Solutions
Math 4400
HOMEWORK #5
1. Consider the eld F = Z 19Z.
(a) Compute all powers 2k for k = 1, . . . , 18 to see that the order of 2 is 18.
Here are the powers of 2 modulo 19.
k
2k
1 2 3 4 5 6 7
Solutions
Math 4400
HOMEWORK #4
1. If d divides n, prove that (d) divides (n). (Here, is the Euler -function.)
Let d be a divisor of n and let
d = pe1 pe2 per
r
12
be its unique factorization into pow
Solutions
Math 4400
HOMEWORK #1
1. Let m be the least common multiple of two positive integers a, b. If M is any other common
multiple of a and b, prove that m divides M . (Hint: use division with rem
Solutions
Math 4400
EXAM #1
1. Find all solutions in Z 12Z = cfw_0, 1, 2, . . . , 11 to the congruence x10 1 (12).
First, if x10 = 1 in Z 12Z, then x9 is the inverse of x. Therefore, any solution to t
M4400 Exam2 Name gals-MS \ UID
1. Consider the modular numbers Z/5Z = {0, 1, 2, 3, 4}.
(i) Fill in the multiplication table for 2/52
Il
The modular numbers Z/SZ is a eld since every non-zero number
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M4400 Exam 1 Name UID
1. Suppose L is a subset of the integers Z, and L satises the following 3 properties:
(i) L is closed under addition, meaning
TyEL=>$+yEL
(ii) L is closed under mul
Numbers, Groups
and
Cryptography
Gordan Savin
Contents
Chapter 1. Euclidean Algorithm
1. Euclidean Algorithm
2. Fundamental Theorem of Arithmetic
3. Uniqueness of Factorization
4. Eciency of the Eucli
Math 4400, Fall 2016
Quiz 13
Name:
Calculators, cell phones, or notes are not allowed.
(1) The continued fraction algorithm for > 1 is purely periodic of period 2
such that [] = [2 ] = [4 ] = 6 and [1
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Math 4400, Fall 2016
Quiz 10
Name:
Calculators, cell phones, or notes are not allowed.
(1) Compute all the elements of T (11) F11 [ 2] and find the generators of T (11).
T (11) = cfw_1, 4 2, 3 2 2, 6
Math 4400, Fall 2016
Quiz 9
Name:
Calculators, cell phones, or notes are not allowed.
(1) Given that 2 is a primitive element of (Z/37Z) find a primitive 3-rd rootof 1 in
Z/37 and a square root of 3 (
Math 4400, Fall 2016
Quiz 6
Name:
Calculators, cell phones, or notes are not allowed. The problems are worth
5 pts.
(1) Solve x27 = 7 modulo 55 for some 0 x 54.
(55) = (5)(11) = 40. Since 1 = 27 3 2 4
Solutions
Math 4400
HOMEWORK #3
1. Let G be a group. Prove that the identity element is unique. (Hint: suppose there are two
identity elements e and e , and then show they are equal by considering the
Solutions
Math 4400
HOMEWORK #2
1. Let a and b be two positive integers such that a + b is a prime number. Prove that gcd(a, b) = 1.
Let d = gcd(a, b) and p = a + b a prime. Since d divides a and d di
Due Friday, August 2nd
Math 4400
EXAM #3
This is a take-home exam. You may use anything from class or the course notes. Do not collaborate
with anyone and do not use any additional resources. You may
Math 4400
EXAM #2
Do exactly 5 of the following 6 problems on the blank paper provided. Show your work.
1. Every ring is a eld, but some elds are not rings. For each of the following rings, determine
Solutions
Math 4400
EXAM #2
Do exactly 5 of the following 6 problems on the blank paper provided. Show your work.
1. For each of the following rings, determine whether or not it is a eld. For each rin
Solutions
Math 4400
REVIEW #2
1. A eld is a special type of ring. What additional conditions must a general ring satisfy in order
to be a eld? For each of these additional conditions, give an example
Math 4400
EXAM #1
Do exactly 6 of the following 7 problems on the blank paper provided. Show your work.
1. Find all solutions in Z 12Z = cfw_0, 1, 2, . . . , 11 to the congruence
x10 1 (12).
2. Find a
Solutions
Math 4400
REVIEW #1
1. Let d be the greatest common divisor of 1326 and 1254.
(a) Use the Euclidean algorithm to compute d.
1326 = 1 1254 + 72
1254 = 17 72 + 30
72 = 2 30 + 12
30 = 2 12 + 6
Math 4400, Fall 2016
Quiz 2
Name:
Calculators, cell phones, or notes are not allowed. Each problem is worth
5 pts.
(1) Let G be a group. Show that the identity of G is unique and inverses
in G are uni