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?
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.
Some fun modular arithmetic, part one:
Verify that $7$ is coprime to $12$ (factor them both).
Compute $\varphi(12)$.
Explain what Euler’s theorem says about powers of $7$ modulo $12$. Why did we check the last two things?
Compute $7^{115} \pmod{12}$ by hand using Euler’s Theorem. Calculator is ok for multiplications and reductions, but show all steps.
Some fun modular arithmetic, part two:
Verify that $59$ is coprime to $26$ (factor them both).
Compute $\varphi(26)$ (it’s important you get this right, check your answer with the Sage Sandbox (the euler_phi function is listed there)).
Compute $59^{(7^{115})} \pmod{26}$ by hand (this may use the previous problem…..)
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).
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?
Catch up on last daily post problems or any other studying.