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Exam 4: Number Theory and Cryptography

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Question 21

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

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Question 22

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

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Question 23

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

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

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

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

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

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

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

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 $\sigma ( 3 ) = 2$

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

Find 18 mod 7.

Find 289 mod 17.

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

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

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

Find −88 mod 13.

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

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