Math 105 (Fall 2026)

October 14: Midterm 1 in class.

October 12: We proved that an integer a is invertible modulo n, i.e., there exists an integer z such that az ≡ 1 (mod n) if and only if gcd(a, n) = 1. In this case, z can be chosen to be a Bezout coefficient for a in the equation r a + s n = 1. We then introduced a complete residue system modulo n.

October 09: We did a worksheet in class. In particular, we proved that if a is congruent to b mod n and c is cognruent to d mod n, then a+c is congruent to b+d mod n, and ac is congruent to bd mod n. 

October 07: We stated the theorem about solutions to a linear diophantine equation Ax + By = C: this has no solution if d:= gcd(A, B) does not divide C, and if d divides C then it has infinitely many solutions which can be parametrized by (x_0 + tB/d, y_0 - t A/d) where (x_0, y_0) is a particular solution. If d = A r + B s, then a particular solution is just x_0 = r C/d and y_0 = s C/d. We then defined Congruences: we say a is congruent to b mod n, if n divides (a - b). So mod 10, 1 is congruent to both 11 and -9. We showed that a is congruent to a mod n (reflexivity), if a is congruent to b mod n, then b is congruent to a mod n (symmetry), and if a is congruent to b mod n and b is congruent to c mod n, then a is congruent to c mod n (transitivity).

October 05: We saw some more corollaries of Bezout's theorem: the set \{r a + s b | r, s \in Z \} is the set of integral multiples of gcd(a, b); that the gcd(ka, kb) = |k| gcd(a,b); and if d = gcd(a,b) then gcd(a/d, b/d) = 1. We then saw some corollaries of the Fundamental Theorem of Arithmetic: if a and b are coprime and ab = c^n there exists integers d, e such that a = d^n and b = e^n; and if a^{1/n} is a rational number, then it is an integer. HW2 is posted.

October 02: We proved Euclid's lemma: if a prime p divides the product ab, then p must divide either a or p must divide b. By induction, we get that if a prime p divides a product a_1 a_2 a_3 ... a_r, then p must divide one of the a_i. This was a key ingredient in our proof of the fundamental theorem of arithmetic.

September 30: We saw the key facts which make the Euclidean algorithm work: gcd(a, 0) = a and gcd(a + bq, b) = gcd(a, b) for all integers a, b, and q. We then saw how to use the Euclidean algorithm to obtain Bezout's theorem: given integers a, b, there exists integers r, s such that gcd(a, b) = ra + sb. The integers r and s are often called Bezout coefficients for a and b; they are not unique.

September 28: Defined prime integers and divisors. Proved
(i) every integer > 1 has a prime divisor,
(ii) there are infinitely many primes, and
(iii) every integer > 1 has a prime factorization [we do not know yet that this prime factorization is unique!]. 

Then defined the greatest common divisor of two integers and ran the Euclidean algorithm on 17 and 5.
Make sure to check carefully that for integers a, b, c, r, s, if a divides both b and c, then a divides rb + sc, and if a divides b and b divides c, then a divides c.

September 25: Introduction to the course; discussed integral solutions of y^2 = 2^x + 3 in detail as well as of x^2 + y^2 = z^2 satisfying z = y+1.

Lecture Time: MWF 2 to 2:50pm
Lecture Location: PETERSON 102
Textbook: Number Theory: In Context and Interactive by Karl-Dieter Crisman. Available via this link. We will use the 2024/6 edition.

Discussion Section:
Section 001: Thursday, 2:00pm-2:50pm at APM 2301

Course Personnel:
Instructor: Karthik Ganapathy Venkitachalam
Teaching Assistant: Tran Duy Anh Le
Note: To contact us, please use Piazza. The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza.

Office Hours:
Karthik Ganapathy: Monday 4:15pm to 5:45pm, Friday 11am to 12:30pm at APM 1230
Tran Duy Anh Le: Wednsday 3pm-5pm at HSS 4062

Homework:
Homework assignments will be posted on Canvas almost every week. Normally you will have one week to submit your completed homework via Gradescope. Your lowest homework assignment will be dropped at the end of the course when computing your grade.

Exam:
There will be two midterm exams and one final exam. No makeup exams will be given in this course.
Note: It is standard Math Department practice to utilize different versions of exams, both within each lecture's exam, and between lectures whose exams are at different times.

Midterm I: October 14, 2026, during class
Midterm II: November 13, 2026, during class
Final exam: December 9, 2026, 3:00pm-5:59pm at PETER 102

Regrades: Regrade requests will be made using the built-in regrade request feature in Gradescope. There will be a limited window of time after the exams are made available during which the regrade request feature will be active. This time window will be announced when the exam scores are released. Only the regrade requests sent within the time window will be considered.

Grading Policy:

The grades will be determined using the formula: 10% Homework, 25% Midterm Exam I, 25% Midterm Exam II, 40% Final Exam. 

The curve for this course will be determined at the end but will be at least as generous as the standard one (97 for A+; 94 for A; 90 for A-; 87 for B+; 84 for B; 80 for B-; 77 for C+; 74 for C; 70 for C-). No D grades will be assigned.

Attendance Policy: You are expected to attend every session of the lectures and discussion section you are enrolled in. Students who do not attend class rarely succeed in this course.  If you must miss class for any reason, it is your responsibility to catch up on the material. You are also responsible for the information given in any announcements made during class. To help you with this, I will record every lectures in the event you are unable to attend. Lecture recordings can be found at UCSD Podcast.

Academic Integrity: Academic dishonesty is considered a serious offense at UCSD. Students caught cheating will face an administrative sanction which may include suspension or expulsion from the university. It is in your best interest to take pride in your work maintain your academic integrity.

Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (https://osd.ucsd.edu/.) Students are required to discuss accommodation arrangements with instructors and OSD liaisons in the department in advance of any exams or assignments.

Here are some resources from other UCSD instructors that you may find useful:
1) Cristian Popescu's course website from Winter 2021
2) Aaron Pollack's course website from Winter 2022