Math 104 A - Homework 7. Due Wednesday, December 10.
The date when you have the knowledge to work on each problem is indicated next to it.
1. (Friday, December 5. On the existence of primitive roots and the optimal order of elements.)
Solve Problem 17 (a)
Math 104A - Homework 2. Due Wednesday, October 22.
1. Leftovers from the previous week. In case you havent done so last week, turn in:
Problem 4(c) of Section 1.4.
Problem 1 of section 2.1.
2. Linear diophantine equations. Section 2.3, solve problems 1,
Math 104A - Fall 2014 - Final Exam Solutions
Problem 1.
Solve the quadratic congruence
2x2 5x 4 0
mod 72 .
Solution:We rst solve the congruence
2x2 5x 4 0
mod 7.
We use the quadratic formula
5 1
5 57
=
= x1 = 1, x2 = 6 41 = 6 2 = 5 in Z7 .
x1,2 =
4
4
We u
Math 104A - Fall 2014 - Midterm II
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation wil
Math 104A - Homework 6. Due Wednesday, December 3.
The date when you have the knowledge to work on each problem is indicated.
1. (Monday, November 24. Primitive roots.) List all primitive roots mod 9 and mod 11,
respectively.
Hint: Once you nd one primiti
MATH 104A: FALL 2016
SOLUTION TO SELECTED PROBLEMS HOMEWORK 1
KUANG SITTIPONG T
Note: Solutions provided below might not be complete. Especially, when
I say (why?), which means that details of the proof should be added.
1.2 - 2a
Let H = aj | j an integ
Exam II 11/18/09
SOLUTIONS
Math 406
1. (15 points) Using the Chinese Remainder Theorem, find all solutions to the following
system:
7x 4 (mod 11)
x 3 (mod 7).
First note that since 71 8 modulo 11, the first equation can be replaced with x 1
modulo 11. The
Math 104A Prof. Rabin Solutions to First Midterm Exam
(1) Find all integer solutions of 70x + 15y = 250. Are there solutions having both x
and y positive?
The equation is equivalent to 14x + 3y = 50, which has integer solutions since
(14, 3) = 1. The Eucl
Math 104A Prof. Rabin Solutions to Second Midterm Exam
(1) In the set of Gaussian integers Z[i], determine whether 3 + 5i is prime. Also
determine whether 19 is prime.
The size of 3 + 5i is s(3 + 5i) = 32 + 52 = 34 = 2 17. Therefore, if 3 + 5i has a
nontr
Math 104A Prof. Rabin Homework Assignment 5
due Thursday, November 3
Do the following problems from the textbook:
Section 3.2, problems 6, 8, 10, 11, 13, 14, 18, 19.
One approach to Problem 8 is to show that j(n) grows faster than some other function
that
1.
(i) Recall that R comes equipped with the norm
N (a + b 37) = a2 + 37b2
and that the norm is multiplicative:
N (zw) = N (z) N (w).
If 2 were reducible, then we could write
2=
where , are not units in R. Let
= a + b 37
for a, b Z. Taking norms we nd
22
Math 104A - Fall 2014 - Midterm I
Name:
Student ID:
Instructions:
Please print your name, student ID.
During the test, you may not use books, calculators or telephones.
Read each question carefully, and show all your work. Answers with no explanation will
Math 104A - Homework 1
Section 1.1 - 2, 4a, 4c, 8, 12a, 12b, 24
2 Find intervals containing solutions to the following equations.
(a) f (x) = x 3x = 0.
Since f (0) = 1 and f (1) = 2 , by the intermediate value theorem,
3
there is some solution of f in the
Solutions
1.
(i) The units in Z36 are obtained by requiring (a, 36) = 1. This yields
U (Z36 ) = cfw_1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35.
(ii) We let x be the inverse of 11. Then
11x 1
mod 36 = 11x = 1 + 36y = 11x + 36(y) = 1.
We run the Euclidean
Math 104A - Homework 5. Due Wednesday, November 19.
