91Ó°ÊÓ

Show that every positive integer can be represented uniquely as the sum of distinct powers of 2 . (Hint: Consider binary expansions of integers.)

Describe the steps that Alice and Bob follow when they use the Diffie-Hellman key exchange protocol to generate a shared key. Assume that they use the prime \(p=101\) and take \(a=2,\) which is a primitive root of \(101,\) and that Alice selects \(k_{1}=7\) and \(\mathrm{Bob}\) selects \(k_{2}=9 .\) (You may want to use some computational aid.)

Use the Euclidean algorithm to find $$\begin{array}{ll}{\text { a) } \operatorname{gcd}(1,5) .} & {\text { b) } \operatorname{gcd}(100,101)} \\ {\text { c) } \operatorname{gcd}(123,277) .} & {\text { d) } \operatorname{gcd}(1529,14039)} \\ {\text { e) } \operatorname{gcd}(1529,14038)} & {\text { f) } \operatorname{gcd}(11111,111111)}\end{array}$$

List five integers that are congruent to 4 modulo 12

Show that a positive integer is divisible by 11 if and only if the difference of the sum of its decimal digits in even-numbered positions and the sum of its decimal digits in odd-numbered positions is divisible by 11 .

Use the Euclidean algorithm to find $$\begin{array}{ll}{\text { a) } \operatorname{gcd}(12,18) .} & {\text { b) } \operatorname{gcd}(111,201)} \\ {\text { c) } \operatorname{gcd}(1001,1331,} & {\text { d) } \operatorname{gcd}(12345,54321)} \\ {\text { e) } \operatorname{gcd}(1000,5040) .} & {\text { f) } \operatorname{gcd}(9888,6060)}\end{array}$$

How many divisions are required to find gcd(34, 55) using the Euclidean algorithm?

Decide whether each of these integers is congruent to 5 modulo \(17 .\) \(\begin{array}{ll}{\text { a) } 80} & {\text { b) } 103} \\ {\text { c) }-29} & {\text { d) }-122}\end{array}\)

Determine how we can use the decimal expansion of an integer \(n\) to determine whether \(n\) is divisible by \(\begin{array}{llll}{\text { a) } 4} & {\text { b) } 25} & {\text { c) } 20}\end{array}\)

Find each of these values. a) \((177 \bmod 31+270 \bmod 31) \bmod 31\) b) \((177 \bmod 31 \cdot 270 \bmod 31) \bmod 31\)