Math 235 (Fall 2009): Assignment 10 Solutions
1.1: 28 = 4 7. Since gcd(4, 7) = 1, using the Euclidean Algorithm (or simply by
guessing), we nd that 1 = 4 2 + 7 (1). We dene e1 := 1 4 2 = 7 and
e2 := 1 7 (1) = 8.
It is easy to check (one by one) that the s
MATH 235: Assignment 5 Solutions
1.1: Writing N = n0 + 10n1 + . . . + 10k nk , we can reduce
fact that 10 is congruent with 1 mod 3, we conclude that
mod 3 and using that
N n0 + 1 n1 + 12 n2 + . . . + 1k nk .
Clearly then N 0 mod 3 i the sum of its digits
ASSIGNMENT 5 - MATH235, FALL 2007 Submit by 16:00, Monday, October 15 (use the designated mailbox in Burnside Hall, 10th floor).
1. To check if you had multiplied correctly two large numbers A and B, A B = C, you can make the following check: sum t
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SUMS OF TWO SQUARES
PETE L. CLARK
1. Sums of two squares
We would now like to use the rather algebraic theory we have developed to solve a
purely Diophantine problem: which integers n are the sum of two squares?
The relevance of the Gaussian integers is n
Solution 1:
By the rst homomorphism theorem we know that
R
=
Z
ker f .
Recall ker f is an ideal in Z. Since Z is a PID we see that ker f = (n) for some integer n.
Case 1: Suppose n = 0.
This implies
R
=
Case 2: Suppose n = 0.
Then (n) = nZ. Thus
R
=
Z.
=
Final Examination
December 17, 1996
189-235A
1.
Say whether the following statements are true or false. You do not need to provide
justi cation.
a The ring Z x of polynomials with integer coe cients is euclidean.
b The ring R x of polynomials with real co
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\lheadcfw_\bf Math 235A: Practice Final
Important theorems about ring homomorphisms and ideals.
In the following propositions, the letters R and S will always denote rings.
1. Suppose that : R S is a ring homomorphism. Then Ker() is an ideal in the ring R
and Im() is a subring of the ring S . (
6
Euclidean Domains, PIDs and UFDs
We now consider three classes of integral domains that have additional structure, allowing us to say a good deal more about there algebraic properties.
These are unique factorization domains, or UFDs, which allow unique
Math 455.1 April 4, 2009
Fermats Little Theorem
For the RSA encryption system, we shall need the following result
Theorem 1 (Fermats Little Theorem). Let p be a prime. Then for each integer
a not divisible by p,
ap1 1 (mod p).
Proof. Let a be an integer f
THE GAUSSIAN INTEGERS
KEITH CONRAD
Since the work of Gauss, number theorists have been interested in analogues of Z where
concepts from arithmetic can also be developed. The example we will look at in this handout
is the Gaussian integers:
Z[i] = cfw_a +
UNIFORM CONTINUITY
FOKKO VAN DE BULT
Uniform continuity is a property of functions which is slightly stronger than just
ordinary continuity. It has many applications, the most important of which for our
purposes is that it implies that continuous function
Continuity and Uniform Continuity
521
May 12, 2010
1. Throughout S will denote a subset of the real numbers R and f : S R
will be a real valued function dened on S . The set S may be bounded like
S = (0, 5) = cfw_x R : 0 < x < 5
or innite like
S = (0, ) =
Uniform Continuity
Recall that if f is continuous at x0 in its domain, then for any > 0, > 0 such that for
all x in the domain of f , |x x0 | < = |f (x) f (x0 )| < . The number will generally
depend on x0 . In the future, we shall omit the phrase for all
DIVISIBILITY AND GREATEST COMMON DIVISORS
KEITH CONRAD
1. Introduction
We will begin with a review of divisibility among integers, mostly to set some notation
and to indicate its properties. Then we will look at two important theorems involving
greatest c
Math 347
Worksheet: Number Theory I
A.J. Hildebrand
Denitions: Divisibility, Primes, Composites, GCDs
Divisibility. Let a, b Z, with a = 0.
Denition: We say a divides b if there exists m Z such that b = ma.
Notation: a | b for a divides b; a b for its n
The Division Algorithm
Theorem (The Division Algorithm). Let n and m be natural numbers. Then (existence
part) there exist integers q (for quotient) and r (for remainder) such that
m = nq + r
and 0 r n 1. Moreover (uniqueness part), if q , q and r, r are
3.2. THE EUCLIDEAN ALGORITHM
53
3.2. The Euclidean Algorithm
3.2.1. The Division Algorithm. The following result is known
as The Division Algorithm :1 If a, b Z, b > 0, then there exist unique
q, r Z such that a = qb + r, 0 r < b. Here q is called quotien
LECTURE 11
Congruence and Congruence Classes
Definition 11.1. An equivalence relation on a set S is a rule or test applicable to pairs of elements
of S such that
(i)
aa
,
(ii)
ab
(iii)
aS
(reexive property)
ba
a b and b c
(symmetric property)
ac
(transiti
Number Theory - Factors, GCD, and Primes
1
Introduction
Number theory is the branch of mathematics that deal with properties of integers. However, it has a
very close tie with computer science. The most notable applications of number theory is in the eld
Fundamental Theorem of Arithmetic and Divisibility Review Mini Lecture
Here we will provide a proof of the Fundamental Theorem of Arithmetic (about prime factorizations).
Before we get to that, please permit me to review and summarize some divisibility fa
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MATH 235, Sample Midterm Solutions
1. Find d = gcd(126, 78), and find some u, v Z so that d = 126u + 78v.
Solution. Applying the Euclidean algorithm, we get
126
78
48
30
18
12
=
=
=
=
=
=
78 1 + 48
48 1 + 30
30 1 + 18
18 1 + 12
12 1 + 6
62+0
We conclude t
McGill University
Department of Mathematics and Statistics
MATH 235 Algebra 1, Fall 2016
Instructors. The instructors for this course are:
Jan Vonk: Burnside Hall 1242, jan.vonk@mcgill.ca
Hao Lee (TA): Burnside 1035, hao.lee@mail.mcgill.ca
Alice Pozzi
Algebra I, MATH 235
Fall 2015
Instructor: Prof. Eyal Goren
Office: Burnside Hall 1108
Office hours: MWF, 14:30 - 15:30
Lecture hours: MWF, 13:35 - 14:25, Maass 112
TA Office hours and tutorial sessions:
Bruno Joyal, Monday 15:00 - 16:00, Friday, 11:00 to