Second exam for Math 463
Prof. Bernardo brego
May 1st, 2014.
Time limit: 75 minutes. Problems 15 are worth 20 points each, Problem 6 is worth 10 extra points. All
your answers must be justified. Good luck!
In the following problems all variables are integ
Math 342 Number Theory
Quiz 1 Solution (Section D102)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Quasi-statement: a sentence which is neither always true nor always false.
Instead, its truth or falsity depends upon some variable.
Seq
Math 342 Number Theory
Solution Quiz 5 (Section D103)
Problem 1 (10 points total)
Dene the following:
For a positive integer m; (m)
(m) =| cfw_k N : 1 k m, (k, m) = 1 |
Primitive roots modulo m
x is a primitive root modulo m if ordm a = (m).
Index of a
Math 342 Number Theory
Solution Quiz 1 (Section D101)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Statement: A statement is a declarative sentence that is either true or false.
Family: A family is a function F where is an index set an
Math 342 Number Theory
Solution Quiz 4 (Section D101)
Problem 1 (10 points total)
Dene the following:
(3 point) Eulers function
If n is a positive integer, (n) is dened to be the number of integers between
1 and n which are coprime to n.
(3 points) Pyth
1
Problem set 5 - D102
Problem 1.1 Prove the following statements:
If p and 2p + 1 are odd primes, prove that (4p + 2) = (4p) + 2.
If p and 2p1 are odd primes, and n = 2(2p1) then prove that (n) = (n+2).
For each natural number n, (3n) = 3(n) i 3 | n.
Abbreviations and Notation
Notation for Sets, Logic, and Number Theory
|A|
AB
AB
A\B
AB
AB
aA
n|m
gcd(m, n)
lcm(m, n)
(n)
(n)
(n)
a b (mod m)
ordm (a)
ak ak1 . . . a0 (b)
S(n)
( f1, f2, . . . , fm )
x
x
cfw_x
ep
pk n
fn
Mn
the number of elements in the
15
Arithmetic Functions
A real or complex valued function of one or several variables where all variables varies in N
is called an Arithmetic function. Therefore a single variable real valued arithmetic function
is a function from N to R.
1
Example 15.1 S
13
Primitive Roots II
Theorem 13.1 (Lagrange) Let p be prime, let n > 0 and let f (x) = an xn +
an1 xn1 + . . . + a1 x + a0 be a polynomial for which an is not divisible by p. Then
f (x) has at most n incongruent roots modulo p.
Proof: We proceed by induc
14
Quadratic Residues
Quadratic Residue: If a and m are relatively prime, then we say that a is a
quadratic residue modulo m if the equation x2 a (mod m) has a solution.
Lemma 14.1 If p is an odd prime and (a, p) = 1 then the equation x2 a (mod p)
has eit
12
Primitive Roots I
Order: If a is relatively prime to m, the order of a modulo m, denoted ordm a, is
the exponent of the smallest power of a congruent to 1 modulo m.
Primitive Root: If a has order (m) modulo m, then we call a a primitive root
modulo m.
9
Diophantine Equations
Diophantine Equation: A polynomial equation, possible in many variables, with
integer coecients. For instance x2 + 7xy 6 = 0 or x2 + y 2 z 2 = 0. A solution to
a diophantine equation is an assignment of integers which solves the eq
8
Congruences
Basic Properties
Congruence: Let a, b be integers, and let m be a positive integer. We write
a b (mod m)
and say a is congruent to b modulo m, if m|(a b). Note that m|(a b) if and
only if there exists k Z so that mk = a b, that is, if and on
10
Eulers Theorem
Eulers Phi or Eulers totient function: For every positive integer n, we let (n)
denote the number of positive integers 1 k n for which (k, n) = 1. I.e.
(n) = |cfw_1 k n | (k, n) = 1|.
&
&
Example: (12) = 4 since it counts the numbers 1,
11
More on Eulers totient function
As we know
(n) = |cfw_1 k n | (k, n) = 1| =
1.
