HW 1, MAT 312/AMS 351, LECTURE 01: SPRING 2016
Do problems #3,6 in section 1.1, as well as the following problems.
Problem I: Find the greatest common divisor (gcd) of the numbers 210
and 56; write this gcd in the form 210m + 56n for integers m,n.
Problem
HW 7, MAT 312/AMS 351: SPRING 2016
Show some work or explain your reasoning, for your solutions of each of
the following prolems.
In problems (1)-(3) below define permutations on 9 letters , S(9) to
be the following products of cycles: = (13869)(27831)(43
SOLUTIONS FOR REVIEW FOR MIDTERM FOR MAT
312/AMS 351: SPRING 2016
The midterm will cover sections 1.1-1.6, 2.1-2.3, 4.1.
(1) Let a,b,p,q denote integers which satisfy: a = bp + 756; b = 756q + 315.
Using only this information compute the greatest common d
HW 1, MAT 312/AMS 351, LECTURE 02: SPRING 2016
Do problems #3 and #6 in section 1.1, as well as the following problems:
Problem I: Find the greatest common divisor (gcd) of the numbers 210
and 56; write this gcd in the form 210m + 56n for integers m, n.
P
ANOTHER REVIEW FOR FINAL EXAM; MAT 312/ AMS
351 FALL 2013
(1) Dene a permutation S6 by = (1, 2, 3)(2, 1)(3, 6)(3, 4, 6). Write
as a product of disjoint cycles. Is the cyclic group < > isomorphic to Z6 ?
(2) Let D5 denote the symmetry group of the pentago
HOMEWORK WEEK 4
05 AUGUST 2014
Deadline: 12 August 2014, TUESDAY 13:00. (Notice the change of day.)
Q. 1. Show whether the groups G10 and G7 are cyclic or not. If so, determine their generators.
Proof. The subgroups generated by the elements of G7 are
[1]
MAT 312/AMS 351, LECTURE 02: SPRING 2016
Topics and Text:
In this course we will discuss elementary results in number theory, set theory and group theory. Also several topics in coding theory are developed,
related to number theory and group theory Our te
MAT 312 - AMS 351
FINAL EXAM
SUMMER II, 14 August 2014
Question 1.
a. Find all four solutions to the equation x2 1 0 mod 35.
(5 pts)
b. Solve the equation [243]n [x]n [1]n for n = 1130.
(7 pts)
c. Find the last three digits of the integer
2001
20032002
.
MAT 312/AMS 351
Solutions to Final Exam.
1. (30 points) a) (10 points) Solve the system of congruences
x = 10
mod 13,
x = 16
mod 19.
mod 13, x = 3
mod 19,
Answer:x = 244 mod 247.
Solutions: 1) Remark that
x = 3
so
x = 3
mod 13 19,
x = 244
mod 247.
2) Sinc
AMS 311, Fall 2014
Homework 3
Due at the beginning of class, Sept 23, 2014
1. (20 points, 5 points each) If P (A) = 0.8, P (B|A) = 0.5, A and B are independent, for
each statement, say True, False, or Cant Tell (if you give some reasoning, you have
a poss
AMS 311, Fall 2014
Homework 2: Suggested Solution Key
1. Sarah has n keys, of which one will open the door.
(a) If she tries the keys at random, discarding those that do not work, what is the
probability that she will open the door on the kth try?
p = P (
AMS 311, Fall 2014
Homework 4
Due at the beginning of class, Oct 7, 2014
1. (20 points) A company is deciding on installing a 3-component or 5-component satellite
system in XYZ town. It is known that the system functions on any given day if a majority
of
AMS 311, Fall 2014
Homework 2
Due at the beginning of class, Sept 16, 2014
1. (10 points, 5 points each) Sarah has n keys, of which one will open the door.
(a) If she tries the keys at random, discarding those that do not work, what is the
probability tha
AMS 311, Fall 2014
Homework 3: Suggested Solution Key
1. (20 points, 5 points each) If P (A) = 0.8, P (B|A) = 0.5, A and B are independent, for
each statement, say True, False, or Cant Tell (if you give some reasoning, you have
a possibility of partial cr
MAT 312/AMS 351 Spring 2014 Review for Midterm 2
1.6 Understand the concept of the multiplicative order of a mod n. Understand Theorem
1.6.2: if a has order k mod n, then ar as mod n if and only if r s mod k. Know how to
prove Fermats Theorem (1.6.3): if
Solutions to Homework Week 2
24 July 2014
Section 1.5. page 59.
1. Exercise 1 (vi). Solve 15x 5 mod 100.
(1 pt)
Since (15, 100) = 5 and 5|5 there are ve solutions. We reduce the congruence to 3x
1 mod 20 and solve it for x.
3x 1 mod 20
7 3x 7 1 mod 20
x
MAT 312
SUMMER II, 2014
Solutions
MIDTERM
Question 1.
a. Find all n N, with n 2, for which the following congruences hold.
