Math 115, Summer 2012
Homework 2
Solution
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 2.1.5) Write a single congruenc
MATH 115, SUMMER 2012
QUIZ 4 SOLUTION
JAMES MCIVOR
There are two problems - one is on the back of this page, 20 points total. Write
clearly and in complete sentences. If you need extra paper, ask me. You may nd
the following denition and theorems useful:
MATH 115, SUMMER 2012
QUIZ 4
JAMES MCIVOR
There are two problems - one is on the back of this page, 20 points total. Write
clearly and in complete sentences. If you need extra paper, ask me. You may nd
the following denition and theorems useful:
Denition
MATH 115, SUMMER 2012
QUIZ 5 SOLUTION
JAMES MCIVOR
There are problems on both sides of this page, 20 points total. Write clearly and
in complete sentences. If you need extra paper, ask me.
(1) (2 points each) True or False. Explain your answer.
(a) There
MATH 115, SUMMER 2012
QUIZ 5
JAMES MCIVOR
There are problems on both sides of this page, 20 points total. Write clearly and
in complete sentences. If you need extra paper, ask me.
(1) (2 points each) True or False. Explain your answer briey.
(a) There are
MATH 115, SUMMER 2012
QUIZ 3
JAMES MCIVOR
There are problems on the back of this page, 20 points total. Write clearly and
in complete sentences. If you need extra paper, ask me.
(1) (8 points) Find all solutions to the congruence x3 + 8x 5 0 mod 53 .
(2)
$0 LUL'VImJ
/
MATH 115, SUMMER 2012
' QUIZ 3 '
JAMES MCIVOR
There are problems on the back of this page, 20 points total. Write clearly and
in complete sentences. If'you need extra paper, ask me. '
(1} (8 points) Find all solutions to the congruence 3:3
MATH 115, SUMMER 2012
QUIZ 1 SOLUTION
THURSDAY, JUNE 21ST
JAMES MCIVOR
Please show your work carefully and write neatly. Make sure to double-check
your answers. There are 20 points total.
(1) (6 points) Calculate the greatest common divisor of 115 and 585
MATH 115, SUMMER 2012
MIDTERM EXAM
WEDNESDAY, JULY 18TH
JAMES MCIVOR
The exam is out of 180 points. You have the full class time (1 hour, fty minutes)
to complete it. No calculators or cheat sheets are allowed, and do not use your
cellphone, etc. You must
MATH 115, SUMMER 2012
QUIZ 1
THURSDAY, JUNE 21ST
JAMES MCIVOR
Please show your work carefully and write neatly. Make sure to double-check
your answers. There are 20 points total.
(1) (6 points) Calculate the greatest common divisor of 115 and 585. Then
ex
MATH 115, SUMMER 2012
QUIZ 2
JAMES MCIVOR
There are problems on the back of this page. Write clearly and in complete
sentences. If you need extra paper, ask me.
(1) (6 points) Solve the system of congruences, if possible. If not possible,
explain why not.
MATH 115, SUMMER 2012
MIDTERM EXAM
WEDNESDAY, JULY 18TH
JAMES MCIVOR
The exam is out of 180 points. You have the full class time (1 hour, fty minutes)
to complete it. No calculators or cheat sheets are allowed, and do not use your
cellphone, etc. You must
MATH 115, SUMMER 2012
WORKSHEET FOR LECTURE 14
JAMES MCIVOR
(1) (8 points) Find all solutions to the congruence x2 4x + 23 0 mod 125.
Solution: Working mod 5, we get the solutions x = 1, x = 3. These are
both nonsingular, since f (x) = 2x 4, so f (1) = 2,
Math 115, Summer 2012
Homework 4
Due Tuesday, July 16th
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 3.1.7) Which of t
MATH 115, SUMMER 2012
HOMEWORK 5
SOLUTION
JAMES MCIVOR
(1) (NZM 3.5.1) Find a reduced form equivalent to 7x2 + 25xy + 23y 2 .
Solution: By applying step 2 with k = 2, and then step 1, we obtain
the reduced form x2 + 3xy + 7y 2 .
(2) (NZM 3.5.4) Show that
Math 115, Summer 2012
Homework 3
Due Thursday, July 12th
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 2.6.2) Solve the
Math 115, Summer 2012
Homework 1
Due Tuesday, June 26th
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) Calculate the gcd of 3
Math 115, Summer 2012
Homework 3
Solution
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 2.6.2) Solve the congruence x5
Math 115, Summer 2012
Homework 2
Due Tuesday, July 3rd
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 2.1.5) Write a sin
MATH 115, SUMMER 2012
HOMEWORK 6
DUE TUESDAY, AUGUST 7TH
JAMES MCIVOR
(1) Show that the equation 12x2 + 13y 2 = 227 has no solutions in integers.
(2) Show that the curve 17x2 + 24y 2 = 221z 2 contains no rational points besides (0, 0, 0).
(3) Consider the
Math 115, Summer 2012
Homework 4
Solution
NZM a.b.c refers to a problem in our text, 5th edition - these may dier slightly from the problems
appearing in other editions, so use the version printed here to be safe).
(1) (NZM 3.1.7) Which of the following c
MATH 115, SUMMER 2012
HOMEWORK 5
DUE TUESDAY, JULY 31ST
JAMES MCIVOR
(1) (NZM 3.5.1) Find a reduced form equivalent to 7x2 + 25xy + 23y 2 .
(2) (NZM 3.5.4) Show that a binary quadratic form f properly represents an
integer n if and only if there is a form
MATH 115, SUMMER 2012
MOCK QUIZ, LECTURE 9
JAMES MCIVOR
Pretend this is a reai quiz and take. It's not for a grade, though. First look for
the ones you already know how to do. Well go over it at the end, but use it as a
guide to see what you should study
SAC/[gen
If
/_
MATH 115, SUMMER 2012
QUIZ 2
JAMES MCIVOR
There are problems on the back of this page.
Write clearly and in complete
sentences. If you need extra paper, ask me.
(1) (6 points) Solve the system of Icongruencesf possible.
If not possible,
exp
Mathematics 115
Professor K. A. Ribet
First Midterm Exam
September 22, 2011
Please put away all cell phones, books, iPads, iPods, laptops, etc., etc. You may consult
a single two-sided sheet of notes. Please write carefully and clearly in complete sentenc
MATH 115 MID-TERM 1
1. (4 pts) a) (3 pts) Given the factorisation 1728 = 26 33 , show that for any integer a which is relatively prime to 7, 13 and 19, we have a1728 a a1728 (hint: use Fermat's little theorem.) b) (1 pt) Given the factorisation 172