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Exam 4: Number Theory and Cryptography

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Question 21

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Decrypt the message EARLYL which is the ciphertext produced by encrypting a plaintext message using the transposition cipher with blocks of three letters and the permutation $\sigma$ of {1,2,3} defined by $\sigma ( 1 ) = 3 , \sigma ( 2 )$ 1 , and $\sigma ( 3 ) = 2$

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Question 22

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Find the check digit of the student id starting with 3179 822.

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Question 23

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Use the Euclidean algorithm to find gcd(300, 700).

Prove or disprove: A positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to 3 modulo 4.

find each of these values - $\left( 12 ^ { 2 } \bmod 17 \right) ^ { 3 } \bmod 11$

Use the Euclidean algorithm to find gcd(144, 233).

Find Icm(2⁸⁹, 2³⁴⁶) by directly finding the smallest positive multiple of both numbers.

find the sum and product of each of these pairs of numbers. Express your answer as a binary expansion. - $( 10\quad1011 ) _ { 2 } , ( 110\quad1011 ) _ { 2 }$

Use the Euclidean algorithm to find gcd(34, 21).

Find $\text { the integer } a \text { such that } a = 89 ( \bmod 19 ) \text { and } - 9 \leq a \leq 9 \text {. }$

Find the first five terms of the sequence of four-digit pseudorandom numbers generated by the middle square method starting with 9361.

Find the smallest $\text { integer } a > 1 \text { such that } a + 1 \equiv 2 a ( \bmod 11 ) \text {. }$

Find integers $a \text { and } b \text { such that } a + b \equiv a - b ( \bmod 5 )$

Find −88 mod 13.

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

Find 18 mod 7.

find each of these values -(123 mod 19 + 342 mod 19) mod 19.

Prove or disprove that 30! ends in exactly seven 0's.

Find gcd(20!, 12!) by directly finding the largest divisor of both numbers.

Find 289 mod 17.