Skip to main content
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}\)

Subsection3.2Homework 1

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

Subsubsection3.2.1Exercises

  1. Exercises 8 #1.
  2. Exercises 8 #3.
  3. Exercises 8 #8.
  4. Exercises 8 #13.
  5. Exercises 8 #18.
  6. Exercises 8 #19.
  7. Exercises 8 #20.
  8. Exercises 8 #25.
  9. Exercises 8 #26.
  10. Exercises 8 #28.
  11. Prove using a direct proof: If there exists an integer \(k\) such that \(a = 6k + 5\text{,}\) then the equation \(ax = 1\) (mod 6) can be solved for \(x\text{.}\)
  12. Prove by contraposition or contradiction: If, for all \(a\) satisfying \(1 \leq a \leq m-1\text{,}\) the equation \(ax \equiv 1 \quad \text{(mod m)}\) can be solved for \(x\text{,}\) then \(m\) is a prime number.