u y v w v y t u
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MATH 1020, Homework 7
Kiumars Kaveh
October 20, 2011
Due date: Wednesday October 26, 2011
Problem 1: Find all positive integers n with (n) = 12.
Problem 2: Show that if a and b are relatively prime positive integers then
a(b) + b(a) 1 (mod ab).
Problem 3:
MATH 1020, Homework 6
Kiumars Kaveh
October 12, 2011
Due date: Wednesday October 19, 2011
Problem 1: Find the remainder of the following numbers modulo the given
number. You can use Fermats or Wilsons theorems.
(i) 6! (mod 7)
(ii) 516 (mod 17)
(iii) 2200
MATH 1020, Homework 4
Kiumars Kaveh
September 22, 2010
Due date: Wednesday September 28, 2011
Refer to the text for the statement of Dirichlets theorem (Theorem 3.3) and
the notions of twin primes (pairs of primes of the form p and p + 2), and the
sieve o
MATH 1020, Homework 5
Kiumars Kaveh
September 30, 2010
Due date: Wednesday October 5, 2011
Problem 1: Use the Euclidean algorithm to nd the greatest common divisor (666, 1414).
Problem 2: Find integers a, b, c such that (a, b, c) = 1 but (a, b) = 1,
(a, c
MATH 1020, HOMEWORK 8
KIUMARS KAVEH
Due date: Monday November 7, 2011
Problem 1: Recall that the Dirichlet product of two arithmetic functions
f and g is dened as:
(f g)(n) =
f (d)g(n/d).
d|n
Compute ( )(10), where is the Euler function and is the Mbius
o
MATH 1020, HOMEWORK 10
KIUMARS KAVEH
Due by Friday December 9, 2011 (either hand it in class on Wednesday or slip it under my door by Friday.)
This is a bonus homework. Only best 8 out of 10 homeworks will count.
So you may not need to hand in HW10 if you
MATH 1020, HOMEWORK 9
KIUMARS KAVEH
Due date: Wednesday November 28, 2011
Problem 1: Find a complete set of (incongruent) primitive roots of 17.
Show your computation.
Problem 2: Find all the integer solutions x of the congruence:
3x 4 (mod 23).
j
Problem
Department of Niasthematics
University of Pittsburgh
MATH 1020 (Number theory)
Midterm 1, Fall 201i!)
Instructor: Kiumars Kaveh
Last Name: Student Number:
First Name:
FOR FULL MARK YO U MUST PRESENT YOUR SOLUTION CLEARLY. I
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Question Iark
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1
Some practice problems for midterm 1
Kiumars Kaveh
October 10, 2011
Problem: (a) What does it mean to say that a numerical algorithm runs in
polynomial time?
Solution: The number of required bit operations to complete the calculation
is at most a constant
MATH 1020, Review topics for midterm 1
Kiumars Kaveh
October 8, 2011
You are responsible to know the homework problems, some problems in
the test will be from homeworks or very similar to them.
There will be proof problems (like the ones in the homework
MATH 1020, Review topics for nal exam
Kiumars Kaveh
December 13, 2011
The focus of the exam is on the material covered in the second half of the
course although of course there will be questions from the rst half too.
The format is similar to the midter
Solutions to selected homework problems
Kiumars Kaveh
October 8, 2011
Problem: Find and prove a formula for the sum of rst n Fibonacci numbers
with even indices, i.e., f2 + f4 + + f2n .
Solution: By looking at the rst few Fibonacci numbers one conjectures
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in som
Calculus III
Preface
Here are my online notes for my Calculus III course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus III or needing a refresher in som
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2005 UW Integration Bee
Preliminary Competition
February 1, 2005
Welcome to the second annual UW Integration Bee! Only a
writing implement will be necessary. Specifically, no textbooks,
notes, tables of integrals, calculators, math T-shirts, tattoos, or
c
) a ( !% B ) a $C17$ (% 0 ) A (% ) (% ) % A 0 ) A ( 7$yy( !1"'!&% A 0 ) a (% ) A (% $ B Q 1ee$ A 6 0 ) a( ) A (% $I$ B Q 1fH"'!&% v % ) % u v (% ) (% 0 w v u )s (% y( !h( !&h5`t$ v % ) (% u v (% ) % 0 w v u )s ( ( !h58V$y( !&`x$Ctf"'!&% Q 1efE5 6 cH$ 0 )
Trigonometry
Dierentiation Formulas
d
k=0
dx
(1)
d
[f (x) g (x)] = f (x) g (x)
dx
(2)
d
[k f (x)] = k f (x)
dx
(3)
d
[f (x)g (x)] = f (x)g (x) + g (x)f (x) (4)
dx
d
dx
f (x)
g (x)
=
g (x)f (x) f (x)g (x)
2
[g (x)]
(5)
y
( 1 ,
2
(
(
3
)
2
2
, 22 )
2
31
, 2
Dierentiation Formulas
Integration Formulas
d k=0 dx d [f (x) g (x)] = f (x) g (x) dx d [k f (x)] = k f (x) dx d [f (x)g (x)] = f (x)g (x) + g (x)f (x) dx g (x)f (x) f (x)g (x) d f (x) = 2 dx g (x) [g (x)] d f (g (x) = f (g (x) g (x) dx dn x = nxn1 dx d s
Calculus II
Preface
Here are my online notes for my Calculus II course that I teach here at Lamar University.
Despite the fact that these are my class notes, they should be accessible to anyone wanting to
learn Calculus II or needing a refresher in some o
MATH 1020, Homework 1
Kiumars Kaveh
September 2, 2011
Numbered problems are from the Kenneth Rosens Elementary number theory and its applications, 6th Ed.
All problems have equal points.
Due date: Monday September 12, 2011
Section 1.1: 4, 40. If you like