User:Marty/UCSC Math 110 Fall 2008/Homework 8

This homework assignment has three problems, and your solutions should be typed using LaTex (which you have installed previously. Follow these steps to complete the homework assignment:
 * Copy and paste the template below into your Tex editor.
 * Save and compile the file, and view the PDF (this will help you see what you are trying to do).
 * Edit the template to complete the computations and proofs.
 * Make sure to change the name and e-mail addresses to your own (you are not Student X).
 * Follow all of the directions in the template, which are given on lines that begin with a % symbol in the body of the text.
 * Save and compile the file now and then, to view the PDF results.
 * After you finish, save and compile the PDF, print the file, and turn it in.
 * The first section is worth 4 points. The second section is worth 3 points.  The third section is worth 3 points.

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\newcommand{\NN}{\mathbb N} \newcommand{\ZZ}{\mathbb Z} \newcommand{\QQ}{\mathbb Q} \newcommand{\RR}{\mathbb R} \newcommand{\CC}{\mathbb C} \newcommand{\Mod}{\mbox{ mod }}

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\title[]{Math 110, Homework 8}% \author{Student X}% \address{Dept. of Mathematics, University of California, Santa Cruz, CA 95064}% \email{studentx AT ucsc DOT edu}%

%\date{}% %\dedicatory{}% %\commby{}% % \maketitle % \section{Cube roots} In this section, we demonstrate how to find ``cube roots'' in modular arithmetic. We prove the following theorem: \begin{thm} Suppose that $p$ is a prime number, and $p \equiv 2$, mod $3$. Suppose that $x \in \ZZ$. Then there exists $y \in \ZZ$ such that $y^3 \equiv x$, mod $p$. \end{thm} % Observe that \equiv is used for the "congruence" symbol (the "triple-equal-sign"). % Observe that the word "mod" is outside of dollar signs, so that it looks good. \proof Let $p$ be a prime number, such that $p \equiv 2$, mod $3$. Examples include: % Give five examples of prime numbers which are congruent to 2, mod 3.

It follows that $p-1 \equiv 1$, mod $3$. % Give a proof of the statement below. % Use the fact that if p-1 is congruent to 1, mod 3, then p-1 is relatively prime to 3. % Then, explain the existence of a multiplicative inverse of 3, modulo p-1.

Therefore, there exists an integer $n$, such that: $$n \cdot 3 \equiv 1, \Mod p-1.$$ % Observe that double-dollar signs lead to an equation on its own line, centered. % The code \Mod, within dollar signs, leads to a good text-looking "mod" with space around it. % If you just wrote "mod" within dollar signs, LaTex would think that there were variables m,o,d, and would format it in an ugly way. % Use Fermat's little theorem (the Fermat-Euler theorem, modulo p) to prove the result below. % Hint... $\phi(p) = p-1$. Let $y = x^n$. Then we find that: $$y^3 \equiv x, \Mod p.$$ \qed

\section{Counting an army}

General Sun Tzu knows that his army contains more than $500$ troops, and less than $1500$. His army is very well-trained. He shouts out ``Sevens'', and his army lines up in rows of seven; there are many rows, and 5 troops left over. Then he shouts out ``Tens'', and his army lines up in rows of ten; there are many rows, and 1 troop left over. Finally, he shouts out ``Elevens'', and his army lines up in rows of eleven; there are many rows, and no troops left over.

In this section, we determine exactly how many troops the general has in his army. Let $N$ be the number of troops. The first two orders of the general provide him with the following information: $$N \equiv 5, \Mod 7.$$ $$N \equiv 1, \Mod 10.$$ It follows that there exist integers $x,y$, such that $N - 5 = 7x$ and $N - 1 = 10y$. Therefore, we find that: $$N = 7x + 5 = 10y + 1.$$ Hence $7x - 10y = -4$. Every solution to this linear Diophantine equation has the form: $$x = -2 + 10a, y = -1 + 7a,$$ for some $a \in \ZZ$. It follows that, for some $a \in \ZZ$, $$N = 7x + 5 = 7(-2 + 10a) + 5= -9 + 70a.$$ Hence, we have found that: $$N \equiv 61, \Mod 70.$$

% Finish this proof as follows: % Consider the two congruences:  $N \equiv 61, \Mod 70$ and $N \equiv 0, \Mod 11$. % Convert this system of two congruences into a single congruence, modulo $770$. % Use the same style of argument given above. % Use the Euclidean algorithm to solve the linear Diophantine equation along the way. % You do not need to type and include the Euclidean algorithm (unless you want to). % Make sure to check your answer to the linear Diophantine equation. % From your solution to the linear Diophantine equation, you should deduce a congruence, modulo 770. % Find a (unique) number between 500 and 1500 that satisfies this congruence modulo 770. % Deduce how many troops the general has. % Write clearly in paragraphs. % Your answer will be a "pretty" number.

\section{Divisibility by eleven}

There is a very convenient trick, to determine whether a number is divisible by eleven: \begin{thm} Suppose that $N$ is a positive integer. Let $d_0$ be its unit-digit, $d_1$ its tens-digit, $d_2$ its hundreds-digit, etc.. In other words, $d_0, \ldots, d_t$ is a finite sequence of integers between $0$ and $9$, such that: $$N = \sum_{i=0}^t d_i \cdot 10^i.$$ Then, $N$ is divisible by $11$ if and only if the alternating sum of the digits of $N$ is divisible by $11$. In other words, $N$ is divisible by $11$ if and only if: $$\sum_{i=0}^t (-1)^i d_i \mbox{ is divisible by } 11.$$ \end{thm} \proof % Write a proof of this theorem. % Use the fact that a number is divisible by 11 if and only if it is congruent to zero, mod 11. % What is 10^i congruent to, mod 11? Use your answer. \qed

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