To Do:

1. Please check the previous daily post (scroll down) and take care of the missing administrative items you haven’t done.
2. Please make sure you are now fully aware of the way daily posts work (you set aside an hour; you upload evidence of your worktime). We will have discussed this in class.
3. The main activity for this daily post is to watch my video “Modular Arithmetic: User’s Manual” (9:22 mins:secs; link above requires CU access, no ads; also on YouTube here), and do the accompanying self-check worksheet. Please be sure to show your work on the last problem. (Note: you don’t need to print; you can work on a separate sheet of paper if desired; in general this type of thing is true, since you are just showing evidence of your work, I’m not picky.)
4. Explain how XOR can be described in terms of modular arithmetic.
5. 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 (except the last three, which we haven’t discussed yet, concerning inverses/dividing) and then compare to the computer output). Note: worksheets on the course website do not save your data.
6. 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.”
7. 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”.
8. Catch up on the last daily post (classical ciphers), if needed. While you’re doing this, think about how each of them can be described in terms of modular arithmetic. You can check your decryptions against the Solutions to Daily Due Aug 24 (AI generated) (problems 9–12).
9. Upload a record of your work today to canvas (as always).