Relevant Sections of Textbook

• 2.2.9 (Invertibility)
• 2.2.10 (unit group)
• 2.4 (Multiplicative dynamics)

To Do:

Added Sept 2: solutions to the daily due Aug 31 are now posted: Solutions to Daily Due Aug 31.

1. In class, we proposed a formula $\varphi(n) = n(1 - 1/p_1)(1-1/p_2)\cdots(1-1/p_s)$ when $n$ is a product of distinct primes $p_i$. Verify this for $n = 15$ and $n=30$ by comparing to your computations of $\varphi$ from the last daily post.
2. Write me a few sentences “check-in”; how do you feel the class is going? What are challenges, what can I do to help with those?
3. Use successive squaring to compute $3^{133} \pmod{1009}$. You can use a calculator for the multiplications/reductions, but show all steps of the algorithm. If you learned about double-and-add (see last daily post), you can use that if you prefer.
4. Some fun modular arithmetic, part one:
   1. Verify that $7$ is coprime to $12$ (factor them both).
   2. Compute $\varphi(12)$.
   3. Explain what Euler’s theorem says about powers of $7$ modulo $12$. Why did we check the last two things?
   4. Compute $7^{115} \pmod{12}$ by hand using Euler’s Theorem. Calculator is ok for multiplications and reductions, but show all steps.
5. Some fun modular arithmetic, part two:
   1. Verify that $59$ is coprime to $26$ (factor them both).
   2. Compute $\varphi(26)$ (it’s important you get this right, check your answer with the Sage Sandbox (the euler_phi function is listed there)).
   3. Compute $59^{(7^{115})} \pmod{26}$ by hand (this may use the previous problem…)
6. Use what you learned from the last few problems to compute $3^{(11^{200006})} \pmod{50}$ “by hand” (this will always mean calculator is ok for multiplications and reductions, but show steps).
7. Observe (verify) that $\varphi(15) = 8$ and $3^8 = 6 \pmod{15}$. (You can check these on Sage or by hand or both.) This looks like it contradicts Euler’s Theorem. Why is it ok?
8. Remember to upload to canvas!