assume tr
391 Homework 1 solutions
Exercises 1.1: 3, 4(a)(b)(c) (I do not insist that you prove these by
induction any logically correct proof will do!).
3. Prove that the square of an even number is even and
Math 391 Midterm
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Instructions
Answer ALL questions (or as many as you have time for).
You may use a calculator if you wish.
READ each question CAREFULLY.
Make sure you JUSTIFY yo