Exercises

1.
Give a multiplication table for the integers modulo 11. From the table, find the inverses of 2 and 3. Verify your answers by evaluating 2-1$\limfunc$mod11 and 3-1$\limfunc$mod11. BITMAPSETAnswer0.2006in0.1859in0ina1

2.
Solve the congruence 5x + 4≡8 $\left(\vphantom{ \limfunc{mod}%
13}\right.$$\limfunc$mod13$\left.\vphantom{ \limfunc{mod}%
13}\right)$. Verify your answer by evaluating 5x + 3$\limfunc$mod13. BITMAPSETAnswer0.2006in0.1859in0ina2

3.
A jar of jelly beans sits on a table. If the jelly beans are evenly divided among five children, there are three jelly beans left over; and if the jelly beans are evenly divided among seven adults, there are five jelly beans left over. How many jelly beans are in the jar? Are other solutions possible? If so, what are they?BITMAPSETAnswer0.2006in0.1859in0ina3

4.
What is the smallest 100-digit prime?BITMAPSETAnswer0.2006in0.1859in0ina4

5.
If p is the smallest 100-digit prime, what is p-1$\limfunc$modp? What is 2(p-1)/2$\limfunc$modp? What about (p-1)/4$\limfunc$modp?BITMAPSETAnswer0.2006in0.1859in0ina5

6.
The matrix M = $\left[\vphantom{
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 4 \\
1 & 4 & 9
\end{array}
}\right.$$\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 4 \\
1 & 4 & 9
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 1 & 1 \\
1 & 2 & 4 \\
1 & 4 & 9
\end{array}
}\right]$ is used as a block cipher modulo 26 to scramble letters in a message, three letters at a time. Assume A↔ 0, B↔1, C↔2, and so forth. Descramble the ciphertext FKBHRTMTU.BITMAPSETAnswer0.2006in0.1859in0ina6

7.
Find an irreducible polynomial of degree 3. Use this polynomial to describe how to calculate sums and products in the field GF27.BITMAPSETAnswer0.2006in0.1859in0ina7

8.
A barge company transports bales of hay and barrels of beer up the Mississippi River. The company charges $2.30 for each bale of hay and $3.00 for each barrel of beer. The bales of hay average 75 pounds and take up 5 cubic feet of space; the barrels of beer weigh 100 pounds and take up 4 cubic feet of space. A barge is limited to a payload of 150,000 pounds and 8,000 cubic feet. How much beer and how much hay should a barge transport to maximize the shipping charges?BITMAPSETAnswer0.2006in0.1859in0ina8