1. (Quadratic congruences.) In class, we have studied congruences of the type
x2
mod p.
We now discuss arbitrary quadratic congruences.
Let p > 2 be a prime and consider the quadratic polynomial
f (x) =
Math 104A - Homework 4
Due 7/14
1 Write a function that takes as input an interval [a, b], a number subintervals
N , and a function f , and outputs the approximate derivative of f at the
nodes xi = a + i h, for i = 0, . . . , N , where h = ba . Use the th
Math 104A - Homework 3
Due 7/7
2.4.6 Show that the following sequences converge linearly to p = 0. How large
must n be before we have |pn p| 5 102 ?
a pn = 1/n.
Since
|pn+1 0|
1/(n + 1)
n
=
=
1
|pn 0|
1/n
n+1
as n , we have that pn converges linearly to 0
Math 104A - Homework 5
Due 7/21
1 Write a code that implements Gaussian quadrature with 3 nodes. The
quadrature is given by
1
f (x)dx a0 f (x0 ) + a1 f (x1 ) + a2 f (x2 ),
1
5
where x0 = 3 , x1 = 0, x2 = 3 and a0 = 9 , a1 = 8 , a2 = 5 . The code
5
5
9
9
s
Math 104A - Homework 4. Due Wednesday, November 12.
1. (Multiplicative functions.) Let k 0, and dene for each positive integer n the function
dk .
k (n) =
d|n,d>0
(i) Show that if (n, m) = 1 then any divisor d > 0 of the product nm can be written as
d = d
Math 104A Prof. Rabin Homework Assignment 4
due Thursday, October 27, 2016
Do the following problems from the textbook:
Section 2.4, problem 1 (obviously a calculator will help.)
Section 2.2, problems 3, 7.
Problem 3 is asking you to do the following. Sho
Math 104A Prof. Rabin Final Homework Assignment
due Thursday, December 1, 2016
Do the following problems from the textbook:
Section 4.1, problems 2, 6, 9, 10.
Hint for problem 2: when are both primitive roots,
show that ()(&'()/* 1( ).
Section 4.2, probl
Math 104A: Fall 2012
Homework 2
Due Friday, 10/12/2012 at 5:00 pm
1. Dene the Fibonacci numbers Fn by F1 = F2 = 1 and Fn = Fn1 + Fn2 for n > 2. Prove
that no two successive Fibonacci numbers Fn and Fn+1 have a common divisor a > 1.
Solution: For n 2, dene
Math 104A: Fall 2012
Homework 4
Due Friday, 11/2/2012 at 5:00 pm
1. Find all pairs (x, y ) Z2 such that 19x + 20y = 1909. Which of these pairs satisfy
x, y > 0?
2. Let p > 0 be prime and let a, b Z. Prove that (a + b)p ap + bp (mod p). (Hint:
Binomial The
Math 104A: Fall 2012
Homework 5
Due Friday, 11/9/2012 at 5:00 pm
1. Show that if m > 2 and if cfw_a1 , . . . , am Z is a complete residue system for Zm , then
cfw_a2 , . . . , a2 Z is not a complete residue system for Zm .
1
m
Solution: Since cfw_a1 , .
Math 104A: Fall 2012
Midterm 2 Solutions
Monday, 11/19/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these inst
Math 104A: Fall 2012
Midterm 2
Monday, 11/19/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these instructions i
Math 104A: Fall 2012
Midterm 1
Wednesday, 10/24/2012
Instructions: Please write your name on your blue book. Make it clear in your blue book
what problem you are working on. Write legibly. This exam is graded out of 100 points.
Following these instruction
Math 104A: Fall 2012
Homework 6
Due Friday, 11/16/2012 at 5:00 pm
1. Let a, b, n Z+ with (a, b) = 1. Prove that there exists x Z such that (ax + b, n) = 1.
(Hint: Consider applying the Chinese Remainder Theorem with moduli given by the set of
primes divid