Remember, you should set aside a studious hour for this. If you don’t finish everything in one hour, continuing now or catching up later or skipping things is at your discretion. Here’s what you should do with your hour before Friday’s class:

- For the course, we will use Sage Mathematics Software for computations. Here’s a Sage worksheet that shows how to use Sage as a calculator for modular arithmetic. Check it out (do each computation by hand and then compare to the computer output). Note: worksheets on the course website
*do not save your data*. - Create an example modulo 10 of residues $a,b,c$ such that $a \not\equiv b \pmod{10}$ and $c \not\equiv 0 \pmod{10}$, but $ac \equiv bc \pmod{10}$. Explain why this means you “can’t always cancel in modular arithmetic.”
- Compute $10^{12} \pmod{11}$ by cleverness. Check your answer with Sage. Verify that $10 \not\equiv 10^{12} \pmod{11}$. Explain why this shows you “can’t take mod up in the exponent”.
- Find our course notes (link to overleaf document at top left of this site!), and do Concept Checks 2.10 and 2.11 (currently page 16). You will need to create an overleaf account for the poster later, so now is a good time to do this.
- Watch this video about modular dynamics (18:46). The goal here is to remind you about injective/surjective/bijective functions and to get you thinking about modular arithmetic in a “dynamical” way, which will help form intuition for later in class.
- Draw a “multiplicative dynamical portrait” for $x \mapsto 2x$ modulo $5$. That means, draw the elements of $\mathbb{Z}/5\mathbb{Z}$ as dots and draw an arrow from each dot to what you get when you double it.
- Draw a “multiplicative dynamical portrait” for $x \mapsto 2x$ modulo $6$.
- What do you observe about these two situations? For each one, is the function $f(x) = 2x$ mod $n$ injective, surjective, both or neither?
- Contact me if you have any concerns about the class being recorded. I *do not* want this to discourage anyone from speaking in class, so talk to me, I can edit the audio if needed.
- Make it part of your daily (“lecturely”) routine to look over what has been done and catch up on anything missing. Remember, the “Archive” tab above has all our materials day-by-day including notes.
- FYI: The best way to contact me is discord. If you want to meet in person just let me know or come to office hours Wednesdays (see “About”). I also have a calendar for one-on-one appointments (link in canvas/discord) but you may need to remind me to add some slots.