Problem 1
What is \(14 \bmod 9\) ? What is \(-1 \bmod 9\) ? What is \(-11 \bmod 9\) ?
Problem 1
What is \(3^{1024}\) in \(Z_{7}\) ? (This is a straightforward problem to do by hand.)
Problem 1
Compute the positive powers of 4 in \(Z_{7}\). Compute the positive powers of 4 in \(Z_{10}\). What is the most striking similarity? What is the most striking difference?
Problem 2
Suppose you have computed \(a^{2}, a^{4}, a^{8}, a^{16}\), and \(a^{32}\). What is the most efficient way to compute \(a^{53}\) ?
Problem 2
Compute the numbers \(1 \cdot{ }_{11} 5,2 \cdot{ }_{11} 5,3 \cdot{ }_{11} 5, \ldots, 10 \cdot_{11} 5 .\) Do you get a permutation of the set \(\\{1,2,3,4,5,6,7,8,9,10\\} ?\) Would you get a permutation of the set \(\\{1,2,3,4,5,6,7,8,9,10\\}\) if you used another nonzero member of \(Z_{11}\) in place of \(5 ?\)
Problem 2
Encrypt the message HERE IS A MESSAGE using a Caesar cipher in which each letter is shifted three places to the right.
Problem 3
Compute the fourth power mod 5 of each element of \(Z_{5}\). What do you observe? What general principle explains this observation?
Problem 3
Determine whether every nonzero element of \(Z_{n}\) has a multiplicative inverse for \(n=10\) and \(n=11\).
Problem 3
A gigabyte is one billion bytes; a terabyte is one trillion bytes. A byte is 8 bits, each a 0 or a 1 . Because \(2^{10}=1024\), which is about 1000 , you can store about three digits (any number between 0 and 999) in 10 bits. About how many decimal digits could you store in five gigabytes of memory (a gigabyte is \(2^{30}\), or approximately one billion bytes)? About how many decimal digits could you store in five terabytes of memory (a terabyte is \(2^{40}\), or approximately one trillion bytes)? How does this compare with the number \(10^{120}\) ? (To do this problem, it is reasonable to continue to assume that 1024 is about 1000 .)
Problem 3
Encrypt the message HERE IS A MESSAGE using a Caesar cipher in which each letter is shifted three places to the left.
