Math 115 - Introduction to Number Theory

Homework Assignments

General rules on homework assignments are:

• For any assigned problem from the book, you may use without proof any earlier exercise in the book.
• For an assigned problem not from the book, you may use without proof any exercise in the book that could have been used without proof in the preceding problem in the homework assignment. (If the assignment contains consecutive problems not from the book, use induction.)
• When doing a problem from the book, you may not use material from subsequent sections of the book (even if the homework assignment includes problems from those sections). (For problems not from the book, use the principle in the previous bullet point.)
• Answers to many of the problems are given in the back of the book. Such problems will be graded under the assumption that you have looked at the answer in the back. (I do not recommend doing this until after you have worked the problem.) In particular, you will be expected to provide more explanation than the answer in the back of the book does.

Homework assignments are due on bCourses on the days indicated below. The lowest three homework scores will be dropped (note that homework scores include two additional scores from clickers—see the section on Clickers, below).

iClickers

We will use iClickers in class. These must be the physical iClickers, not the iClicker app. The latter is not allowed, because use of cell phones during class can be very distracting.

iClickers can be obtained from the bookstore, or you may be able to buy a used one from another student.

The grades from iClicker use will be incorporated into the homework portion of the course grade.

Clicker grades will be determined as follows. For each class day (other than the first two), you will receive one clicker point for participation if you have provided answers (correct or not) to at least 50% of the clicker questions on that day, and one clicker point for correctness if you answer at least one clicker question correctly on that day. These will be reported to bCourses separately, but will be combined for the purposes of the next steps.

I will then drop the lowest four sessions (class days), and then combine that information with the homework scores by creating two ``fake'' assignment scores based on the total clicker scores.

The clicker points are not meant to be a big challenge. The vast majority of you should be receiving two clicker points on every day that you attend. The point of clickers is that you should be checking your own understanding of the material (including the reading) as the course progresses.

The clicker grading plan is subject to change.

Course Handouts

Exams (Generally)

Policies for exams are as follows.

• No calculators or other portable electronic devices are allowed. Arithmetic on the exam questions will be kept to a minimum (as was done on the sample exams).
• Exams are closed book. No notes or "cheat-sheets" are allowed.
• Do not bring bluebooks. Exams will have enough space for an answer to be written on the exam itself, and scrap paper will be available as needed.

One more thing. If you are computing gcd(a,b) or finding x and y such that gcd(a,b)=xa+yb, and if |a|<20 and |b|<20, then you can use any method you want and don't have to include the details. If |a| or |b| is at least 20, though, you are expected to use the Euclidean Algorithm and to show your work.

The two midterms will be given during the normal class hours (11:10–12:00), and will be in our normal classroom (Physics 385).

Generally, about a week before each exam, a sample exam will be distributed in class and posted on bCourses. This will usually be an exam from an earlier Math 115 class that I've taught. Sample exams should be used to get a general idea of the likely length of an exam and the general nature of questions to be asked (e.g., the balance between computational and more theoretical questions). However, one should not (for example) observe that a sample exam contains questions on material from Sections 1.5, 2.1, 2.7, 3.1, 3.4, etc., and expect to see questions from those sections on the actual exam.

Exams are cumulative, so the second midterm may have questions from material prior to the first midterm. Of course, the final exam will cover the whole course, but will have increased emphasis on the material not covered on the midterms.

Here is a link "How to lose marks on math exams" (by a former GSI Andrew Critch).

The Math Department maintains an archive of old exams (usually without answers). Here is the link for Math 115.

And finally, a word about regrades: Grade calculation errors are welcome for discussion or review. Whether this solution should be worth 4 or 5 points is not.

First Midterm

The first midterm was given on Friday, October 1, from 11:10 to 12:00 Pacific Time, in our usual classroom (Physics 385).

It covered:

• the textbook, up to and including Example 11 and its solution on page 88;
• lectures, up to and including Friday, September 24;
• all handouts so far, except for the most recent one (on the general version of Hensel's lemma); and
• homeworks, up to and including Homework 4.

• the content on the distribution of prime numbers (from Theorem 1.18 through the end of Section 1.3); and
• the example of lack of prime factorization discussed starting on the bottom of page 21 and continuing through the middle of page 23.

As noted under the heading “Exams (Generally),” this was a fully closed-book exam.

A sample midterm was distributed in class on September 24, and is available on bCourses. Solutions to the sample midterm have also been posted on bCourses.

Solutions to the midterm itself are also now available on bCourses.

Second Midterm

The second midterm was given on Wednesday, November 3, from 11:10 to 12:00 Pacific Time, in our usual classroom (Physics 385).

It covered:

• the textbook, up to and including Section 3.3;
• lectures, up to and including Friday, October 22;
• all handouts, up to and including Handout 8 (Slides from the lecture of September 29); and
• homeworks, up to and including Homework 8.

• topics listed as skipped in the section on Midterm 1; and
• any part of Chapter 2 beyond the first two paragraphs of Section 2.9.

As noted under the heading “Exams (Generally),” this was a fully closed-book exam.

A sample midterm was distributed in class on October 27, and is available on bCourses. Solutions to the sample midterm are also now available on bCourses.

Solutions to the midterm itself are also now available on bCourses.

RRR Week

We will meet on Monday, Wednesday, and Friday of RRR week (at the usual time and place). We will review the course. We will not use clickers during RRR week.

Office hours will continue on their usual schedule during RRR week.

Final Exam

As noted above, the final exam was given on Monday, December 13, 11:30 am – 2:30 pm, in the regular classroom (Physics Building 385).

The exam started at 11:30, not 11:40 (so, not “Berkeley time”).

It covered the whole course (all of the reading, homework problems, and lectures). The list of sections covered is listed in “Syllabus” above.

We skipped, and therefore the exam did not cover:

• topics listed as skipped in the sections on Midterm 1 and Midterm 2; and
• any part of Section 3.7 beyond the first seven lines on page 174.

A sample final exam was distributed in class on December 8. Both the sample exam and solutions to it are available on bCourses.

Solutions to the final exam itself have also now been posted on bCourses.

Homework Score

The overall homework score is now posted on bCourses.

Note that the homework score has not been computed on a percentage basis.

Instead, the aggregate homework score was computed as follows. First, the clicker scores were turned into two fake homework scores. (As it turned out, almost everyone got maximal clicker points, so this didn't really have an effect on the aggregate homework score.) Homework scores were first adjusted so that the mean on each homework (among those turned in) was the same across homeworks. Then the bottom three were dropped. The resulting scores were added. Scores in roughly the bottom half of the class were brought up, so that homework would not have a disproportionate effect on the final grade. The resulting score was adjusted to make its standard deviation closer to a number proportionate to the standard deviations of the midterms and finals (as if it were a hypothetical exam worth 150 points). I also made it so that the top score was 150.

As a result, the percentage you get by dividing your aggregate homework score by 150 will be approximately an increasing function of the average that you might compute on your own, but it will not be the same value, nor will it be linear.

Last updated 17 December 2021