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 iClickers, below).

The homework score has now been uploaded to bCourses. It is an integer in the range from 0 to 150. It was computed as follows. Each student has 12 homework scores (Homeworks 1–12) and two additional number from their clickers scores. The lowest three of these 14 numbers were discarded, and the remaining 11 were added to give a number in the range from 0 to 330. This was then multiplied by 150/330 and rounded to the nearest integer to give a score from 0 to 150. That is the score that was uploaded to bCourses.

Due to constraints on hiring of graders for our homework, only three problems from each assignment were graded. (This was in effect for all of the homework assignments.)

iClickers

We will use iClickers in class. You may use the iClicker Cloud app (preferred), or the older physical iClickers.

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

Links:

• Getting started with iClicker Cloud
• How to Participate in a Polling Session with an iClicker Remote

The grades from iClicker use will be incorporated into the homework portion of the course grade as two homework assignments, "Click 1" and "Click 2".

Clicker grades will be determined as follows. For each class day (other than the first one or two), you will receive one clicker point for participation if the iClicker web site grants you a participation point for that day, and one clicker point for correctness if you answer at least one clicker question correctly on that day. These will not be uploaded to bCourses individually, but their sum is available from the iClickers web site (as your total score for that day, capped at 2).

I then dropped the lowest three sessions (class days), and added up the remaining scores (for a maximum total of 38), multiplied by 60/38 and rounded to the nearest integer, giving a score from 0 to 60.

If that score was < 30, then "Click 1" was set to that score divided by 3 and rounded to the nearest integer.

If that score (call it S) was >= 30, then "Click 2" was set to 30 - (60 - S) / 3, rounded to the nearest integer.

Whichever of Click 1 or Click 2 was not defined above was then set so that they added up to S. These scores have now been uploaded to bCourses.

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

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 (9:40–11:00 am), and will be in our normal classroom (Wheeler 102).

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 Tuesday, October 8, from 9:40 to 11:00 am Pacific Time, in our usual classroom (Wheeler 102).

It covered:

• the textbook, up to and including page 88, except as indicated below;
• lectures, up to and including the time indicated on Tuesday, October 1;
• all handouts so far; 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 October 1, 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.

The (very rough) curve for the midterm is:

The median was 58, the mean was 52.8, and the standard deviation was 17.15.

Second Midterm

The second midterm was given on Thursday, November 14, from 9:40 to 11:00 am Pacific Time, in our usual classroom (Wheeler 102).

It covered:

• the textbook, up to and including page 160 (except for portions not covered);
• lectures, up to and including Tuesday, November 5;
• handouts, up to and including Handout 12 (Slides from the lecture of November 5); and
• homeworks, up to and including Homework 9.

• 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 November 7, and is available on bCourses. Solutions to the sample midterm are also available on bCourses.

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

The (very rough) curve for the midterm is:

The median was 52, the mean was 49.6, and the standard deviation was 22.0.

RRR Week

We will meet on Tuesday and Thursday 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 Tuesday, December 17, 3:00–6:00 pm PT, in the regular classroom (Wheeler 102).

The exam started at 3:00, not 3:10 (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 Tuesday, December 10, and was also be posted on bCourses. Solutions to the sample final were posted on bCourses on Thursday December 12.

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

The (very rough) curve for the final exam is:

The median was 115, the mean was 114.6, and the standard deviation was 43.0.

Last modified 25 December 2024