MAT 312/AMS 351, Spring 2011
Solutions to Homework Assignment 7
Maximal grade for HW7: 100 points
Section 4.2 1. Determine the order and sign of each of the following
permutations:
i) (10 points)
(1 2 3 4 5)(8 7 6)(10 11);
Answer: Sign=(-1), order=30.
Sol
MAT 312/AMS 351, Spring 2011
Solutions to Homework Assignment 1
Maximal grade for HW1: 100 points
For each of the following pairs a, b of integers, nd the greatest common
divisor d of a and b and express d in the form ar + bs.
1.1. i) (10 points) a = 7 an
MAT 312/AMS 351, Spring 2011
Homework Assignment 11
Due Before 11:45 am Wednesday 5/11.
Please read Section 5.4 thoroughly before starting on the problem set.
Written Assignment: Section 5.4: exercises 2, 3, 4; Problem E.
Show your work: all answers shoul
MAT 312/AMS 351, Spring 2011
Homework Assignment 10
Due Before 11:45 am Wednesday 5/4.
Please read Sections 5.15.4 thoroughly before starting on the problem
set.
Written Assignment: Section 5.1: exercise 10 (Gn = Z );
n
Section 5.3: exercises 1, 4; Sectio
MAT 312/AMS 351, Spring 2011
Homework Assignment 9
Due Before 11:45 am Wednesday 4/27.
Please read Sections 5.1 and 5.2 thoroughly before starting on the problem
set.
Written Assignment: Section 5.1: exercises 2, 3, 4(iii); Section 5.2:
exercise 5 (G20 =
MAT 312/AMS 351, Spring 2011
Homework Assignment 8
Due Before 11:45 am Wednesday 4/6.
Please read Section 4.2 thoroughly before starting on the problem set.
Written Assignment: Section 4.2: exercises 2, 4, 5, 10, 13 ;
Show your work: all answers should be
MAT 312/AMS 351, Spring 2011
Homework Assignment 7
Due Before 11:45 am Wednesday 3/30.
Please read Section 4.2 thoroughly before starting on the problem set.
Written Assignment: Section 4.2: exercises 1, 3, 6, 7, 11 (i);
Show your work: all answers should
MAT 312/AMS 351, Spring 2011
Solutions to Homework Assignment 3
Maximal grade for HW3: 100 points
Section 1.4. 5. (15 points) Show that no integer of the form 8n + 7 can
be written as a sum of three squares.
Solution: Let us describe the squares of congru
MAT 312/AMS 351 Fall 2010 Notes on Hamming Codes 1. Error-detecting matrices. Suppose a group code is generated by a matrix
G=
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 0 0
0 1 1 0
0 1 1 1
.
We can write this matrix as G = (I4 |A) where A is the 4 3 matrix
HW 6: MAT 312/AMS 351, SPRING 2015
(1) Do problems 8,10,11 in section 2.2. Also do problem 2(a)(b)(d)(e) in
section 2.3.
(2) Let m,n denote relatively prime integers greater than 1. The Chinese
remainder theorem states that for each [a]m Zm and [b]n Zn th
APPLIED ALGEBRA MAT 312 AND AMS 351: SPRING
2015
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 theor
REVIEW PROBLEMS FOR MIDTERM :MAT 312/AMS 351,
SPRING 2015
(1) Dene a subset D of the positive integers by
D = cfw_18a + 15b | a, b Z, 18a + 15b > 0.
Show that D is just the collection of all positive integers which are divisible
by 3.
(2) Is the cube root
HW4:MAT 312/AMS 351, SPRING 2015
(1) Find all the solutions (mod 126) to the linear congruence
21x = 35 mod 126.
(2) Find all the solutions (mod 2520=8 15 21) to the 3 simultaneous
linear congruences
x = 5 mod 8
3x = 9 mod 15
x = 13 mod 21
(3) Compute the
HW5:MAT 312/AMS 351, SPRING 2015
(1) How many integers mod n, [a]n , are invertible if n = 23 62 132 ?
