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 exam
since there are only 6 questions and you have 1 h
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 observe:
Any subring of a eld is an integral domain.
(Pro
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 dened Z: it is a ring, it is ordered, and it satises th
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 have that a1 = for 2 b
b
(p 2) and moreover b = a. This
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 (26 )10 .25 . But 26 = 64 1 (mod 65)
so (26 )10 1 (mod 6
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 = k , i.e. k + 4 < 2k . Consider
the inequality 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 true for n = k , i.e. k 3 k is a multiple
of 3. Consider
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 the square
of an odd number is odd.
By denition, an int
Math 391 Midterm
Full name:
ID number:
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 your answers that way, I can give some credit even for wr