Chapter 3, Euclidean Algorithm Video Solutions, Applied Algebra: Codes, Ciphers and Discrete Algorithms

02:15

Problem 1

Find the base 5 representation of the decimal number 9374 .

Melissa Munoz

Numerade Educator

Give the addition and multiplication tables for the integers modulo 3 , where $a \oplus b=(a+b) \bmod 3$ and $a \otimes b=a b \bmod 3$. Use the tables to solve the equations $2 \oplus x=1$ and $2 \otimes x=1$.

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Problem 3

Give the addition and multiplication tables for the integers modulo 4 , where $a \oplus b=(a+b) \bmod 4$ and $a \otimes b=a b \bmod 4$. Can you use the tables to solve the equations $2 \oplus x=1$ and $2 \otimes x=1$ ? Why or why not?

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Problem 4

Give the addition and multiplication tables for the integers modulo 6 , where $a \oplus b=(a+b) \bmod 6$ and $a \otimes b=a b \bmod 6$. If $a$ and $b$ are in the set $\{0,1,2,3,4,5\}$, can you always solve the equation $a \oplus x=b$ ? For which choices of $a$ and $b$ can you solve the equation $a \otimes x=b$ ?

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04:39

Problem 5

Give the addition and multiplication tables for the integers modulo 7, where $a \oplus b=(a+b) \bmod 7$ and $a \otimes b=a b \bmod 7$. If $a$ and $b$ are in the set $\{0,1,2,3,4,5,6\}$, can you always solve the equation $a \oplus x=b$ ? For which choices of $a$ and $b$ can you solve the equation $a \otimes x=b$ ?

Anas Venkitta

Numerade Educator

Problem 6

Give the addition and multiplication tables for the integers modulo 13, omitting 0 from the multiplication table. Describe the patterns that appear in the two tables. How are the patterns similar? How are they different?

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01:16

Problem 7

Consider the alphabet as represented by the integers modulo 26 , using the conversion table
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F} & \mathrm{G} & \mathrm{H} & \mathrm{I} & \mathrm{J} & \mathrm{K} & \mathrm{L} & \mathrm{M} \\
\hline 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline \hline \mathrm{N} & \mathrm{O} & \mathrm{P} & \mathrm{Q} & \mathrm{R} & \mathrm{S} & \mathrm{T} & \mathrm{U} & \mathrm{V} & \mathrm{W} & \mathrm{X} & \mathrm{Y} & \mathrm{Z} \\
\hline 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 \\
\hline
\end{array}
$$
Describe how you would design a word scramble that is based upon addition and/or multiplication modulo 26 .