Exam 4: Number Theory and Cryptography

determine whether each of the following "theorems" is true or false. Assume that a, b, c, d, and m are integers with m > 1. - $\text { If } a \equiv b ( \bmod m ) \text {, then } 2 a \equiv 2 b ( \bmod m )$

What is the decryption function for an affine cipher if the encryption function is $f ( x ) = ( 3 x + 7 ) \bmod 26 ?$

Convert (1 1101)2 to base 16.

Find an inverse of 5 modulo 12.

Find the first five terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 1357.

Find $\operatorname { div } m \text { and } a \bmod m \text { when } a = 76 , m = 52 \text {. }$

Prove or disprove: The sum of two irrational numbers is irrational.

Convert (1 1101)2 to base 10.

Convert (BC1)16 to base 2.

Encrypt the message KING using the RSA system with n=43 . 61 and e=13 , translating each letter into integers (A=00, B=01 , ... ) and grouping together pairs of integers.

Find three integers $m \text { such that } 13 \equiv 7 ( \bmod m )$

Prove or disprove: There exist two consecutive primes, each greater than 2.

Find gcd(2⁸⁹, 2³⁴⁶) by directly finding the largest divisor of both numbers.

Use the Euclidean Algorithm to find gcd(128, 729).

Prove or disprove: For all integers $a , b , c \text {, if } a \mid c \text { and } b \mid c \text {, then } a b \mid c ^ { 2 }$

Use the Euclidean Algorithm to find gcd(900, 140).

Given that gcd(620, 140) = 20, write 20 as a linear combination of 620 and 140.

Convert (2AC)16 to base 10.

find the sum and product of each of these pairs of numbers. Express your answer as a hexadecimal expansion. - $( 2 \mathrm {~A} ) _ { 16 , } , ( \mathrm { BF } ) _ { 16 }$

Find the prime factorization of 111,111.