Problem 4
Find all numbers, if any, \(a \in Z_{9}\) different from 1 and 8 (notice that \(-1 \bmod 9=8\) ) such that \(a^{8} \bmod 9=1\).
Problem 4
The numbers 29 and 43 are primes. What is \((29-1)(43-1) ?\) What is \(199 \cdot 1111\) in \(Z_{1176}\) ? What is \(\left(23^{1111}\right)^{199}\) in \(Z_{29}\) ? In \(Z_{43}\) ? In \(Z_{1247}\) ?
Problem 5
What is \(16+2318 ?\) What is \(16 \cdot 2318 ?\)
Problem 5
Use a spreadsheet, programmable calculator, or computer to find all numbers \(a\) different from 1 and 32 (which equals \(-1\) mod 33 ) with \(a^{32} \bmod 33=1\). (This problem is relatively straightforward to do with a spreadsheet that can compute mods and that will let you "fill in" rows and columns with formulas. However, you do have to know how to use the spreadsheet in this way to make it straightforward!)
Problem 5
Given an element \(b\) in \(Z_{n}\), what can you say in general about the possible number of elements \(a\) such that \(a \cdot_{n} b=1\) in \(Z_{n}\) ?
Problem 6
How many solutions with \(x\) between 0 and 34 are there to the system of equations $$ \begin{aligned} &x \bmod 5=4 \\ &x \bmod 7=5 ? \end{aligned} $$ What are these solutions?
Problem 6
If \(a \cdot 133-m \cdot 277=1\), what can you say about all possible common divisors of \(a\) and \(m\) ?
Problem 6
How many digits does the \(10^{120}\) th power of \(10^{100}\) have?
Problem 7
Compute each of the following. Show or explain your work. Do not use a calculator or computer. a. \(15^{96}\) in \(Z_{97}\). b. \(67^{72}\) in \(Z_{73}\). c. \(67^{73}\) in \(Z_{73}\).
Problem 7
If \(a\) is a 100 -digit number, is the number of digits of \(a^{10^{120}}\) closer to \(10^{120}\) or \(10^{240}\) ? Is it a lot closer? Does the answer depend on what \(a\) actually is rather than the number of digits it has?
