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, 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
REVIEW PROBLEMS FOR MIDTERM IN
MAT312/AMS351; FALL 2013
(1)
(a) Compute the greatest common divisor (223,776).
(b) Find integers m,n such that (223,776)=223m + 776n.
(c) Is 223 invertible mod 776? If so, nd its inverse mod 776.
(2)
(a) Compute the cardina
HW 9, MAT 312/AMS 351: SPRING 2017
Problem 1. In Section 4.1 do problems #: 1 (only 1 2 , 2 3 , 1 3 ), 2,
3, 4
Problem 2. In Section 4.2 do problems #: 1, 2
In the problems below define permutations on 9 letters , S(9) to be
the following products of cycl
HW 2, MAT 312/AMS 351, LECTURE 01: SPRING 2017
Problem 1. Let a and b be two positive integers. Suppose that there exist
integers s, t and s0 , t0 such that as+bt = 2017 and as0 +bt0 = 2016. Compute
(a, b).
Problem 2. Do Excercises #2, 5, 6, 7 of Section
HW 7, MAT 312/AMS 351: SPRING 2016
Problem 1. Section 2.1 Ex. 2, 3, 6, 8.
Problem 2. Section 2.2 Ex. 1, 2, 3, 6, 10.
Problem 3 (Recall that two sets X and Y are said to have the same
cardinality (notation: |X| = |Y |) if there is a bijection f : X Y .). L
MAT 312/AMS 351 Homework 5 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
March 5, 2017
Problem 1
Do Problems 4 and 5 in Section 1.5 of the text:
4. a) Show that the polynomial x4 + x2 + 1 has no integer roots, but that it has a root modulo 3,
a
HW 3, MAT 312/AMS 351, LECTURE 01: SPRING 2017
Problem 1. For each of the following a and n, compute the remainder of
the division of a by n.
(a) a = 20172017 , n = 3.
(b) a = 6!, n = 7.
(c) a = 2424 , n = 10.
Problem 2. For each of the following numbers
MAT 312/AMS 351 Homework 5 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
March 6, 2017
Problem 1
Do Problems 4 and 5 in Section 1.5 of the text:
4. a) Show that the polynomial x4 + x2 + 1 has no integer roots, but that it has a root modulo 3,
a
HW 3, MAT 312/AMS 351, LECTURE 01: SPRING 2017
Problem 1. Write a formula for the sum of the first n integers that are
divisible by 3, then prove its validity.
Problem 2. Let a and b be two positive integers. Show using Corollary
ab
1.3.5 that lcm(a, b) =
HW 12, MAT 312/AMS 351: SPRING 2017
Problem 1.
(1) Set H = cfw_4a + 6b | a, b Z Z. Show that H is a
subgroup of Z.
(2) Is H cyclic?
Problem 2. Write down all the cyclic subgroups of the following groups
G6 ,G7 , G8 , G11 , G12 . Which of these groups is c
HW 10, MAT 312/AMS 351: SPRING 2017
Problem 1. Compute each of the powers.
(a) (15)1031 (where (15) is a 2-cycle in S(6).
(b) (285)277 (where (285) is a 3-cycle in S(8).
Problem 2. Write the following permutations in S(6) as a product of (not
necessarily
HW 5, MAT 312/AMS 351, LECTURE 01: SPRING 2017
Problem 1. Do problems #4 and #5 in section 1.5 of the text book; be sure
to show your work and explain your reasoning.
Problem 2. Find all integer solutions to the following linear congruences (if
possible).
MAT 312/AMS 351 Homework 8 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
April 22, 2017
Problem 1
Section 2.3 Ex. 1,2,5,6.
Proof. 1. See back of text
2. (a) Suppose a relation R on X is both symmetric and antisymmetic. Suppose to the contrary
t
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
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
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
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
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
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
MAT 312/AMS 351 Homework 9 Solutions
Giulia Sacca, Yuanqi Wang, Benjamin Sokolowsky
April 17, 2017
Problem 1
Do exercises 1,2,3, and 4 in section 4.1.
See back of the textbook.
Problem 2
Do exercises 1 and 2 in Section 4.2
See back of the textbook
Problem
HW 12, MAT 312/AMS 351: SPRING 2017
Problem 1.
(1) Set H = cfw_4a + 6b | a, b Z Z. Show that H is a
subgroup of Z.
(2) Is H cyclic?
Problem 2. Write down all the cyclic subgroups of the following groups
G6 ,G7 , G8 , G11 , G12 . Which of these groups is c
HW 11, MAT 312/AMS 351: SPRING 2017
Problem 1. Let G be a group.
(a1 )1 = a.
Show that for every element a G,
Problem 2. Let (G, ) be a group. Show that if a c = b c, then a = b.
Problem 3. Let G denote a group with 4 members: denote these members
by e (t
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, 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
Solutions to Homework Assignment 9
Maximal grade for HW9: 90 points
Section 5.1. 2. (10 points) Let a, b be elements of a group G. Fin (in
terms of a and b) an expression for the solution x of the equation axba1 = b.
Answer: x
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)