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Assignment #3, Due 20 Sep

Show your work on all problems. (In particular, show all steps of the Euclidean Algorithm.) Feel free to use a calculator.

1.

Exercise 5.1(a) on page 28 of A Friendly Introduction .

2.

Use the Euclidean Algorithm to find an integer solution to the equation

105x+121y=1.

3.

On page 35 of A Friendly Introduction , we see that the equation

12453x+2347y=1

has a solution

$\begin{displaymath} x=304,\quad y=-1613.\end{displaymath}$

Use this information to find an inverse of 2347 modulo 12453. Then use that to find a solution to the equation

$\begin{displaymath} 2347x \equiv 2 \pmod{12453}.\end{displaymath}$

4.

Use the Euclidean Algorithm to decrypt the following message written in the affine cipher [11,16] (multiply by 11 and then add 16):

CABZKQDMN

5.

Exercise 9.1(a) on page 55 of A Friendly Introduction .

6.

Use Euler's Formula to find a number $0 \leq a < 10$ such that

$\begin{displaymath} a\equiv 7^{1000} \pmod{10}.\end{displaymath}$

(Note that this is the same as the last digit of the decimal expansion of 7¹⁰⁰⁰.)

About this document ...

Assignment #3, Due 20 Sep

This document was generated using the LaTeX2HTML translator Version 97.1 (release) (July 13th, 1997)

The command line arguments were:
latex2html -link 0 -split +0 wksheet1.

The translation was initiated by Joshua Holden on 9/14/2000

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Joshua Holden
9/14/2000