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 solutions or show
that there are no integral solutions.
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 sufficient.
I will give you a problem for your group to
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 integers.
(a) 15, 35
(c) 11, 121
(e) 99, 100
(b) -12, 18
(d) 0
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 sufficient.
I will give you a problem for your group to
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 5, but that
is divisible by 3.
2. (4.3.4a) Find all sol
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 sufficient.
I will give you a problem for your group to
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: psweed@math.ucdavis.edu
Class: MTWR 8:00AM - 9:40AM in OLSON 223
Prerequisites: MAT 21B, but I will a
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) Show that for all x R, we have bxc + bx + 0.5c = b2xc.
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 ] + [n/p3 ] + . . .
Proof. We consider the product form o
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 that
fn+1
An
1 = lim
lim
n!1 An+1
n!1 fn
An
fn+1
= lim
n!1
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 sufficient.
I will give you a problem for your group to
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 and gcd(, 26) = 1. You
will then create your cipher C P +
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, find a reduced residue system modulo n.
2. (6.3.4) Let
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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 identity: ab = (22n + 2n )A1 B1 + 2n (A1 A0 )(B0 B1 ) + (2n +
A brief note on Hensels lemma: its statement and examples
In class we proved
Theorem (Hensels lemma). Let p be a prime, k 2 an integer, and f (x) a polynomial with integer
coecents. Suppose r is an (integer) solution to the congruence f (x) 0 mod pk1 . Th
MATH 115A Number Theory, HW1 Solutions
1. Let T be the set
T := cfw_a bk | k Z, a bk > 0.
Note that T is nonempty: when k = 0,
a bk = a 0 = a
and a is positive by assumption, thus a T . Thus T is nonempty. By denition, everything in T is of
the form a bk
MATH 115A Number Theory, HW1
1. R1.1.2
2. R1.1.4
3. R1.1.6
4. R1.1.34. [Heres a hint (i.e., this is how I did this problem): for every positive integer n, dene
qn := min cfw_a | b such that |a b|
1an
1
.
n
Now argue the following:
(a) qn exists for all n
Fast modular exponentiation, or, how to compute residues of
numbers bigger than the number of atoms in the universe
Suppose youre given a large power of some number, like 5321 , and some
other number, like 123. Is it possible to quickly nd the remainder r