1kn
(k,n)=1
Notice that for any set E, |E| denotes the number of elements of E and we have
|E| =
1.
xE
Examples: (1) = (2) = 1, (3) = 2 and in general for any prime p, (p) =
7
More on gcd, lcm and Primality
Theorem 7.1 Suppose that a and b have prime power factorizations a = pa1 pan
1
n
and b = pb1 pbn . Then
1
n
mincfw_a1 ,b1
gcd(a, b) = (a, b) = p1
mincfw_a
pn n ,bn
and
maxcfw_a1 ,b1
lcm(a, b) = [a, b] = p1
maxcfw_an ,b
Math 342 Number Theory
Solution Quiz 2 (Section D101)
Problem 1 (10 points total)
(a) (6 points) Dene the following:
Divisor: For two integers a and b, we say that a is a divisor of b if there exists
some m Z such that b = am.
lcm of a sequence of integ
Math 342 Number Theory
Solution Quiz 5 (Section D101)
Problem 1 (10 points total)
Dene the following:
(3 point) Primitive root Modulo m
If a has order (m) modulo m, then we call a a primitive root modulo m.
(3 points) Order of an integer relatively prim
Math 342 Number Theory
Solution Quiz 6 (Section D101)
Problem 1 (15 points total)
Dene the following:
(3 point) Quadratic Residue mod a prime p
We say that an integer a is a quadratic residue modulo p if there exists an
integer x such that x2 a (mod a).
g
Kibronlnc.
gua/Oi
User's Guide Kibron AquaPi
A Portable Microtensiometer
User's Guide
May 19, 2009
Revision E
Copyright 2010 Kibron Inc. All rights reserved.
Information in this manual is subject to change without prior notice. International
copyright l
850
WILLIAM D. HARKINS AND ROY W. WAMPLER
VOl. 53
[CONTRIBUTION
FROM KENTCHEMICAL
LABORATORY
OF THE UNIVERSITY
OF CHICAGO
1
THE ACTIVITY COEFFICIENTS AND THE ADSORPTION OF
ORGANIC SOLUTES. I. NORMAL BUTYL ALCOHOL IN
AQUEOUS SOLUTION BY THE FREEZING POINT
Chem 366W
LAB MANUAL
Dipole
Page VII - 1
Moment
ea7b8ad42bc15652ad6eb57d369e912b3942e052.docxx
EXPERIMENT VII
DIPOLE MOMENT OF POLAR MOLECULES IN SOLUTION
(S&G 5th ed. Expt 31, 6 th and 7th eds. Expt. 30, 8 th ed.
Expt.29)
The heterodyne-beat frequency me
Chem 366W
LAB MANUAL
C /C
p
v
faad95f787225899d33e552a669892dad709af15.docxx
Page IV- 1
Ratio
EXPERIMENT IV
Cp/Cv RATIO
(S&G 5th, 6 th, 7th & 8 th eds. Experiment 3)
In this experiment the heat capacity ratio Cp/Cv, is obtained by the
Sound Velocity metho
MATH 342
Assignment 4
Due Monday June 20 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Xie
Jianchao
301252416
Please attach this cover sheet and provide answers on additional sheets in order.
(1) How many incongruen
Selected Solutions from Assignment Problems
June 23, 2016
Question 1.4. Prove that if a and b are positive integers satisfying (a, b) =
[a, b], then a = b.
Proof. Let d = (a, b) = [a, b]. Then, as d is a multiple of a and b, d = ax =
by, for some integers
MATH 342
Assignment 1
Due Friday May 27 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Please attach this cover sheet and provide answers on additional sheets in order.
(1) Find the greatest common divisor d of 481 a
MATH 342
Assignment 3
Due Monday June 13 by 4pm in the assignment box near the AQ 4000 area.
Last Name:
First Name:
Student #:
Please attach this cover sheet and provide answers on additional sheets in order.
(1) Show that if n is a positive integer with
Solutions to 18.781 Problem Set 2 - Fall 2008
Due Tuesday, Sep. 23 at 1:00
1. (Niven 1.3.39) Prove that
1
1
1
1
1
1
1 1
+ +
=
+
+ +
.
2 3
2007 2008
1005 1006
2008
You may find it easier to prove a general statement!
This problem hints at a general stateme
Math 463 Elementary Number Theory
Homework 4 Solutions
Prof. Arturo Magidin
1. Find all solutions to the congruence 20x 4 (mod 32) by solving the corresponding diophantine
equation (of the form ax + by = c).
Solution. The congruence 20x 4 (mod 32) corresp