(i) 13 7 mod n,
(ii) 1 6 mod n,
(iii) 0 3 mod n.
b. Find all n N such that (n) = 12.
c. Prove that if an odd prime number can be exp
REVIEW FOR FINAL EXAM FOR MAT 312 AND AMS
351: FALL 2013
(1)
(a) Use the Euclidean algorithm to compute gcd(a, b) for the integers
a = 148 and b = 127.
(b) Is [127]148 invertible in Z148 ? Explain.
(c) Is Z148 a eld (with respect to the usual addition mul
ADDITIONAL PROBLEM FOR HW 4
Problem: Recall that for each integer n 2 we let Gn Zn denote the
subset of all integers mod n which are invertible mod n. Let m, n denote
integers satisfying m, n 2 and (m, n) = 1. Dene a mapping
f : Zm Zn Zmn as follows: set
HW 9 ADDITIONAL PROBLEM FOR
MAT312/AMS351; FALL 2013
Problem: Let In denote the n n identity matrix; and for each 1 i n
let Ci denote the ith column of In thus In = [C1 , C2 , C3 , ., Cn ]. Dene
a function f : Sn GL(n, R) from the set of permutations on n
ADDITIONAL PROBLEM FOR HW 1
Problem: For positive integers a, b set
D = cfw_am + bn : m, n Z, am + bn > 0; and let d denote the minimum
integer in D. In class we discussed why d is equal to the greatest common
divisor of a, b. In this problem you are aske
Homework Week 1
8 July 2014
Part A is for credit only and worth 25 points. It is optional to solve Part B. If you hand
in the answers for Part B, I will check them too.
PART A
1. (Exercise 1.1.1(ii) Find the greatest common divisor of a = 28 and b = 63 an
MAT 312 - AMS 351
PRACTICE QUESTIONS for FINAL EXAM
SUMMER II, 2014
Q. 1. Prove that the square of any odd integer always leaves a remainder of 1 when divided
by 8.
Proof. Let n = 2k + 1 for some k Z. Then
n2 = (2k + 1)2 = 4k 2 + 4k + 1 = 4k(k + 1) + 1.
E
Section 1.5
3) b) 0 10 1 1 1 1 = 0 1 10 1 0 d) 11 1 0 10 5) The null space of H is given by the equations. 0 = c1 + c3 0 = c1 + c2 + c4 0 = c1 + c5 Thus, c3 , c4 , and c5 are check bits, while c1 and c2 are free. We know that (0, 0, 0, 0, 0) is in the nul
Section 2.1
1b) The four symmetries, and the associated permutations of the vertices, are as follows: The identity map (A A, B B , C C , D D); rotation through 180 degrees (A C , B D, C A, D B ); reection through the horizontal axis (A D, B C , C B , D A)
Section 2.6
1) a) Yes. b) No. Two elements sent to C. c) Yes. d) No. Two elements sent to A. 2) a) ABCD CDBA d) ABCDEF DFABEC 3) a) ABCDE DABEC c) ABCDEF CDEFBA 4) a) Not a permutation group. For instance: ABC CAB which is not in the set. c) This is a per
Section 3.2
1) a) a b is true because b = a c. b) a c is false. c) b e is false. d) b c is false. 3) The left cosets of H are as follows: (0, 0) + H (0, 1) + H (0, 2) + H (0, 3) + H = = = = cfw_(0, 0), (1, 2) cfw_(0, 1), (1, 3) cfw_(0, 2), (1, 0) cfw_(0,
Section 3.3
4) 1. By Burnsides Theorem, there are
1 6
(2 + 2 + 2) = 1 equivalence class. This agrees.
2. By Burnsides Theorem again, there are 1 (3 + 3 + 2 + 2 + 2 + 6) = 3 equivalence 6 classes. This agrees. 5) Label the three vertices of the triangle A,
HW 6, MAT 312/AMS 351, LECTURE 01: SPRING 2017
Problem 1. Do Problems #1, 2, 3, 5, 6 of Section 1.6
Problem 2. Compute the (multiplicative) order of each [a]n .
(a) [7]55
(b) [2]23
(c) [5]27
(d) [63]101
Problem 3. Compute each power ([a]n )k express this
MAT 312/AMS 351 Homework 5 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
March 18, 2017
Problem 1
Do problems 1,2,3,5,6 in Section 1.6
1. Find the orders of:
i)2 modulo 31
ii)10 modulo 91
iii)7 modulo 51
iv)2 modulo 41.
Answer: The orders are 5
PRACTICE PROBLEMS FOR, MAT 312/AMS 351: SPRING
2017
The midterm will cover Sections 1.1-1.6 and 2.1-2.2.
From the book, you can practice with the following problems (some of
them you have done in the HW). If you are comfortable with these problems
you don
SOLUTIONS TO PRACTICE PROBLEMS FOR, MAT
312/AMS 351: SPRING 2017
Problem 1. Find all integers x such that
35x 20
mod
85
Solution 1. Dividing this conguence by 5 yields
(i) 7x 4
mod
17
Note that [7]1
17 = [5]17 (show work). So if we multiply the congruence
MAT 312/AMS 351 Homework 1 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
February 13, 2017
Problem 1
Use proof by contradiction to show that the following set X of positive rational numbers does not
have a smallest element:
X=cfw_
1
| a N Q>0
a