91Ó°ÊÓ

Does 17 divide each of these numbers? $$\begin{array}{llll}{\text { a) } 68} & {\text { b) } 84} & {\text { c) } 357} & {\text { d) } 1001}\end{array}$$

Show that 15 is an inverse of 7 modulo 26.

Convert the decimal expansion of each of these integers to a binary expansion. \(\begin{array}{llll}{\text { a) } 231} & {\text { b) } 4532} & {\text { c) } 97644}\end{array}\)

Prove that if \(a\) is an integer other than \(0,\) then a) 1 divides \(a . \quad\) b) \(a\) divides 0 .

Convert the decimal expansion of each of these integers to a binary expansion. \(\begin{array}{llll}{\text { a) } 321} & {\text { b) } 1023} & {\text { c) } 100632}\end{array}\)

A parking lot has 31 visitor spaces, numbered from 0 to \(30 .\) Visitors are assigned parking spaces using the hashing function \(h(k)=k\) mod \(31,\) where \(k\) is the number formed from the first three digits on a visitor's license plate. a) Which spaces are assigned by the hashing function to cars that have these first three digits on their license plates: \(317,918,007,100,111,310 ?\) b) Describe a procedure visitors should follow to find a free parking space, when the space they are assigned is occupied. Another way to resolve collisions in hashing is to use double hashing. We use an initial hashing function \(h(k)=k \bmod p,\) where \(p\) is prime. We also use a second hashing function \(g(k)=(k+1) \bmod (p-2) .\) When a collision occurs, we use a probing sequence \(h(k, i)=(h(k)+i \cdot g(k)) \bmod p .\)

Encrypt the message WATCH YOUR STEP by translating the letters into numbers, applying the given encryption function, and then translating the numbers back into letters. a) \(f(p)=(p+14) \bmod 26\) b) \(f(p)=(14 p+21) \bmod 26\) c) \(f(p)=(-7 p+1) \bmod 26\)

Convert the binary expansion of each of these integers to a decimal expansion. \(\begin{array}{ll}{\text { a) }(11111)_{2}} & {\text { b) }(1000000001)_{2}} \\\ {\text { c) }(101010101)_{2}} & {\text { d) }(110100100010000)_{2}}\end{array}\)

Find an inverse of \(a\) modulo \(m\) for each of these pairs of relatively prime integers using the method followed in Example \(2 .\) a) \(a=4, m=9\) b) \(a=19, m=141\) c) \(a=55, m=89\) d) \(a=89, m=232\)

Decrypt these messages encrypted using the shift cipher \(f(p)=(p+10) \bmod 26 .\) a) CEBBOXNOB XYG b) LO WIPBSOXN c) DSWO PYB PEX