(2) Find a positive integer k such that ([a]n )k = [1]n holds for n as in
problem (1) above and for a=5,35,77.
(3) Do #3,7,8,9 in section 1.6.
(4) If X1
HW3:MAT 312/AMS 351, SPRING 2015
(1) Compute the addition table for Z9 .
(2) Compute the multiplication table for Z9 . For each element in G9 compute its inverse.
(3) Find the following inverses if they exist: [15]1 ; [15]1 ; [8]1 .
157
69
68
(4) Do probl
MAT 312/AMS 351, Spring 2011
Homework Assignment 6
Due Before 11:45 am Wednesday 3/23.
Please read Section 4.1 thoroughly before starting on the problem set.
Written Assignment: Section 4.1: exercises 1, 2, 3;
Show your work: all answers should be justied
MAT 312/AMS 351, Spring 2011
Homework Assignment 5
Due Before 11:45 am Wednesday 3/16.
Please read Sections 2.12.3 thoroughly before starting on the problem
set. It is longer than usual since it covers 2 weeks.
Written Assignment: Section 2.1: exercises 2
MAT 312/AMS 351 Fall 2010 Homework 3 1. Prove (by induction, or otherwise) that ( a b ) | an b n for any integers a > b. A proof by induction would start with a2 b2 = (a b)(a + b), note that a3 b3 = (a b)(a2 + ab + b2 ), use this form to conjecture the ge
MAT 312/AMS 351 Fall 2010 Homework 7 1. Explain why if n > 2 then (n) is even. 2. Fill in this table according to the map Z Z Z calculated from 5 12 60 the algorithm in the proof that (ab) = (a)(b). Show your work. Note that vertically this table will not
MAT 312/AMS 351 Fall 2010 Homework 9 1. Calculate the order of the permutation = 12345678 46182573 .
Hint: write it rst in cycle notation. 2. Same question for = 12345678 41576382 .
3. Give a conjugacy relating 1 = (1547)(263) to 2 = (123)(4567), so that
MAT 312/AMS 351 Fall 2010 Homework 11 1. A linear fractional transformation f (x) is a function of the form f ( x) = ax + b cx + d
where a, b, c, d, are real numbers satisfying ad bc = 1. Show that if f (x) as above and g (x) = (ex + f )/(gx + h) are line
Calculator program giving remainders: :Prompt M :Lbl P :Prompt A :A-M*int(A/M)->R :Disp R :Goto P Prompt, Input, Disp are on the I/O menu from the PRGM button. Lbl, Goto are on the CTL menu from the PRGM button. int (Integer part) is on the NUM menu from
MAT 312/AMS 351 Fall 2010 Review for Midterm 1 1.2. Understand how to use induction to prove that a statement P (n) holds +1) for every integer n. Example: P (n) is the statement 1 + 2 + . . . + n = n(n2 . n Problem 2 p.14. Example: The binomial coecients
MAT 312/AMS 351 Fall 2010 Review for Final Revised 12/9/10. NOTE: Final is cumulative. Use review sheets 1 and 2 as well as this one. Also review all homework, as well as midterms 1 and 2. 3.2 Given a permutation S (n), know how to compute its order (Deni
MAT 312/AMS 351 Midterm exam #1 with SOLUTIONS Tuesday 10/8/02
1. Prove by induction that for all positive integers n,
n
i(i 1) =
i=1
n(n2 1) . 3
SOLUTION: The base case n = 1 is true since both sides of the equality to be established produce 0. So assume
MAT 312/AMS 351 Midterm exam #2 with solutions Thursday 11/14/02
1. (a) State precisely Eulers theorem. SOLUTION Let n be a positive integer and let a be an integer that is relatively prime to n. Then a(n) 1 mod n.
(b) Compute the last two digits of 7523
MAT 312/AMS 351, Spring 2011
Solutions to Homework Assignment 8
Maximal grade for HW8: 100 points
Section 4.2 2. (20 points) Give an example of two cycles of lengths r
and s respectively whose product does not have order lcm(r, s).
Answer: (1 2) and (2 3)