General Information

Class times: Mon + Wed + Fri from 09:00–09:50 AM [Alpine 212]
Book: Elementary Number Theory, Second Edition (Dover Books on Mathematics); by Underwood Dudley.
Syllabus: Syllabus for Math 102

Instructor: Dr. Joshua Wiscons (he/him/his)
Email: joshua.wiscons@csus.edu
Office: Brighton Hall (BRH) Room 144
Student office hours:

• In person: Monday 2–3 PM and Wednesday 4–5 PM [Brighton 144]
• Virtual: Thursday 12:30–1:30 PM [only via Zoom]
• Also by appointment. Just send me an email.

Homework Assignments

Due dates and submissions are managed in Canvas. Let me know if you have any questions!

Jump to specific assigment: HW01 | HW02 | HW03 | HW04

Homework 01

Problems to submit for grading

These are the problems to turn in. Please organize your work, justify your steps, and write in complete sentences with correct punctuation. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

• Problem 1: Suppose that $a$, $b$, and $c$ are arbitrary integers. Prove that if $a\mid b$ and $b\mid c$, then $a\mid c$.
  • Hint: try to start like this: “Assume that $a\mid b$ and $b\mid c$. Since $a\mid b$, there is some integer $x$ such that $b = ax$. Also since $b\mid c$, there is some integer $y$ such that…” Now keep going to show that $a$ divides $c$; your goal is to show that $c$ is $a$ times some integer.
• Problem 2: Suppose that $a$, $b$, and $d$ are arbitrary integers. Prove that if $d\mid a$ and $d\mid b$, then $d^2\mid ab$.
• Problem 3: Let $a,b\in \mathbb{Z}$ be integers. Prove that if $x^2 + ax + b = 0$ has an integer solution $r$, then $r$ divides $b$.
  • Hint: Try starting like this: “Assume that $x^2 + ax + b = 0$ has an integer solution $r$. Then $r^2 + ar + b = 0$.” Now keep going to show that $r$ divides $b$ according to the definition of divides.
  • Note: the assumption $a,b\in \mathbb{Z}$ only allows you to treat $a$ and $b$ as arbitrary integers, so make sure not to give them specific values.

Problems for extra practice

Please try to solve these problems, but you do not turn them in. I’m happy to talk about these problems (or any others) if you have questions. Problem numbers refer to our book Elementary Number Theory, Second Edition.

• Page 3: Exercise 3
• Page 9: Problem 7

Homework 02

Problems to submit for grading

• Problem 1: Compute $\gcd(4144,7696)$ using the Euclidean Algorithm. Please show all work.
• Problem 2: This problem has two parts. First compute $\gcd(314,159)$ using the Euclidean Algorithm. Then continue with the “extended” Euclidean Algorithm to find ind integers $x$ and $y$ so that $314x +159y = \gcd(314,159)$. Please show all work.
• Problem 3: Let $a,b,c\in \mathbb{Z}$, and let $d = \gcd(a,c)$. Prove that if $c\mid ab$, then $c\mid db$.
  • Hint: consider using Theorem 4; the proof of Corollary 1 may provide extra inspiration.
• Problem 4: Assume that $p$ is prime number and that $p = n^3 - 1$ for some positive integer $n$. Prove that $p=7$.
  • Hint: think about how you can factor $n^3 - 1$? But, $p$ is prime, so what are its possible factors? What does this imply about $n$?

Problems for extra practice

• Page 4: Exercises 4,5
• Page 9: Problems 4,7
• Page 12: Exercise 3
• Page 19: Problem 8

Homework 03

Problems to submit for grading

• Problem 1: Let $n\in \mathbb{Z}$ be a positive, composite number. Assume that $p$ is the smallest prime factor of $n$, and write $n = mp$ for some $m\in \mathbb{Z}.$ Prove that if $m$ is also composite, then $p \le \sqrt[3]{n}$.
  • Hint: If $m$ is composite, then you can factor $m$ as $m = ab$ for some integers $a$ and $b$ with $1 < a,b < m$. What can you say about the relationship between $p$ and $a$? About $p$ and $b$? What does this tell you about $n$?
• Problem 2: Let $n\in \mathbb{Z}$ with $n\ge 2$. Let $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ be the prime-power decomposition for $n$. Prove that if every exponent $e_1,\ldots,e_k$ is even, then $n$ is a perfect square (i.e. $n=m^2$ for some integer $m$).
• Problem 3: Determine if each equation has integer solutions or not. Please explain your answers, but you do not need to actually find the solutions.
  1. 14x + 34y = 90
  2. 14x + 35y = 91
  3. 14x + 36y = 93

Problems for extra practice

• Page 19: Problems 5, 9, 10
• Page 22: Exercise 2
• Additional Problem: Suppose that $n$ is any four-digit integer and that $\operatorname{gcd}(n,100!) = 1$. Prove that $n$ must be prime.
  • Hint: What happens if you assume $n$ is not prime? Then Lemma 4 of Section 2 tells you something about the prime divisors of $n$. But, $\operatorname{gcd}(n,100!) = 1$ also tells you something about the prime divisors of $n$; think about what the prime divisors of $100!$ are.

Homework 04

Problems to submit for grading

• Problem 1: Find all integer solutions to each one of the following equations. Please show all work.
  1. $10x+15y = 14$
  2. $2x+6y = 20$
• Problem 2: Find all integer solutions to the system of equations: \[\begin{aligned}x+y+z &= 31 \\ x+2y+3z&=41 \end{aligned}\]
  • Hint: Try combining the two equations to get a new equation in two variables, which you can use the methods we developed to solve. Then plug your solutions into one of the original equations to solve for the remaining variable.
• Problem 3: Let $p$ be an arbitrary prime with $p\ge 5$. Prove that $p\equiv 1$ (mod 6) or $p\equiv 5$ (mod 6).
  • Hint: write out what $p\equiv a$ (mod 6) means according to the definition. Then use that $p$ is prime to explain why $p$ must not be equivalent to any of 0,2,3,4 when working mod 6.
• Problem 4: Find the least residue of each of the following without using a calculator. Please show your work.
  1. $(9876543)^3$ (mod $5$)
  2. $2^{113}$ (mod $5$)
     • Hint: $2^2 \equiv -1$ (mod $5$)

Problems for extra practice

• Page 26: Problems 6
• Page 32: Problems 2, 4
  • Hint: on Problem 4, you should disprove the statement by giving a concrete example of $a$, $b$, and $m$ for which it fails. (There are many such examples, so everyone can find their own.)