MAT 115A
HW2
Joshua Sumpter
HW 2 Solutions
1
book problems
2.3.18) Use identity (2.2) with n = 4, and then with n = 2, to multiply (10010011)2 and (11001001)2 .
Solution:
We have the following identit
PRACTICE PROBLEMS
1.
Part 1: Greatest Common Divisor, Fundamental Theorem of
Arithmetic
1. Prove or disprove: (a1 , ., ai ), (ai+1 , ., an ) = (a1 , ., an ).
2. Describe the set cfw_275n + 625mm, n Z
STUFF TO REMEMBER 2
1.
On Polynomials over Z/pZ
Proposition 1. Suppose f (x) is an integer-coecient polynomial. x0 is a root of
multiplicity n if and only if f (x0 ) = f (x0 ) = . = f (n1) (x0 ) = 0,
LECTURE 14
1. Arithmetic Progression: a Summary
So far, we have proved various cases of the statement that there are innitely
many prime numbers of the form
aq + r,
for specic
a, r
such that
(a, r) =
STUFF TO REMEMBER
Proposition 1.
Proposition 2.
Proposition
3.
(a, b) = mincfw_ma + nbm, n Z, ma + nb > 0.
cfw_ma + nbm, n Z = (a, b). (n)
For any integer
n
is the set of all multiples of
n.)
that is
LECTURE 17
1. Warm-up
eg.
of
Classify all units of
12
Z/13Z
by their orders.
(13) = 12
and the positive factors
are 1,2,3,4,6,12.
Orders
Number of Units with this Order
1
2
3
4
6
12
You have to consid
MAT 115A
HW1
Joshua Sumpter
HW 1 Solutions
1b) Given that fn An , show that lim
n!1
fn+1
= .
fn
fn+1
An
= 1 and that lim
= 1.
n+1
n!1 A
n!1 fn
Proof. By our conjecture, we know that lim
It follows tha
MAT 115A
HW3
Joshua Sumpter
HW 3 Solutions
1
Book Problems
3.5.12) Let n be a positive integer. Show that the power of a prime p occurring in the prime factorization of n! is
given by [n/p] + [n/p2 ]
MAT 115A, SSI16
Homework 1
This homework is due on Tuesday, June 28 in class. Your homework must be typed in LATEX.
1. (1.1.5) Prove that
3 is irrational using the Well-Ordering Principle.
2. (1.1.12)
MAT 115A, SSI16
Homework 5
This homework is due on Tuesday, July 26 in class. Your homework must be typed in LATEX.
1. (6.3.1abc)
(a) For all n cfw_6, 9, 10, calculate (n).
(b) For all n cfw_6, 9, 10,
MAT 115A, SSI16
Discussion Worksheet 4
Get into groups of four, and then break into pairs.
1. In this exercise, you will create an affine cipher.
(a) With your partner, choose , N cfw_0 with 0 , 25 an
MAT 115A, SSI16
Discussion Worksheet 1
In groups of 3-4, try to solve as many of these problems as you can. Make sure that everyone in
your group understands the solution! An outline of a solution is
MAT 115A, SSI16
Homework 3
This homework is due on Tuesday, July 12 in class. Your homework must be typed in LATEX.
1. (3.7.2) For each of the following linear Diophantine equations, either find all s
MAT 115A, SSI16
Discussion Worksheet 3
In groups of 3-4, try to solve as many of these problems as you can. Make sure that everyone in
your group understands the solution! An outline of a solution is
MAT 115A, SSI16
Homework 2
This homework is due on Tuesday, July 5 in class. Your homework must be typed in LATEX.
1. (3.3.1) Find the greatest common divisor of each of the following pairs of integer
MAT 115A, SSI16
Discussion Worksheet 4
In groups of 3-4, try to solve as many of these problems as you can. Make sure that everyone in
your group understands the solution! An outline of a solution is
MAT 115A, SSI16
Homework 4
This homework is due on Tuesday, July 19 in class. Your homework must be typed in LATEX.
1. (4.3.2) Find an integer that leaves a remainder of 1 when divided by either 2 or
MAT 115A, SSI16
Discussion Worksheet 2
In groups of 3-4, try to solve as many of these problems as you can. Make sure that everyone in
your group understands the solution! An outline of a solution is
Math 115A: Number Theory
Summer Session I 2016
Instructor: Patrick Weed
Office: MSB 2131
Office Hours: MTW 10:00AM - 11:00AM, or by appointment
Email: [email protected]
Class: MTWR 8:00AM - 9:
LECTURE 13
1.
Application to Cryptography
As one important application of modular arithmetic, in this section we introduce
the basic idea behind cryptography.
1.1.
Caesar Cipher.
A most basic idea beh
LECTURE 9
Problem Session
1.
eg.
Since
eg.
In
q7 (9) = 2
and
3
2 = 1,
we get
Find a positive integer
Z/8Z, 3 3 = 1;
eg.
In
q7 (910000 ).
Find
n
such that
so one can take
Find a positive integer
m
3333